Elastic Waves and Metamaterials: The Fundamentals 9819902045, 9789819902040

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Yoon Young Kim

Elastic Waves and Metamaterials: The Fundamentals

Elastic Waves and Metamaterials: The Fundamentals

Yoon Young Kim

Elastic Waves and Metamaterials: The Fundamentals

Yoon Young Kim Department of Mechanical Engineering Seoul National University Seoul, Korea (Republic of)

ISBN 978-981-99-0204-0 ISBN 978-981-99-0205-7 (eBook) https://doi.org/10.1007/978-981-99-0205-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book introduces wave phenomena in elastic solids and metamaterials to undergraduate/graduate students and researchers with little or no background in this field. Despite the prevalence of wave phenomena in daily life, the practical applications of waves require understanding wave physics and their intricate manipulation. The concept of metamaterial has emerged as a novel technique for manipulating waves. For instance, the retroreflection of a longitudinal elastic wave from a free boundary cannot be achieved with natural materials because it requires specific exotic mechanical properties, which cannot be found in natural materials. Metamaterials can now realize such exotic properties. Not chemically synthesized, metamaterials are composed of internal structural patterns embedded in single or multiple materials; an example of a metamaterial is an aluminum plate with repeatedly patterned or machined voids. Scientists interpret the observed intriguing phenomena occurring in metamaterials using their homogenized “effective” material properties. In general, the effective properties are frequency-dependent, and theoretically, they can be negative or even infinitely positive. They can be extraordinarily anisotropic and not observable in natural substances. This book intends to assist readers in comprehending the fundamental physics of elastic waves and exploring the fascinating metamaterials necessary for unprecedented wave manipulation. Many books on metamaterials focus primarily on electromagnetic waves; books on elastic metamaterials, which are helpful for beginners in this field, are uncommon. However, most existing books on elastic waves do not cover metamaterials. Metamaterials introduce nonconventional concepts such as negative mass and stiffness, making it challenging to streamline learning from fundamental elastic wave phenomena to wave manipulation using elastic metamaterials. Assuming that the readers of this book have little or no exposure to elastic waves, let alone metamaterials, this book is pedagogically written. To deliver the fundamental concepts on the subject of this book and avoid complicated and lengthy analysis, this book primarily focuses on one-dimensional wave phenomena. However, some two-dimensional topics are also discussed when necessary. Nearly every topic is self-explanatory; although references to the topic covered in this book are provided, the vast majority of this book can be studied without delving into the cited references. I believe that v

vi

Preface

this is the best way for a beginner to learn a new subject, such as elastic metamaterials. I hope undergraduates, first-year graduate engineering/science students, and field engineers/scientists find this book the right book to become acquainted with the exciting field of elastic waves and elastic metamaterials. The materials covered in this book are based on my lectures on elastic waves and metamaterials at Seoul National University. Some of the research outputs of my former Ph.D. students and postdoctoral fellows are included in this book—they are Profs. J. H. Oh, X. Yang, J. S. Lee, and J. Yang, as well as Drs. H. J. Lee, J. M. Kwon, K. Kim, H. Lee, C. Piao, B. Ahn, Y. K. Ahn, S. Yoo, I. K. Lee, W. Lee, C. I. Park, M. K. Lee, P. S. Ma, J. H. Park, and W. W. Yoon, and I acknowledge their contributions. In addition, I am grateful to some of my current graduate students, J. Lee (who helped write Chap. 10), S. Y. Kim, H. J. Kim, S. H. Kim, C. W. Park, J. Jeon, S. Choe, J. A. Park, Y. B. Oh, J. H. Cho, and G. Kim, who assisted in preparing some figures, problem sets, and portions of this book. Dr. M. S. Kim (from the Ulsan National Institute of Science and Technology) contributed to Chap. 9. Professors J. K. Yang of Seoul National University, W. Jeon of the Korean Advanced Institute of Science and Technology, and J. H. Oh of the Ulsan National Institute of Science and Technology are to be commended for their valuable inputs to the recent developments in topological insulators, acoustic black holes, and nonlinear metamaterials. I also appreciate the feedback and inputs provided by Profs. B. Assour (CNRS), M. L. Hussein (University of Colorado), H. Genkai (Beijing Institute of Technology), G. Huang (University of Missouri), J. Li (HKUST), K. J. Song (Pusan National University), and R. Zhu (Beijing Institute of Technology). I thank Prof. M. J. Choi at Jeju National University and Prof. Y. Joung at Sookmyung Women’s University for their hospitality in hosting me during my sabbatical semester; I was able to focus on completing this book during my visits to their institutes. The encouragement of Director H. Lee, Dr. T. Choi, and Dr. K. H. Hwang of the Center for Advanced Meta-Materials also helped me start writing this book. I’d also like to express my extraordinary gratitude to my family, J. S. Seo, S. G. Kim, and S. H. Kim, for their unwavering encouragement in times of difficulty. Most of the technical results presented on metamaterials were obtained by the research project supported by the National Foundation of Korea (NRF) Grant (No. 2014M3A6B3063711), the Global Frontier R&D Program on Center for Wave Energy Control based on Metamaterials. The Seoul National University Grant in 2021 supported the writing of this book. Seoul, Korea (Republic of)

Yoon Young Kim

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Mechanical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Waves Versus Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 PCs and Metamaterials for Advanced Wave Manipulation . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 19

2

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Undamped Free Vibration of 1-DOF System . . . . . . . . . . . . . . . . . 2.2 Damped Free and Forced Vibration of 1-DOF System . . . . . . . . . 2.3 Impedance and Power in 1-DOF System . . . . . . . . . . . . . . . . . . . . . 2.4 Vibration of Undamped 2-DOF System and Effective Mass Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Dynamic Vibration Absorber: Resulting Physical Phenomena . . . 2.6 Dynamic Vibration Absorber Interpreted by Effective Mass . . . . 2.7 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 30 33

3

Longitudinal Waves in 1D Monatomic Lattices . . . . . . . . . . . . . . . . . . . 3.1 Governing Equation and General Solution . . . . . . . . . . . . . . . . . . . 3.2 Phase, Energy and Group Velocities . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Characteristic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dispersion Relation for ω ≥ ωcutoff . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 60 67 71 73 74

4

Longitudinal Waves in 1D Diatomic Lattices . . . . . . . . . . . . . . . . . . . . . 4.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dispersion Relation in Passband . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dispersion Relation in Stopband . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Transition from Diatomic Lattice to Monatomic Lattice . . . . . . . . 4.5 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 83 84 88 89

38 42 44 48

vii

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Contents

5

Effective Material Property Manipulation in 1D Lattice Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.1 Frequency-Dependent Effective Mass . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Frequency-Dependent Effective Stiffness . . . . . . . . . . . . . . . . . . . . 96 5.3 Doubly Negative Effective Material Properties . . . . . . . . . . . . . . . . 99 5.4 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6

Metamaterials: Effective Property Realization . . . . . . . . . . . . . . . . . . . 6.1 Metamaterial Modeling via Spring–Mass System . . . . . . . . . . . . . 6.2 Evaluation of Frequency-Dependent Effective Mass . . . . . . . . . . . 6.3 Evaluation of Frequency-Dependent Effective Stiffness . . . . . . . . 6.4 Metamaterial with Frequency-Dependent Effective Mass and Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 111 114

Longitudinal Waves in 1D Continuum Bars . . . . . . . . . . . . . . . . . . . . . . 7.1 Governing Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 General Solution of 1D Wave Equation . . . . . . . . . . . . . . . . . . . . . . 7.3 Characteristic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Reflection and Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Energy Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Impedance Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Fabry–Pérot Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Dispersive Longitudinal Waves-Metamaterial Interpretation . . . . 7.10 Transfer Matrix Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Bloch–Floquet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 Dispersion Analysis of Periodic Continuum Body . . . . . . . . . . . . . 7.13 Analysis of Stop Band in Periodic Continuum Body . . . . . . . . . . . 7.14 Periodic Bar Structure Containing Quarter-Wave Stacks . . . . . . . 7.15 Material Characterization Using S Parameters . . . . . . . . . . . . . . . . 7.16 Lowered Effective Impedance by Resonators (Advanced Topic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.17 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 130 133 134 138 143 148 150 153 159 162 165 170 172 175

Flexural Waves in a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Wave Analysis by Euler–Bernoulli Beam Theory . . . . . . . . . . . . . 8.2 Reflection and Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Wave Analysis by Timoshenko Beam Theory . . . . . . . . . . . . . . . . . 8.4 Flexural Waves in a Beam with Distributed Resonators . . . . . . . . 8.5 Flexural Waves in Periodic Discrete System . . . . . . . . . . . . . . . . . . 8.6 Enhanced Actuation or Sensing by Resonators (Advanced Topic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 213 218 223 226

7

8

117 126

181 198 202

232

Contents

ix

8.7 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9

Fundamentals of Elastic Waves in 2D Elastic Media . . . . . . . . . . . . . . 9.1 Governing Field Equation in Elastic Media . . . . . . . . . . . . . . . . . . . 9.2 Dispersion Relations in 2D Anisotropic–Isotropic Media . . . . . . . 9.3 Fundamentals of Wave Reflection and Transmission in Isotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 State Vector Representation for Transmission and Reflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Reflection and Transmission across Two Dissimilar Isotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Reflections from Traction-free Boundary . . . . . . . . . . . . . . . . . . . . 9.7 Anomalous Reflection from Traction-free Boundary: Metasurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Perfect Transmission Across 2D Different Media Using Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Wave Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Scattering Parameters for Anisotropic Media . . . . . . . . . . . . . . . . . 10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Design of Anisotropic Metamaterial Layers . . . . . . . . . . . 10.3.3 Metamaterial Realization and Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Perfect Mode-Converting Transmission at Normal Incidence by Metamaterial Matching Layer . . . . . . . . . . . . . . . . . . 10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 247 253 263 269 271 280 284 291 292 295 297 299 307 307 311 315 318 330 342 344

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Chapter 1

Introduction

Although mechanical, aerospace, structural, and automotive engineers encounter vibration phenomena in various contexts, the phenomena are typically viewed from the perspective of standing waves that do not spatially transport mechanical energy. If the events are viewed from the perspective of mechanical energy-transporting waves that propagate through space, it may yield new insights. However, the current undergraduate and graduate engineering curricula devote little time to propagating waves. This book will assist readers in comprehending the phenomena from a wave perspective and in learning cutting-edge techniques for manipulating waves. This chapter begins with a comparison of mechanical waves and vibrations. Then, various types of mechanical waves are discussed. Following this, current wave manipulation techniques utilizing “phononic crystals” (PCs) or “metamaterials” will be discussed. After introducing the concept of PCs and metamaterials, several pertinent topics and examples are discussed to stimulate interest in mechanical wave research. Later chapters provide more comprehensive theoretical explanations so that readers of this book can acquire a “concrete” understanding of various wave phenomena and wave manipulations utilizing PCs and metamaterials. In this book, linear wave phenomena will be mainly studied.

1.1 Mechanical Waves A wave is a phenomenon or disturbance that moves from one place to another and transfers energy (power). It can take the form of a wiggle in both time and space. Think of the taut string in Fig. 1.1 as an illustration of a mechanical wave. When a disturbance starts at the string’s left end, it spreads to the right and impacts the fixed right end. The wave then propagates leftward after being reflected from the end. These wave phenomena can be studied in both time and space, as seen in the figure. We’ll discover how the upward disturbance that began at the left end spreads

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_1

1

2

1 Introduction

Fig. 1.1 Wave propagation and reflection. A transverse wave initiated at the left end of a taut string is reflected from the fixed right end (t: time with ti+1 > ti )

and transforms into a downward disturbance when it reaches the fixed boundary condition at the right end. The transfer of mechanical wave energy requires a “medium” such as a string, liquid, or solid. The most common types of mechanical waves are elastic waves in solids, sound waves in air (or water), and transverse waves that travel along a taut string. As depicted in Fig. 1.1, transverse waves cause particles in a medium to vibrate perpendicular to the direction of their propagation. Figure 1.2 depicts longitudinal waves in which particles oscillate parallel to the wave transmission direction. Although elastic materials can permit the propagation of a three-dimensional elastic wave, this book focuses on one-dimensional waves and discusses two-dimensional ones briefly. One-dimensional wave models can accurately describe some wave motions in a three-dimensional space. Consider, for instance, the situation depicted in Fig. 1.2, in which all particles in the same cross-section oscillate simultaneously in phases with the same amplitude; in this case, it may be possible to represent the fundamental physical characteristics of the wave using a one-dimensional wave model. Bending (or flexural) wave propagation in a beam1 of thickness t, as depicted in Fig. 1.3, is another significant wave that can be analyzed using a one-dimensional model. The bending wave involves the transverse displacement and the rotation of the normal to the beam’s centerline, resulting in the longitudinal motions of fibers parallel to the beam’s centerline. The rotation is brought about by an out-of-plane moment known as the bending moment. Chapter 8 is devoted to the analysis of flexural waves in a beam.

1

A beam is a long, slender member with bending as its predominant mode of deformation. Beams and strings both experience transverse displacements, but beams exhibit a structural resistance known as bending (or flexural) rigidity, whereas strings do not.

1.2 Waves Versus Vibrations

3

Compressions

Dilatations Direction of P wave

Fig. 1.2 Snapshot of a longitudinal wave propagating in a rectangular bar

Fig. 1.3 Snapshot of a bending wave in a beam of thickness t

1.2 Waves Versus Vibrations Waves and vibrations are both time-dependent oscillatory phenomena, but their interpretations differ considerably. Let us consider a traverse displacement v(x, t), expressed as   v(x, t) = A sin kx sin ωt or v(x, t) = Re Aeiωt sin kx ,

(1.1)

v(x, t) = A sin(ωt − x/c)],c = ω/k   − kx) = A sin[ω(t , or Re i Aei(ωt−kx) = Re i Aeiω(t−x/c)

(1.2)

or

where the symbol c is defined as c=

ω . k

(1.3)

4

1 Introduction

The meaning of c will be examined below. In Eqs. (1.1) and (1.2), t denotes time and x indicates the spatial coordinate. Symbol A denotes the amplitude of the displacement v, whereas ω and k represent the angular frequency and wavenumber. The wavenumber can be interpreted as spatial frequency. In Eqs. (1.1) and (1.2), the cosine functions or the combination of cosine and sine functions can be used instead of the sine functions, but we considered only sine functions for simplicity. Figure 1.4 depicts the snapshots of the traverse displacement at various instants, t = ti or t = ti∗ . Using Eq. (1.1), spatial locations known as nodes can be identified; at the nodes, the displacements are always zero at any given instant. Similarly, there are antinodes at which the amplitude of the displacement varies from zero to its maximum. Although the displacement amplitude at a certain point x varies with time, the sinusoidal waveform does not travel along the x-axis. In contrast, the displacement representation given by Eq. (1.2) depicts the time-dependent movement of the sinusoidal waveform along the x-axis with time. Compared with the images in Fig. 1.4a, the images in Fig. 1.4b depict the energy (or disturbance) transport along the space, herein, along the x-axis. The zero-displacement point denoted by P shifts to the right at each increment of time. Notably, if sin(ωt + kx) is used instead of sin(ωt − kx) in Eq. (1.2), the energy transport will occur along the negative x-axis. The wavelength λ illustrated in Fig. 1.4b is related to the wavenumber k as kλ = 2π.

(1.4)

If v(x, t) is given by Re[Aei(ωt−kx) ], the displacement at x + λ becomes v(x + λ, t) = Re{Aei[ωt−k(x+λ)] } = Re{Aei(ωt−kx) e−ikλ } = Re{Aei(ωt−kx) e−i2π } = v(x, t).

Fig. 1.4 Snapshots of mechanical motion described by a the vibration approach using v(x, t) = A sin kx sin ωt in Eq. (1.1), and b the wave approach using v(x, t) = A sin(ωt − kx) in Eq. (1.2)

1.2 Waves Versus Vibrations

5

Because v(x +λ, t) at a location displaced by λ from a current location x recovers the same value as v(x, t), λ and k represent the spatial period and frequency, respectively. The magnitude of v(x, t) depends on the phase φ, which is defined as, φ = ωt − kx. Therefore, the magnitude of v(x, t) can be the same if there is no phase change φ even when x and t vary. Setting φ equal to zero, we can find φ = ωt − kx = 0 →

ω x = . t k

Taking the limit of t and x, i.e., replacing t and x by dt and dx, respectively, we can define the wave speed c as: c

dx ω = . dt k

(1.5)

The wave speed defined in Eq. (1.5) is called the phase speed. To understand the physical significance of c, let us examine the displacement at point x + ct when time t is varied to t + t: v(x + ct, t + t) = Re{Aei[ω(t+t)−k(x+ct)] } = Re{Aei(ωt−kx) eit (ω−kc) } = Re{Aei(ωt−kx) } = v(x, t). (1.6) We used Eq. (1.5) to obtain Eq. (1.6). The result in Eq. (1.6) shows that the amplitude of the displacement remains unchanged if we trace a point moving at the speed of c; this provides the physical meaning of the phase speed. Note that the phase speed c, as given by Eq. (1.5), depends on the relation between ω and k, called the dispersion relation. As we shall see in Chaps. 7 and 8, the phase speeds in a one-dimensional bar carrying longitudinal waves and a one-dimensional beam carrying flexural (bending) waves are given by In bar carrying longitudinal waves:  c=

E , ρ

(1.7)

In beam carrying flexural waves:  c=

EI ρA

1/4

√ ω,

(1.8)

6

1 Introduction

where ρ and E denote the density and Young’s modulus (stiffness) of a medium constituting a bar or beam, respectively. The symbols A and I represent the area and moment of inertia of a beam, respectively. In contrast to the wave speed of longitudinal waves in a bar, the wave speed of flexural waves in a beam is frequencydependent. Therefore, the phase speed c can be independent of ω only if there is a linear relationship between ω and k. Based on the preceding discussions, it is evident that Eq. (1.1) is useful for describing spatially nonpropagating vibrations in space. For example, the displacement sketched in Fig. 1.4a is considered to represent a time-harmonic vibration motion in a string satisfying the boundary conditions (v(x, t) = 0 at x = 0 and x = L). It represents an oscillatory motion in time, but there is no transmission of vibratory energy along the x-axis. Therefore, Eq. (1.1) is useful for describing the vibratory motions in a medium (such as a string) defined in a finite domain. In contrast, Eq. (1.2) is used to analyze the energy transport or propagation of a disturbance, especially in semi-infinite and infinite media. Since it describes the movement of a disturbance from one location to another, Eq. (1.2) represents the motion of a wave. Not surprisingly, the expressions in Eqs. (1.1) and (1.2) are actually related to each other as follows: A [cos(ωt − kx) − cos(kx + ωt)], 2

(1.9)

A sin(ωt − kx) = A(sin ωt cos kx − cos ωt sin kx).

(1.10)

A sin kx sin ωt =

For instance, the standing wave form or nonpropagating vibratory motion expressed in Eq. (1.1) can be constructed by an appropriate superposition of a rightwardpropagating wave and a leftward-propagating wave, as expressed in Eq. (1.9). In contrast, a propagating wave can be constructed by superposing two standing waves, as expressed by Eq. (1.10). This indicates that either expression can be used to describe propagating or nonpropagating time-harmonic oscillatory motion. It is notable, however, that nonpropagating oscillatory phenomena can be described more conveniently using Eq. (1.1) and the propagating waves can be represented using Eq. (1.2). Since this book is primarily concerned with the phenomena of propagating waves, it is preferable to employ the complex expression presented in Eq. (1.2).

1.3 PCs and Metamaterials for Advanced Wave Manipulation This section provides an overview of phononic crystals (PCs) (Brillouin 1953) and metamaterials. Banerjee (2011) gave a comprehensive treatment of metamaterials on elastic/acoustic (and electromagnetic) waves. We study PCs and metamaterials because they can be used to manipulate waves in unprecedented ways. The readers

1.3 PCs and Metamaterials for Advanced Wave Manipulation

7

may consult Deymier (2013) to comprehend earlier research developments primarily pertaining to elastic waves. See also Hussein et al. (2014), Christensen et al. (2015), Pai and Huang (2015), Ma and Sheng (2016), Zhou et al. (2012), Phani and Hussein (2017), and Jin et al. (2021). Hou and Assouar (2009) investigated various elastic wave modes in PCs. Since this book is concerned with elastic waves, research on electromagnetic waves will not be reviewed unless absolutely necessary. Indeed, there are numerous references on related topics within the field of electromagnetic waves. A phonon is the quantum (minimum amount of any physical property) of the quantized energy of a lattice vibration (Kittel 2004), and a crystal is a solid whose constituents (such as atoms) are arranged in a specific pattern. In the context of manipulating elastic waves, PCs exhibit a periodic structure with multiple constituents comprised in its unit cell. As reported by Brillouin (1953), one of the well-known phenomena occurring in PCs is the bandgap phenomenon, which indicates that certain frequency bands cannot pass through a PC. This phenomenon will be briefly explained using Fig. 1.5. As described in chapter 1 of Deymier’s (2013) book, a similar phenomenon was discovered for optical wave, known as the photonic bandgap (John 1987; Yablonovitch 1987). Photonic crystals are the counterpart of PCs in the electromagnetic wave field (Joannopoulos et al. 2008). Earlier works on the elastic PCs are reported in Kafesaki et al. (1995), Kafesaki and Economou (1995), and Kushwaha et al. (1993). To explain an interesting wave phenomenon occurring in PCs, consider an aluminum plate shown in Fig. 1.5. It contains a region of regularly spaced void holes, a phononic crystal. Although a PC should theoretically have extended indefinitely, the finite-sized perforated plate region depicted in Fig. 1.5 can be considered a PC (or a PC plate) because a unique phenomenon occurring in PCs, such as the band gap, can be observed. As shown in Fig. 1.5, a 500 kHz incident harmonic plane wave from the base aluminum plate passes through the phononic crystal, whereas a 700 kHz incident wave does not; a stopband is formed in the PC. The wave attenuation depicted in Fig. 1.5b is not the result of damping (or absorption) but rather multiple interferences caused by the periodic arrangement of holes. Figure 1.5a and b depict, respectively, the constructive and destructive interferences. Interference depends on the hole size, the distance between the holes (which serve as scatterers that scatter the incident waves), and the wavelength of the incident wave. In the figure, the distance between a bright-colored vertical line and another bright-colored line in the base plate represents the wavelength (λ) of a harmonic wave. Notably, the period of the PC should be in the same order as the wavelength if unique wave phenomena, such as the formation of a stopband, are to be observed in PCs. Consequently, if PCs were used to manipulate low-frequency waves, a massive structure would be required. This restricts the use of PCs for low-frequency stopband formation. Recent research has utilized metamaterials to overcome this PCrelated obstacle. For instance, Oh et al. (2018) demonstrated that a stopband can be formed from a near-zero frequency without an excessively large unit cell size if a metamaterial combining a spin motion with a longitudinal wave motion is utilized.

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1 Introduction

(a) Pass band (at 500 kHz)

(b) Stop band (at 700 kHz)

(c) Pass band Stop band Pass band

Fig. 1.5 Wave phenomena occurring in a phononic crystal plate. a Wave passing through a phononic crystal at 500 kHz, b wave stopping through PC at 700 kHz, and c dispersion curve (relating frequency and wavenumber)

The curve presented in Fig. 1.5c is called the “dispersion curve” of the PC plate corresponding to the perforated plate region shown in Fig. 1.5a, b. As depicted in Fig. 1.5c, the dispersion curve represents the relationship between the wavenumber k and frequency. Based on this curve, one can determine the wavenumbers for a given frequency. Interestingly, Fig. 1.5c depicts a frequency band for which there is no real-valued wavenumber. Actually, there are complex-valued or imaginary wavenumbers in the frequency band, but they are not plotted in the figure. Chapter 3 provides additional pertinent information. The frequency band with no real-valued wavenumber is referred to as a “stopband” because no wave can propagate in this band. The phenomenon can be explained by the Bragg scattering (Bragg and Bragg 1913), as explained in Chap. 7. Consequently, the corresponding stopband is referred to as the Bragg stopband. Referring back to Fig. 1.5b, a 700 kHz incident wave falls within the stopband of the PC plate. This indicates that the elaborate design of PCs (using inclusions of proper size and shape) can adjust the location and bandwidth of the stopband. (Although not plotted here, stopbands can also exist at higher frequency ranges.) In the sculpture by Eusebio Sempere depicted in Fig. 1.6a, Martinez-Sala et al. (1995) described an intriguing application of PCs for sound attenuation. The twodimensional periodic square steel tubes can attenuate noise (sound waves) of specific frequencies coming from certain directions due to the formation of stopbands. The

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Fig. 1.6 a Sculpture by Eusebio Sempere at Juan March Foundation in Madrid, b sound attenuation as a function of frequency (Martínez-Sala et al. 1995)

sound attenuation peak appearing at 1,670 Hz shown in Fig. 1.6b is caused by the formation of a bandgap. In addition to controlling wave passage, PCs can manipulate the waves for a variety of purposes. Negative refractions (Bonello et al. 2010; Bucay et al. 2009; Lee et al. 2011; Sukhovich et al. 2008; Yang et al. 2004) and selfcollimation (Park et al. 2015; Pérez-Arjonal et al. 2007) can be achieved, among other phenomena. We will discuss negative refractions realized by metamaterials below. Phononic crystals can be also used in various other applications, such as mode separation (Ma et al. 2011), multiple beam splitting (Lee et al. 2015, 2016a, b), active wave guiding (Oh et al. 2011), one-sided transmission (Oh et al. 2012), wave attenuation (Oh et al. 2013), waveguide dispersion tailoring (Ma et al. 2013), multifrequency wave filtering (Lee and Jeon 2020), etc. Negative refraction, one of the interesting phenomena made possible by PCs, will now be examined in some detail. The negative reflection depicted in Fig. 1.7b is contrasted with that of the typical positive refraction depicted in Fig. 1.7a. Negative phase velocity (c) at the selected frequency in the prism PC plate causes negative refraction, as indicated in the caption of Fig. 1.7. Recall the law of refraction (Snell’s law) which states sin θprism /cprism = sin θ/cal where the subscript “prism” refers to both the nominal and PC prisms in Fig. 1.7. The symbol θprism denotes the angle of incidence from the nominal or PC prism plate and θ , the angle of refraction. The superscript al refers to aluminum. Note that the nominal prism plate in Fig. 1.7a is made of a material distinct from the base aluminum plate, and that the prism PC plate has circular voids that are periodically arranged in the base plate. Because cprism for the nominal prism is positive at the chosen frequency of 25.5 kHz, the reflected angle θ is positive as shown in Fig. 1.7a. In contrast, when the PC prism is used at the same frequency, the phase velocity cprism of the PC prism is negative. As a result, the angle of refraction θ becomes negative, as depicted in Fig. 1.7b. Metamaterials also allow extraordinary or anomalous wave manipulation. A metamaterial is an artificially engineered material with effective material properties not observed in natural materials. This material is engineered by arranging repetitive

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1 Introduction

Fig. 1.7 Wave refraction through prisms of dissimilar media in a plate. (The shear-horizontal wave at 25.5 kHz was used). a Positive refraction (θ > 0) for which 0 < cprism < cal . b Negative refraction (θ < 0) for which 0 > cpc and 0 < cal . (cprism, cal , and cpc : phase velocities in the prism, aluminum plate, and prism phononic crystal plate)

patterns in a base material. Typically, the pattern scale is smaller than the wavelengths of interest. If the wavelength of a considered wave is much longer than the unit cell size (say, 10 times longer) of the engineered material, it can be viewed as a metamaterial. An example is the perforated structure shown in Fig. 1.8a. Depending on the chosen frequency, the frequency-dependent effective material properties of the metamaterial can vary from a negative value to an extremely positive value. Although metamaterials use periodically-arranged unit cells, periodicity may not be necessarily essential. The primary differences between the PCs and metamaterials are in the relative size of the unit cell with respect to the wavelength of interest and working principles. In particular, the unit cell scale of PCs is on the order of a wavelength whereas that of metamaterials is subwavelength, allowing their mechanical behavior to be analyzed using their effective homogenized material properties. The same wave phenomena can be realized by different principles in PCs and metamaterials. For instance, consider a stopband. In PCs, a stopband is due to the Bragg scattering resulting from periodicity, but in metamaterials, a stopband can be formed by local resonances of employed local resonators. Because the Bragg scattering occurring in PCs is the result of constructive or destructive interferences of waves reflected from scatters at multiple locations along a wave propagation direction, a Bragg stopband cannot be conveniently formed at a low frequency unless the unit cell size is unrealistically large. In the case of metamaterials, a low-frequency stopband can be formed if the local resonance frequency is tuned at a low frequency.

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Fig. 1.8 a Single-phase elastic metamaterial prism consisting of 256 unit cells hosted in a steel plate with simultaneously negative effective density and stiffness at certain frequencies, b zoomed unit cells (scale bar: 10 mm), c zoomed single unit cell (scale bar: 2 mm), d amplitudes of the in-plane velocity field at the center frequency of 30 kHz, yielding positive refraction, e amplitudes of the in-plane velocity field at the center frequency of 43.8 kHz, yielding negative refraction (Zhu et al. 2014a)

When local resonance mechanisms are used for metamaterials, the “negative” effective material properties at certain frequencies can be conceptually defined. Veselago (1967) introduced the concept of negative material properties (permeability and permittivity) in the field of electromagnetic waves. Subsequently, Pendry (2000) proposed a super lens with doubly negative electromagnetic material properties, and numerous studies on metamaterials have been conducted for decades. A wave cannot propagate through a medium if either the effective permeability or permittivity (or, equivalently, the effective density or stiffness) is negative at a certain frequency. Nonetheless, if both effective permeability and permittivity are simultaneously negative, waves can propagate through the medium with a negative effective phase velocity. In the early years, the realization and application of negative effective material properties in electromagnetic waves was of paramount importance (Liu et al. 2003; Luo et al. 2002; Pendry 2004; Shelby et al. 2001; Smith et al. 2000; Valanju et al. 2002). In the case of acoustic waves (in fluids), the localized resonant structures (Liu et al. 2000) were proposed to produce an acoustic metamaterial with effective negative elastic constants. The investigations by Li and Chan (2004), Fang et al. (2006), Lee et al. (2010), and Koo et al. (2016), Lee and Jeon (2018), Park et al. (2021), and Jang et al. (2022) are a few examples employing negative acoustic material properties. A more comprehensive review of acoustic metamaterials can be found in Ge et al. (2018). In addition, the metamaterial concept is incorporated into studies on the efficient sound absorption of porous layers under layer size constraints (Lagarrigue et al. 2016, 2013; Yang et al. 2015, 2016; Yoon et al. 2020). Only one type of wave mode is carried by electromagnetic or acoustic waves (transverse waves for electromagnetic and longitudinal waves in acoustic wave in fluids). In contrast, elastic waves (or acoustic waves in solids) carry both longitudinal and transverse waves (Achenbach 1976; Auld 1973; Miklowitz 1978). They become generally coupled unless they propagate through an isotropic medium in an infinite

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1 Introduction

space. This coupling complicates the wave analysis and produces phenomena that may be more intriguing than those in the acoustic field (in fluids). Therefore, manipulating waves with elastic metamaterials can be fascinating and difficult. However, the fundamental principles and concepts established in electromagnetic and acoustic waves can be applied to the wave manipulation of elastic metamaterials. Goffaux et al. (2002), who were motivated by a study on an acoustic metamaterial (Liu et al. 2000), investigated elastic metamaterials employing local resonances. The local resonance-induced effective negative properties in elastic media were subsequently investigated in greater detail (Huang et al. 2009; Huang and Sun 2010; Larabi et al. 2007; Liu et al. 2002, 2005; Wang et al. 2004; Yao et al. 2010). For instance, a chiral structure was proposed to create an elastic metamaterial with negative effective mass and stiffness (Liu et al. 2011), along with a continuum theory (Liu et al. 2012) and an application in beam vibration suppression (Zhu et al. 2014b). Negative refraction of an elastic wave through an elastic metamaterial prism, as depicted in Fig. 1.8, occurs at a frequency at which its effective properties are doubly negative, i.e., its effective density and stiffness are simultaneously negative (Zhu et al. 2014a). At this frequency, the phase velocity is negative due to the double negativity. In contrast, a stopband can be created at a frequency at which only one of the two effective material properties is negative. However, it should be noted that the mechanism of stopband formation due to a singly negative effective property is distinct from that of the Bragg bandgap. There are a number of studies concerning negative density and stiffness of elastic metamaterials (Hou and Assouar 2015; Lai et al. 2011; Liu et al. 2011; Oh et al. 2016a, b, 2017; Oudich et al. 2014; Wu et al. 2011; Zhou and Hu 2009; Zhu et al. 2014a). Simple mass–spring models can be used to describe the mechanism of negative-effective material properties resulting from local resonances (Milton and Willis 2007; Movchan and Guenneau 2004). The experimental realization of negative effective properties also attracted considerable interest (see, for instance, Yao et al. 2008). In Chap. 5, the one-dimensional elastic wave model is described in detail to explain the effective (negative/positive) material properties of metamaterials equipped with local resonators as well as the resulting wave phenomena. As shown in Fig. 1.9, an elastic metamaterial can be utilized to create a subwavelength imaging elastic lens (Lee et al. 2016a, b; Oh et al. 2014; Zhu et al. 2018a, b). According to established knowledge (Abbe 1873), the resolution of a conventional lens is always limited to approximately half the wavelength (λ) by diffraction. This implies that a conventional lens cannot distinguish objects smaller than 0.5λ. Figure 1.9a demonstrates that two sources separated by a distance smaller than 0.5λ cannot be distinguished in a regular plate in the far field. In contrast, a metamaterial hyperlens embedded in an aluminum plate (Fig. 1.9b) can resolve two sources within half the wavelength. The elastic hyperlens depicted in Fig. 1.9b was designed (Oh et al. 2014) using elastic metamaterials capable of exhibiting a “hyperbolic” equifrequency curve, which is required for subwavelength imaging at a particular frequency. (An equifrequency contour is a contour plotted on a two-dimensional wavevector plane for a particular frequency.) Wave focusing in elastic solids can be

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Fig. 1.9 Subwavelength source imaging using an elastic “hyperlens” composed of elastic metamaterial exhibiting a hyperbolic equifrequency contour. Simulations performed at 15 kHz (wavelength λ = 346.66 mm in aluminum plate under plane stress condition) for two sources (size of each source: 0.43λ) apart by 0.48λ, shorter than half of wavelength. Time-harmonic simulations a in a homogeneous aluminum plate, and b in a aluminum plate with hyperbolic metamaterial lens embedded before two sources (Oh et al. 2014)

also an interesting application of metamaterials; see, for example, Yan et al. (2013) and Oh et al. (2017). Cloaking is another fascinating phenomenon that can be realized by metamaterials. Elastic metamaterials can be used to conceal an object from elastic waves (Farhat et al. 2009; Kadic et al. 2012; Lee and Kim 2016; Norris and Shuvalov 2011; Stenger et al. 2012). The theories of cloaking by a transformation method were established earlier for electromagnetic waves (Leonhardt 2006; Pendry et al. 2006) and elastic waves (Milton et al. 2006). As shown in Fig. 1.10, Farhat et al. (2009) proposed a metamaterial cloak composed of concentric coating filled with piecewise constant isotropic elastic material to shield an object from elastic bending waves. If a circular, clamped obstruction (in white) is concealed by an elastic metamaterial, an incident wave (at a particular frequency) propagates through the obstacle as if it does not exist. Recently, an elastic metamaterial employing passive asymmetry has been proposed for cloaking against elastic waves (Zhang et al. 2020). The transformation method is also used to manipulate elastic waves in a number of different ways (Lee et al. 2021; Liu et al. 2017; Zhu et al. 2018a, b). A metasurface is a reduced-dimensional version of metamaterials that can be used to circumvent some limitations of conventional metamaterials. It has a 1D layout in 2D problems and a 2D layout in 3D problems from a geometric standpoint. Initially, the concept of a metasurface was created to disobey Snell’s law of reflection and refraction (Yu et al. 2011). Figure 1.11a depicts the classical law of reflection in which the reflected angle θrL is equal to the incident angle θiL in an elastic plate, i.e., θrL = θiL . Figure 1.11b depicts the situation in which the reflected angle is not equal to the incident angle, i.e., θrL = θiL . To make θrL unequal to θrL , an attachment known as a metasurface is installed on the plate’s free surface. Figure 1.12 depicts the effects of a metasurface attached to the free boundary of a thin aluminum plate, where a longitudinal wave is incident at θiL =10◦ . As shown in Fig. 1.12a, the incident longitudinal wave splits into a reflected longitudinal wave

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1 Introduction

Fig. 1.10 Elastic wave cloak using metamaterial. a Displacement scattered by a rigid clamped obstacle of radius 0.25 m for an incoming plane wave at 250 kHz. b Displacement scattered by the same rigid clamped obstacle surrounded by the metamaterial cloak (composed of 60 isotropic multilayers) with inner and outer radii of 0.3 and 0.6 m for the same incoming wave (Farhat et al. 2009)

Fig. 1.11 Comparison of a the conventional law of reflection (θiL = θrL ) and b generalized law of L

reflection (θiL = θ r ) realized by an elastic metasurface composing rods of various lengths. Symbol L stands for longitudinal waves. (Although s transverse wave is reflected in general, it is not plotted for simpler explanation)

with a reflected angle of θrL = 10◦ and a reflected transverse2 (or shear) wave with a reflection angle of θrT =6◦ . The reflected angle of the transverse wave can be calculated from sin θrT = (cT /c L ) sin θiL , where c L and cT denote the wave speeds of longitudinal and transverse waves in a plate, respectively (See Chap. 9 for additional information.). Overall, the angle of the reflected longitudinal wave is equal to the angle of the incident longitudinal wave. Figure 1.11b depicts how the reflected angle of a longitudinal wave can be changed when a metasurface is attached to the free boundary of a plate. In fact, the angle of the reflected wave can be altered at our discretion. Various elastic metasurfaces were developed to manipulate the wave propagation directions, as may be found in Cao et al. (2018a, b, 2020), Kim et al. (2018b, 2020), Colombi et al. (2017), etc. The application of the metasurface is further discussed in Chap. 9. “Perfect” wave mode conversion from a normally incident longitudinal mode to a transverse mode (or from a shear mode to a longitudinal mode) within the same medium or across dissimilar media is another intriguing achievement of elastic 2

Transverse (T) and shear (S) waves will be interchangeably used. Shear waves may be preferred if the deformation during wave propagation is of significance; otherwise, the transverse waves may be preferred if the direction of particle motion is of more concern.

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Fig. 1.12 Comparison of wave field a without and b with an elastic metasurface (as illustrated in Fig. 1.11b) attached on the free boundary of a thin aluminum plate. The incident wave is a longitudinal wave. Harmonic wave analysis was performed at 100 kHz. (|∇ · u| and |∇ × u|: divergence and curl fields of displacement u, which respectively denote the longitudinal and transverse wave field (Kim et al. 2018b)

metamaterials (Kweun et al. 2017; Yang et al. 2018, 2019; Yang and Kim 2018a, b). The phenomenon of perfect mode conversion realized by elastic metamaterials was utilized for ultrasonic flow measurement (Piao et al. 2020) and pure mode-converting diodes (Yang et al. 2021). A more general theory involving perfect mode-preserving or mode-conversion transmission of an obliquely incident longitudinal wave across dissimilar media has recently been presented (Lee et al. 2022a) and applied to the retroreflection of an obliquely incident longitudinal wave (Lee et al. 2022b). Figure 1.13 depicts the perfect mode conversion of a longitudinal (L) wave to a transverse (T) wave through a monolayer elastic metamaterial embedded in an aluminum plate. Unique oblique void slits in the monolayer metamaterial contribute to the conversion of a longitudinal (or transverse) mode to a transverse (or longitudinal) mode. At the chosen frequency of 104 kHz, the L-to-T mode-converting efficiency (TT ) is 99.58 percent, which is close enough to the perfect mode-converting transmission to be claimed. Current techniques generating transverse waves in a solid medium are inefficient and expensive. Considering the fact that transverse waves are used in a variety of applications, an efficient method to generate a transverse wave in a solid medium can be critically useful. The mode-conversion phenomenon is unique to elastic waves, as electromagnetic or acoustic waves (in fluids) contain only a single wave mode. The noninvasive ultrasound measurement of the flow velocity of a fluid flowing through a pipe is a practical application of the perfect longitudinal-to-transverse wave (L-to-T) mode conversion (Piao et al. 2020). The working principle of noninvasive ultrasonic flowmeter employing ultrasound as well as related developments can be found in Lynnworth and Liu (2006). The noninvasive measurement implies that no pipe drilling or pipe damage is necessary to measure the flow velocity of a fluid inside a pipe, as ultrasound can penetrate the pipe wall and transmit through a fluid in motion. The key mechanical component of a noninvasive ultrasonic flowmeter is a wedge-based ultrasonic transducer. Figure 1.14b shows a metal wedge containing

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1 Introduction

Fig. 1.13 Full-power mode-converting transmission realized by a monolayer single-phase anisotropic metamaterial. a Displacement field under an incident longitudinal (L) wave. Horizontal displacement of unit magnitude is prescribed at the leftmost boundary. Perfectly matched layers (not displayed in figure) are attached to the two ends. b Displacement field obtained with the monolayer metamaterial replaced by the homogenized effective medium. c Unit cell geometry: l 1 = 1.04a, h1 = 0.09a, l 2 = 0.24a, h2 = 0.08a, and θ = 34.5° with a = 2.5 cm. Size d is estimated as 1.775a. d Power ratios of L-to-L transmission (TL ), L-to-T transmission (TT ), L-to-L reflection (RL ), and L-to-T reflection (RT ). Lines and triangles denote results obtained using the actual model with detailed microstructures and the homogenized model with effective properties, respectively (Yang et al. 2019)

a monolayer of metamaterial; it depicts a monolayer with the same void patterns as the unit cell in Fig. 1.13. In contrast, the wedge used in conventional wedge-based transducers (not shown) is composed of a plastic material such as PEEK without any embedded slits or voids. Due to the use of metamaterial in its construction, the wedge shown in Fig. 1.14b is referred to as a metawedge and the transducer unit comprising the metawedge and the PZT is referred to as a meta-transducer. Figure 1.14a depicts the experimental setup used to evaluate the performance of the meta-transducer. The measured ultrasound signals are presented in Fig. 1.15, from which it can be concluded that the meta-transducer outperforms the conventional transducer for estimating the flow velocity; the signal output of the meta-transducer is nearly seven times that of the conventional transducer. This example illustrates a practically advantageous application of elastic metamaterials. Further information on the subject is provided in Chap. 10. Recent attention has also been paid to topological insulators, Willis metamaterials, and nonreciprocal metamaterials for more complex manipulation of elastic waves.

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Fig. 1.14 a Experimental setup for ultrasound-based flow velocity measurement, and b details of installed meta-transducers (a transducer equipped with a metamaterial-slab embedded wedge) (Piao et al. 2020)

Fig. 1.15 Output signals obtained using the experimental setup portrayed in Fig. 1.14a. Results obtained at 420 kHz when using a a pair of conventional PEEK wedge transducers, and b a pair of proposed meta-transducers (Piao et al. 2020)

Let’s begin with a quick overview of topological insulators. The fundamental advantages of employing topological insulators in wave manipulation are that they are highly robust and defect-free. A topological insulator is a novel type of material that insulates energy flow within but permits energy transfer along its surfaces (Hasan and Kane 2010). Such two-dimensional (2D) surface modes of wave propagation are observed in the case of topological insulators (3D). In lower dimensions, a comparable mechanism can be found. For instance, edge modes (1D) can be generated by planar (2D) or volumetric (3D) topological insulators, and corner modes (0D) can be found in a chain (1D), plate (2D), or bulk (3D) structure (Chaunsali et al. 2017; Chen et al. 2021; Schindler et al. 2018). These topological modes of energy transfer take place at the interface between two media with distinct topological properties, which are characterized by topological invariants. This topological nature, and

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1 Introduction

consequently topological invariance, is dictated by the bulk property of the medium. This is so-called bulk-boundary correspondence (Kane and Lubensky 2014). This concept, which originated in condensed matter physics, has been effectively utilized and demonstrated in electronics, options, acoustics, and mechanics (Chaunsali et al. 2018; Fan et al. 2019; Li et al. 2018; Liu and Semperlotti 2018; Miyazawa et al. 2022; Vila et al. 2017; Wang et al. 2017; Wu et al. 2021; Yu et al. 2018). Consequently, physical phenomena in quantum mechanics, including quantum Hall, quantum spin Hall, and valley Hall effects, have been realized in acoustic and mechanical lattices. This has created new opportunities for controlling and managing the flow of energy in mechanical metamaterials. Chen et al. (2019) provide a pedagogical explanation of various topological insulators using a mechanical honeycomb lattice. A recent review of topological elastic metamaterials can be found in Huang et al. (2021). Willis metamaterials also have unique properties. In contrast to materials analyzed by the classical theory of elasticity, Willis materials exhibit cross-coupling between strain and velocity, i.e., coupling between stress and velocity or momentum and stress. This coupling is referred to as Willis coupling. Consequently, metamaterials realizing Willis coupling (Milton and Wills 2007; Willis 1981, 1997) offers a novel method to realize asymmetric wave phenomena such as one-way propagation. Refer to Muhlestein (2016) for a comprehensive discussion of Willis coupling. The experimental demonstration of Willis coupling and the realization of elastic Willis metamaterials can be found in Muhlestein et al. (2017), Liu et al. (2019), Hao et al. (2022), and Chen et al. (2020). Nonreciprocity is a more general method for breaking the symmetry between action and reaction (Nassar et al. 2020). For instance, tensegrity can be used to realize nonreciprocity (Wang et al. 2020). It is noted that Wills coupling can result in nonreciprocity under certain conditions. While the wave propagation phenomena introduced above are based mainly on linear dynamics, nonlinear types of PCs and metamaterials have also been actively explored in recent decades. In nonlinear wave mechanics, different wave phenomena occur depending on wave amplitudes. If nonlinearity is not severe, a perturbation approach (Narisetti et al. 2010) can be effective for analyzing nonlinear wave phenomena, such as the amplitude-dependent-shift of the Bragg and resonance bandgaps (Fang et al. 2017). Recently, the phenomenon of amplitude-induced bandgap was also revealed (Bae and Oh 2020)—unique in nonlinear metamaterials. It was also shown that nonlinear elastic metamaterials can be used to tune bandgaps at extremely low frequencies, resulting in high suppression of largeamplitude vibrations and weak suppression of small-amplitude vibrations (Bae, and Oh 2022). Exotic settings of wave propagation media, such as granular crystals (Li et al. 2014), woodpile lattices (Kim et al. 2015), tensegrity architecture (Fraternali et al. 2014; Wang et al. 2018), 3D-printed lattices (Kim et al. 2018a), and recently origamiinspired structures (Yasuda et al. 2019), can produce another class of nonlinear wave phenomena. These nonlinear media interact with their constituent elements differently than their linear counterparts. Nonlinear particle contact (e.g., Hertzian contact Nesterenko 2001), nonlinear linkage (e.g., the Su-Schrieffer–Heeger model (Chen et al. 2014)), and planar folding mechanisms (e.g., origami polyhedron Yasuda

References

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et al. 2016) can be used to describe the interactions. Consequently, unique wave phenomena, such as solitary waves (Nesterenko 2001), breathers (Chong et al. 2014), nanoptera (Kim et al. 2015), dispersive shock waves and rarefaction waves (Yasuda et al. 2017), can be observed with linear PCs or metamaterials. Recently, the effect of disorder has also been considered in conjunction with nonlinearity, which has led to interesting discoveries such as the topological Anderson insulator (Meier et al. 2018; Shi et al. 2021) and sub- and superdiffusive energy transport (Kim et al. 2018a). Although nonlinear PCs and metamaterials exhibit intriguing wave phenomena, this book focuses primarily on linear wave dynamics. This is due to the fact that they form the basis for understanding PCs and metamaterials. Despite the fact that acoustic black holes (ABHs) may not be directly related to PCs and metamaterials, it is noteworthy that ABHs offer an alternative way to manipulate wave behavior. The ABH is a wedge-shaped structure with a power-law thickness profile. It is possible to reduce the group velocity of an incident elastic wave propagating toward the tip of an ABH. However, the amplitude of its vertical displacement can increase, resulting in a concentration of wave energy at the tip (Mironov 1988). A small amount of viscoelastic materials can be attached to the tip of an ABH in order to effectively attenuate the highly localized wave. Consequently, an ABH can reduce the vibration of a beam or plate structure when attached to a target structure (Krylov and Tilman 2004; Krylov and Winward 2007). Lee and Jeon (2019, 2021) developed a beam-based ABH theory and investigated the damping performance of this theory. For additional related research, see Lee and Jeon (2017, 2021), Park et al. (2019), Park and Jeon (2021), and Park et al. (2022). Creating anechoic terminations for shock-testing devices with ABH is an example of a practical application of ABH (Seo et al. 2020). A review of the theory and application of ABHs can be found in Pelat et al. (2020).

References Abbe E (1873) Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung. Arch Mikrosk Anat 9:413–468 Achenbach JD (1976) Wave propagation in elastic solids. North-Holland Auld BA (1973) Acoustic fields and waves in solids, vol 1 & 2, John Wiley and Sons Bae MH, Oh JH (2020) Amplitude-induced bandgap: new type of bandgap for nonlinear elastic metamaterials. J Mech Phys Solids 139:103930 Bae MH, Oh JH (2022) Nonlinear elastic metamaterial for tunable bandgap at quasi-static frequency. Mech Syst Signal Process 170:108832 Banerjee B (2011) An introduction to metamaterials and waves in composites. CRC Press Bonello B, Belliard L, Pierre J, Vasseur JO, Perrin B, Boyko O (2010) Negative refraction of surface acoustic waves in the subgigahertz range. Phys Rev B 82:104109 Brillouin L (1953) Wave propagation in periodic structures: electric filters and crystal lattices. Dover Bragg WH, Bragg WL (1913) The reflection of X-rays by crystals. Proc Royal Soc London. Series A, Contain Papers Math Phys Char 88:428–38

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Chapter 2

Fundamentals

Before discussing wave phenomena, we will first examine vibrations. The primary purpose of studying vibrations is to acquaint readers with time-dependent phenomena, particularly time-harmonic phenomena; a fundamental understanding of vibrations aids comprehension of the physics of waves in the following chapters. Using local-resonance-related vibration phenomena, we also introduce the concept of effective material properties, such as effective mass (density) and stiffness (longitudinal and shear moduli). The introduction of the effective material concept is primarily motivated by the fact that it facilitates the interpretation of the unusual wave phenomena produced by metamaterials. This concept will be explained in this chapter using a dynamic vibration absorber and will be utilized in subsequent chapters. A dynamic vibration absorber is a tuned spring-mass system attached to a vibrating mechanical system, which was originally created to eliminate unwanted vibrations at a particular location of the vibrating system. It is tuned for operation at a particular frequency. If the physics of the dynamic vibration absorber is interpreted using effective material properties, we will gain a great deal of insight.

2.1 Undamped Free Vibration of 1-DOF System We start with the simplest case of the undamped free vibration of a single-degree-offreedom (DOF) system. A single-DOF system comprising the lumped elements of mass m, stiffness s, and mechanical resistance (damper) c is illustrated in Fig. 2.1a. Although the undamped free vibration is considered herein, Fig. 2.1a includes an external force f (t) applied to the mass, as we will consider forced vibrations later. Based on the free-body diagram in Fig. 2.1b, the equation of motion can be expressed as m u(t) ¨ + cu(t) ˙ + su(t) = f (t), © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_2

(2.1)

27

28

2 Fundamentals

(a)

(b)

Fig. 2.1 a Single degree-of-freedom vibratory system. b Free-body diagram

where u(t) represents the displacement from an equilibrium state and (. ) denotes the derivative with respect to time t, i.e., d()/dt. The equation for the undamped free vibration can be derived from Eq. (2.1) with f (t) = 0 and c = 0, such as m u(t) ¨ + su(t) = 0.

(2.2)

To determine the general solution to Eq. (2.2), the following complex form of u(t) can be considered: u(t) = Ueiωt ,

(2.3)

where U denotes a complex-valued coefficient and ω should be determined to allow a nontrivial solution. As u should be real-valued from the physical perspective, Eq. (2.3) represents a complex-embedded form. This indicates that the real component of the complex-embedded form should be considered as the true solution, i.e., u = Re{Uexp(iωt)} to obtain the real-valued solution. Based on this understanding, we will not use symbol Re unless there is a danger of confusion. If Eq. (2.3) is substituted into Eq. (2.2), (−mω2 + s)Ueiωt = 0,

(2.4)

from which we can find ω as  ω = ±ω0 , with ω0 =

s m

(2.5)

The symbol ω0 is called the resonance frequency, eigenfrequency, or natural frequency of an undamped single degree-of-freedom mass–spring system (or oscillator). The general solution of Eq. (2.2) becomes

2.1 Undamped Free Vibration of 1-DOF System

29

u = U1 eiω0 t + U2 e−iω0 t ,

(2.6a)

u = U1 cos ω0 t + U2 sin ω0 t = U cos(ω0 t + φ)

(2.6b)

or

where Eq. (2.3) represents the complex form of the solution and Eq. (2.6b) represents the general solution explicitly written in a real-valued form. In Eq. (2.6b), U1 , U2 , and U are real-valued constants, and φ denotes the phase. Upon using the solution form in Eq. (2.6b), velocity v(t) and acceleration a(t) can be expressed as v(t) =

 π du , = −ω0 U sin(ω0 t + φ) = ω0 U cos ω0 t + φ + dt 2

(2.7)

d2 u = −ω02 U cos(ω0 t + φ). dt 2

(2.8)

a(t) =

The displacement, velocity, and acceleration in time with φ = 0 are comparatively presented in Fig. 2.2. This figure (or Eqs. (2.6)–(2.8)) indicates that the velocity ◦ ◦ (speed) leads the displacement by 90 and the acceleration is 180 out-of-phase with respect to the displacement. As observed in Eqs. (2.6)–(2.8), we should use both the sine and cosine functions if real-valued quantities are used. This raises inconvenience in analysis, so we prefer to use the complex-embedded solution in Eq. (2.6a). In this case, the calculation of velocity and acceleration can be facilitated as du = iωUeiωt , dt

(2.9)

d2 u = −ω2 Ueiωt . dt 2

(2.10)

v(t) = a(t) =

Fig. 2.2 General solution for undamped free vibration (with φ = 0)

30

2 Fundamentals

For instance, if U = U (real-valued), the actual real-valued velocity can be simply determined as   v(t) = Re iωU eiωt = −U ω sin ωt. Note that we used the same symbols u, v, and a, despite representing real- or complex-valued quantities. If there is a danger of confusion, we will use bold-faced symbols such as u and v to denote complex-valued fields. The complex-embedded form will be extremely beneficial if a (harmonic) force is applied to a system of interest such as the mass–spring system shown in Fig. 2.1a.

2.2 Damped Free and Forced Vibration of 1-DOF System If damped free vibration is considered, we simply consider f = 0 in Eq. (2.1) to obtain m u(t) ¨ + cu(t) ˙ + su(t) = 0.

(2.11)

The general solution to Eq. (2.11) can be assumed as u = Ueγ t .

(2.12)

The substitution of Eq. (2.12) into Eq. (2.11) yields mγ 2 + cγ + s = 0.

(2.13)

Equation (2.13) has the following solutions  c 2 s c ± − γ =− 2m 2m m  2 2 ≡ −β ± i ω0 − β ≡ −β ± iωd ,

(2.14)

where  β= ωd =



c , 2m

(2.15a)

ω02 − β 2 .

(2.15b)

As ω02 > β 2 in typical vibration applications, the expression in the second line in Eq. (2.14) is useful. Using β and ωd , the general solution can be expressed as

2.2 Damped Free and Forced Vibration of 1-DOF System

31

  u = e−βt U1 eiωd t + U2 e−iωd t .

(2.16)

For the real component of u in Eq. (2.16) with U1 = 1 and U2 = 0, the solution behavior as a function of time t is plotted in Fig. 2.3. For damped systems, the decay time constant τ is often used, and it refers to the time required for the original magnitude of displacement to reduce to e−1 : e−βτ = e−1 ≈ 0.3679, τ =

2m 1 = . β c

(2.17)

Evidently, a highly damped system (with a large c values) exhibits a smaller decay time. Let us investigate the forced vibration of a 1-DOF damped system using Eq. (2.1). Upon assuming a harmonic force excitation at an excitation frequency ω, f (t) can be expressed in the following complex-embedded form: f (t) = Feiωt ,

(2.18)

the resulting displacement u(t) is put into the complex-embedded form as u(t) = Ueiωt .

(2.19)

Upon substituting Eqs. (2.18) and (2.19) into Eq. (2.1), we can obtain 

 −mω2 + iωc + s Ueiωt = Feiωt .

(2.20)

Eliminating eiωt , Eq. (2.20) reduces to

Fig. 2.3 Displacement for 1-DOF undamped free vibration as a function of time t. (β =  and ωd = ω02 − β 2 )

√

c/2m

32

2 Fundamentals

  −mω2 + iωc + s U = F.

(2.21a)

For the undamped forced vibration with c = 0, Eq. (2.21a) becomes 

 −mω2 + s U = F.

(2.21b)

From Eq. (2.20), the (complex-valued) magnitude U of displacement can be expressed as U=

F . −mω2 + iωc + s

(2.22)

If writing velocity v(t) as v(t) = V exp(iωt) and using the definition of velocity v = du/dt, one can write v(t) = Veiωt = iωUeiωt .

(2.23)

Therefore, V = iωU =

F . c + i ωm − ωs 

(2.24)

Equation (2.24) can be equally expressed as   s  V ≡ Z(ω)V, F = c + i ωm − ω

(2.25)

where Z (ω) is called the (mechanical) impedance defined as  s Z(ω) = c + i ωm − . ω

(2.26)

As defined in Eq. (2.25), the impedance represents the ratio of the complex representation of a sinusoidal (or time-harmonic) force to the complex representation of the corresponding sinusoidal (or time-harmonic) velocity. Equation (2.26) shows how the system parameters (m, s, and c) affect Z(ω). To identify Z(ω) of an unknown mechanical system, a time-harmonic force can be applied to the system and the velocity response can be measured over a range of frequencies. (Here, the system is assumed to behave as a single degree-of-freedom system.) The concept of impedance originates from the electric field where the impedance draws relationship between the voltage field V and current field I, which represent the complex-valued magnitudes of the time-harmonic voltage and current, respectively (For direct current, Z reduces to resistance.). This impedance concept can be beneficial for analyzing power (product of force and velocity), as demonstrated in the following section.

2.3 Impedance and Power in 1-DOF System

33

2.3 Impedance and Power in 1-DOF System In Sect. 2.2, the mechanical impedance Z is derived as Eq. (2.26). As it is generally complex-valued, Z can be expressed as Z = R + i X.

(2.27)

The real part R is called the (mechanical) resistance, and the imaginary part X is called the (mechanical) reactance. They are explicitly given by R = c, X =ωm −

(2.28a) s . ω

(2.28b)

Note that the resistance R is independent of frequency, whereas the reactance X is frequency-dependent. If we introduce the damping factor ζ , c ζ = √ , 2 ms

(2.29)

R = c = 2ζ mω0 ,

(2.30a)

R and X can be expressed as

X =ωm −

 

 √ s m s 1 = ms ω − ω s mω



  ω 1 ω0 = mω0 − . = mω0 − ω0 ω

(2.30b)

In Eq. (2.30b), denotes the dimensionless frequency defined as =

ω . ω0

(2.31)

As Z is generally complex-valued, it can be conveniently expressed in terms of magnitude Z and phase as Z(ω) = Z (ω)ei (ω) ,

(2.32)

where tan (ω) =

(mω − s/ω) (mω2 − s) 2 − 1 X = = = . R c cω 2ζ

(2.33)

34

2 Fundamentals

For further use, we can express cos and sin in terms of R, X , and Z as cos =

X R , sin = . Z Z

(2.34)

According to Eq. (2.33), < 0 if ω < ω0 ( < 1) and > 0 if ω > ω0 ( > 1). The significance of the frequency-dependent phase variation can be clarified considering the displacement output. To this end, we will derive an explicit expression for u(t) for the forced vibration below. Using Eqs. (2.19), (2.24), and (2.25), the displacement u(t) can be expressed as Veiωt Feiωt v(t) = iπ/2 = iπ/2 jω e ω e ωZ ei

F i(ωt− −π/2) F i(ωt−φu ) = ≡ , e e ωZ ωZ

u(t) =

(2.35)

where

2  π π −1 − 1 + . φu = + = tan 2 2ζ 2

(2.36)

Therefore, the real-valued (true) displacement u(t) can be obtained by considering the real component of the result in Eq. (2.35):

 F i(ωt−φu ) F u(t) = Re(u(t)) = Re = e cos(ωt − φu + φ F ) ωZ ωZ F cos(ωt − φu ) for φ F = 0, = ωZ (ω)

(2.37)

where φ F denotes the phase of F, typically assumed as zero. Unless specified otherwise, φ F = 0 throughout this book. As remarked earlier, the same symbol u is used to represent both the complex-embedded and real-valued forms for avoiding the introduction of excessive symbols. Therefore, u(t) appearing first in Eq. (2.37) is real-valued, whereas u(t) appearing inside Re( · ) is complex-embedded. To investigate the frequency behavior of displacement for forced vibration, it is convenient to normalize the displacement with respect to the static displacement u static = F/s, which is the displacement under an external static force of magnitude F:

 F s u(t) cos(ωt − φu ) /(F/s) = cos(ωt − φu ) = u static ωZ ωZ mω02  = 2 cos(ωt − φu )  ωmω0 (2ζ )2 + − 1

2.3 Impedance and Power in 1-DOF System

= 

1   ω (2ζ )2 + − ω0

=

1

 1 2

35

cos(ωt − φu )

 2 cos(ωt − φu ) ≡ Au cos(ωt − φu ), (2ζ )2 + 2 − 1

(2.38)

where Au represents the displacement amplification factor defined as U(ω) 1 =  Au =  2 , ( = ω/ω0 ). u static (2ζ )2 + 2 − 1

(2.39)

The amplification factor Au and phase φu are plotted as a function of the dimensionless in Fig. 2.4. The following observations can be made in accordance with Fig. 2.4. 1) The factor Au increases monotonically

(to infinity, if c = 0) as → 1 (ω → ω0 ). The maximum of Au occurs at = 1 − 2ζ 2 . 2) If < 1(ω < ω0 ), φu always smaller than 90◦ . In contrast, φu always larger than 90◦ if > 1(ω > ω0 ). The phase φu change by 180◦ for c = 0 at = 1 and for > 1 mass m moves in the direction opposite to that of the applied force.

Fig. 2.4 a Displacement amplification factor Au ( ) and b displacement phase delay φu ( ). ( = ω/ω0 )

36

2 Fundamentals

This extraordinary behavior can be observed only in dynamic problems and not in static problems. The relative phase of the velocity v(t) with respect to the applied force f (t) depends on the frequency. Using Eqs. (2.25) and (2.32), v(t) can be written as (with φ F = 0) v(t) =

F cos(ωt − ). Z (ω)

(2.40)

Referring to Eq. (2.33), < 0 if < 1 and > 0 if > 1. Therefore, v(t) leads f (t) for < 1 (∵ v(t) = (F/Z ) cos(ωt + | |)) and v(t) lags behind f (t) for > 1 (∵ v(t) = (F/Z ) cos(ωt − | |). For the 1-DOF system in Fig. 2.1a under a time-harmonic force excitation f = F exp(iωt), the instantaneous power Pi supplied to the system can be defined as the product of the instantaneous driving force with the resulting instantaneous velocity v(t): 

 iωt  F iωt e Pi = Re( f (t)) · Re(v(t)) = Re Fe · Re Z 



 iωt  F F i(ωt− ) = (F cos ωt) e cos(ωt − ) , = Re Fe · Re Z (ω) Z (ω) (2.41) where F = F. Equation (2.41) simply states that Pi varies temporally as cos(ωt −

) cos(ωt). To extract more meaningful information, the time-averaged power Pa is utilized over a period T = 2π/ω. For a later use, the time-averaging operator •T is introduced as  1 T •T = (2.42) (•) dt. T 0 Using •T , Pa can be calculated as   2π/ω 2 F 1 T ω cos ωt cos(ωt − )dt Pa = Pi T = Pi dt = T 0 2π 0 Z

  ω F 2 2π/ω 1 1 2π ω F2 = · cos · (cos 2ωt + cos )dt = 2π Z 0 2 2π Z 2 ω 2 F cos . (2.43a) = 2Z Using Eqs. (2.34) and (2.43a) can be simplified as Pa (ω) =

F2 R 1 F2 R 1 = . 2 Z (ω)2 2 R 2 + X (ω)2

(2.43b)

2.3 Impedance and Power in 1-DOF System

37

The time-averaged power Pa (ω) in Eq. (2.43b) denotes the power supplied to the system to maintain the harmonic motion oscillating at the excitation frequency ω. If R = 0, i.e., no damping or resistive element exists in the system, then Pa = 0, implying zero net power supply for maintaining harmonic motion in case of no damping. For R = 0 (i.e., c = 0), an excitation force must supply a net power to the system for compensating the dissipative power by the resistive (damping) element. The examination of Eq. (2.43b) shows that Pa is maximized at the frequency corresponding to X (ω) = 0. The frequency rendering X (ω) = 0 is termed the resonance frequency. Upon setting X (ω) = 0, the resonance frequency becomes equal to the eigenfrequency of the undamped system. s X (ω) = ωm − = 0 → ω = ω



s = ω0 m

(2.44)

At the resonance frequency ω0 , the magnitude Z of the impedance becomes realvalued and minimized as Z = R (= c) and = 0 at ω = ω0 .

(2.45)

Consequently, the maximum value of Pa becomes 1 F2 at ω = ω0 . 2 c

Pa |max =

(2.46)

For ω = ω0 , the velocity v(t) and displacement u(t) become F cos(ω0 t), R

(2.47)

F cos(ω0 t). ω0 R

(2.48)

v(t) = u(t) =

If d|v(t)|/dω = 0 in Eq. (2.40) is set, the maximum of the velocity magnitude is found to occur at ω = ω0 and the corresponding v(t) is given by Eq. (2.47). In contrast, the displacement magnitude |u(t)| is not maximized at ω = ω0 . To determine the frequency maximizing the magnitude of u(t), we set dAu /d = 0, where Au denotes the displacement magnification factor defined in Eq. (2.39), thereby yielding =



1 − 2ζ 2 < 1 or ω = ω0 1 − 2ζ 2 < ω0 .

(2.49)

Therefore, the maximum displacement occurs at a frequency smaller than the resonance frequency in the damped system as shown in Fig. 2.4a. However, the maximum power (dissipated by c) occurs at ω = ω0 .

38

2 Fundamentals

Fig. 2.5 Time-averaged power Pa as a function of frequency ω

In Fig. 2.5, Pa is plotted as a function of frequency. In plotting Pa in Fig. 2.5, Eq. (2.43) is used in combination with Eq. (2.30): F 2c Pa ( ) =   2 c2 + m 2 ω02 −

 1 2

( = ω/ω0 ).

(2.50)

In the plot, the frequencies ωl and ωu are called half-power point frequencies at which Pa /Pa |max = 0.5. As a measure of sharpness of the curve in Fig. 2.5, a quantity Q called the quality factor is introduced: Q=

ω0 m ω0 ω0 = . ≡ ωu − ωl ω c

(2.51)

For large Q, the system response is localized near the resonance frequency in the frequency domain. Therefore, for a higher Q, the bandwidth is narrower near the resonance frequency.

2.4 Vibration of Undamped 2-DOF System and Effective Mass Concept In this section, we consider an undamped 2-DOF system shown in Figure 2.6. Upon considering the vibration behavior of an undamped 2-DOF system, the concept of “effective mass” is introduced as an essential concept that is useful in interpreting extraordinary wave phenomena realized by the metamaterials to be discussed in the later chapters. Referring to Fig. (2.6), the displacements of masses m 1 and m 2 from their equilibrium positions are denoted by u 1 (t) and u 2 (t), respectively. Upon considering the free-body diagrams for m 1 and m 2 , the equations of motion for the system can be

2.4 Vibration of Undamped 2-DOF System and Effective Mass Concept

39

Fig. 2.6 Undamped 2-DOF system. Harmonic excitation force applied to mass m 1

expressed as m 1 u¨ 1 (t) = −s1 u 1 (t) − s2 (u 1 (t) − u 2 (t)) + f (t),

(2.52)

m 2 u¨ 2 (t) = −s2 (u 2 (t) − u 1 (t)),

(2.53)

where s1 and s2 denote the stiffnesses, as depicted in Fig. 2.6. To determine a solution of Eqs. (2.52) and (2.53) under a harmonic force excitation, f (t) and u i (t) (i = 1, 2) are put into the following form: f (t) = Feiωt ,

(2.54)

u 1 (t) = U1 eiωt ,

(2.55a)

u 2 (t) = U2 eiωt .

(2.55b)

Upon substituting Eqs. (2.54) and (2.55) into Eqs. (2.52) and (2.53), we can obtain −m 1 ω2 U1 = −s1 U1 − s2 (U1 − U2 ) + F,

(2.56)

−m 2 ω2 U2 = −s2 (U2 − U1 ).

(2.57)

To facilitate the subsequent analysis, the eigenfrequency ω2 of the m 2 − s2 system is introduced and a dimensionless frequency is normalized with respect to ω2 as  ω2 = = The solution of Eq. (2.57) for U2 yields

s2 , m2

ω . ω2

(2.58a) (2.58b)

40

2 Fundamentals

s2 (s2 /m 2 ) U1 = U1 2 s2 − m 2 ω (s2 /m 2 ) − ω2 ω2 1 = 2 2 2 U1 = U1 1 − 2 ω2 − ω

U2 =

(2.59)

Using Eq. (2.59), U1 − U2 can be expressed as U1 − U2 = −

2 ω2 U , or U − U = − U1 . 1 1 2 1 − 2 ω22 − ω2

(2.60)

The substitution of Eq. (2.60) into Eq. (2.56) yields

− m1 +

 s2 ω2 U1 + s1 U1 = F. ω22 − ω2

(2.61)

Let us compare Eq. (2.61) representing the equation of motion for a 2-DOF undamped system and Eq. (2.21b) representing the equation of motion for a 1-DOF undamped system. When the following effective mass m eff 1 (ω) is introduced, the 2-DOF system expressed by Eq. (2.61) may be treated as if it were a 1-DOF system: m eff 1 (ω)

 (m 2 /m 1 )(s2 /m 2 ) s2 = m1 + 2 = m1 1 + ω2 − ω2 ω22 − ω2



 rm ω2 rm  m1 1 + 2 2 2 = m1 1 + 1 − 2 ω2 − ω

2  12 − 2 ,  m1 1 − 2

(2.62)

where the dimensionless parameters 12 and rm are defined as rm =

m2 m1 + m2 , 212 = 1 + rm = . m1 m1

(2.63)

Using the effective mass m eff 1 (ω) defined in Eq. (2.62), Eq. (2.61) is rewritten as 2 −m eff 1 (ω)ω U1 + s1 U1 = F.

(2.64)

The frequency dependency of m eff 1 (ω), as shown by Eq. (2.62), looks peculiar. However, we attempt to treat the 2-DOF system as an effective 1-DOF system using the effective mass because the effective mass concept gives some new insights to us. Upon assuming F = F, Eq. (2.64) becomes U1 =

1 1 F = U1static , 2 m eff (ω)ω2 1 (ω)ω s1 1 − m eff 1− 1 s1

s1

(2.65)

2.4 Vibration of Undamped 2-DOF System and Effective Mass Concept

41

Fig. 2.7 Reinterpretation of the undamped 2-DOF system in Fig. 2.6. a 2-DOF system described by a mass-in-mass model and b effective 1-DOF system with effective mass m eff 1 (ω)

where the static displacement of m 1 is denoted by U1static =

F . s1

For further analysis, we write Eq. (2.65) as U1 1 = . eff static m (ω)ω2 U1 1 − 1 s1

(2.66)

We will examine further m eff 1 (ω) defined in Eq. (2.62). First, the effective mass increases indefinitely as approaches 1. Second, the effective mass can be negative if > 1. As any physical mass should be nonnegative in the real world, the negative mass does not make sense at all. However, we treat it as “effective” mass, because we are treating the two-DOF system in Fig. 2.6 as an equivalent single-DOF system described in Fig. 2.7b. After treating the two serially connected mass–spring systems in Fig. 2.6 as a mass-in-mass system in Fig. 2.7a, the m 2 − s2 system is placed inside mass m 1 because the force is applied only to mass m 1 . Therefore, m 1 with the m 2 −s2 system in its interior as a whole can be regarded as an effective mass, as illustrated in Fig. 2.7b and the system in Fig. 2.7b can be regarded as a single-DOF system if the concept of effective mass is introduced. Owing to the internal vibration of the m 2 − s2 inside m 1 , the effective mass becomes inevitably frequency-dependent. We can generalize the concept of effective mass to cover all material properties, mass (or density) and stiffness (elastic coefficients) in the mechanical system.1 Once the effective property concept is introduced, certain extraordinary physical phenomena can be more conveniently interpreted giving some physical insight. In later chapters, the effective material properties are explored in various wave systems.

1

In electromagnetic systems, effective permittivity and permeability can be considered.

42

2 Fundamentals

2.5 Dynamic Vibration Absorber: Resulting Physical Phenomena √ If the eigenfrequency ω2 = s2 /m 2 of the m 2 − s2 system in Fig. 2.6 is set to be equal to the eigenfrequency of the m 1 − s1 system as ω22 =

s2 s1 ≡ , m2 m1

(2.67)

the m 2 − s2 system will function as a dynamic absorber, making that U1 = 0 at ω = ω2 . We aim to investigate the physical phenomenon both from the classical and metamaterial perspectives over a wide range of frequency if Eq. (2.67) is satisfied. According to the classical perspective, the magnitude of the displacement U1 can be directly examined as a function of ω( ). If the metamaterial perspective is used, the physical behavior will be interpreted using the effective mass concept. Under the condition (2.67), we will simplify U1 in Eq. (2.64) as F F   2 = rm 2 s1 − m eff (ω)ω 1 + ω2 2 s − m 1 1 1 1− 2 F/s1  ( ∵ m 1 /s1 = ω22 )  = rm 2 1 − 1 + 1− 2   (F/s1 ) 1 − 2 = 4 − (2 + rm ) 2 + 1

U1 =

(2.68)

Using U1static = F/s1 , Eq. (2.68) can be written as U1 1 − 2 . = 4 static − (2 + rm ) 2 + 1 U1

(2.69)

Substituting Eq. (2.69) into Eq. (2.59b) yields U2 as U2 1 . = 4 static − (2 + rm ) 2 + 1 U1

(2.70)

Notably, as no damping element exists in the present system, U1 and U2 are realvalued with positive or negative signs. The two resonance frequencies for which U1 and U2 grow indefinitely can be derived from 4 − (2 + rm ) 2 + 1 = 0.

(2.71)

For the subsequent discussion, we simply considered rm = 0.3 and worked with specific equations corresponding to rm = 0.3. However, the following discussion should be valid for other values of rm .

2.5 Dynamic Vibration Absorber: Resulting Physical Phenomena

43

Fig. 2.8 Magnitude |U1 | of the displacement as a function of = ω/ω2 (with r m = 0.3). Symbol U1static = F/s1 denotes the static displacement of m 1

The solutions to Eq. (2.71) are: R,1 = 0.762 and R,2 = 1.311. Using Eq. (2.69), U1 ( )/U1static is plotted in Fig. 2.8. Certainly, U1 ( )/U1static grows indefinitely at = R,1 and = R,2 . More importantly, at = 1 (ω = ω2 ), U1 ( = 1) = 0, U1static = −3.333U1static rm

 m1 F F/s1 F m2 m2 used =− . =− =− = m 2 /m 1 m 2 s2 s1 s2 s2

(2.72)

U2 ( = 1) = −

(2.73)

Equations (2.72) and (2.73) indicate that at = 1, no motion of m 1 exists and m 2 oscillates 180◦ out-of-phase with F. If we rewrite Eq. (2.73) as F = −s2 U2 = −m 2 ω2 U2 ,

(2.74)

where the last expression in Eq. (2.74) is obtained using Eq. (2.57) with U1 = 0 at = 1. Thus, the force F applied to mass m 1 is completely absorbed by the attached m 2 − s2 system, which is the phenomenon involving a dynamic damper tuned following Eq. (2.67). Let us consider Fig. 2.8 where 1 and 2 2 are defined as |U1 ( i )| = 1 (i = 1, 2). U1static 2

They will be explicitly determined in the next section.

44

2 Fundamentals

Fig. 2.9 Serially connected system consisting of n 2-DOF undamped units

Figure 2.8 shows that r = U1 ( )/U1static < 1 for 1 < < 2 .

(2.75)

The significance of Eq. (2.75) may be appreciated considering the serially connected system shown in Fig. 2.9. Assume that the system consists of 2-DOF undamped units subject to a force F exp[iωt] at its right end. If the excitation √ frequency = ω/ s2 /m 2 lies between 1 and 2 for which r < 1, the displacement at the n-th unit from the right end will diminish as r n . If n → ∞, the disturbance due to the applied force will eventually disappear. It should be noted that even though there is no damping in this system, the displacement of the mass m 1 of every 2-DOF unit decreases away from the loaded end. Because the transmission of the disturbance in the serially connected system can be viewed as wave propagation, this phenomenon can be regarded as stopped wave propagation along the negative x-axis. This phenomenon can occur because of the resonance of the m 2 − s2 system. We will investigate various stopped wave phenomena, including the one discussed here, in more detail in Chap. 5 from the wave propagation perspective.

2.6 Dynamic Vibration Absorber Interpreted by Effective Mass Herein, we attempt to interpret the physical phenomena observed in Sect. 2.5 using the effective mass concept. To this end, we start with Eq. (2.64), the equation of motion in the frequency domain, with F = 0: 2 −m eff 1 (ω)ω + s1 = 0.

(2.76)

Using (2.67), s1 is expressed as s1 = m 1

s2 = m 1 ω22 . m2

Substituting Eq. (2.77) into Eq. (2.76) yields

(2.77)

2.6 Dynamic Vibration Absorber Interpreted by Effective Mass

m eff 1 (ω) ==

m 1 ω22 . ω2

45

(2.78)

Using the dimensionless frequency defined in Eqs. (2.58b) and (2.78) becomes

m1 1 +

rm 1 − 2

 =

m1 . 2

(2.79)

In writing Eq. (2.79), we intentionally maintained m 1 to indicate that its left-hand side implies the effective mass. We will now examine the left- (LHS) and right-hand (RHS) sides of Eq. (2.78b) separately,

( ) = m LHS = m eff 1 1+ 1

 rm m1 , RHS = 2 , 2 1−

and plot them in Fig. 2.10. In addition, the displacements U1 and U2 are plotted in Fig. 2.11. Evidently, the intersections of RHS and LHS yield the resonance frequencies R,1 and R,2 . The following observations can be drawn from Figs. 2.10 and 2.11. 1) The phenomenon that U1 = 0 at = 1 can be interpreted as the consequence of infinitely large mass (m eff 1 ( ) → ±∞ as → 1); if a harmonic force is applied to an infinitely large mass, it will not move. 2) The effective mass m eff 1 ( ) becomes m 1 (1 + r m ) = m 1 + m 2 at = 0. This result is expected because the total mass of the 2-DOF system is m 1 + m 2 . In this limit, both U1 and U2 become U1static as only s1 resists against the applied force. 3) In contrast, m eff 1 ( ) → m 1 as → ∞. Therefore, the 2-DOF system functions as if it were a single DOF system with mass m 1 and stiffness s1 . The displacements Fig. 2.10 Plots of RHS (i.e., 2 m eff 1 ) and LHS (i.e., m 1 / ) of Eq. (2.78b)

46

2 Fundamentals

Fig. 2.11 Behavior of U1 /U1static and U2 /U2static as a function of

of masses m 1 and m 2 become zero in this limit because the masses cannot respond sufficiently quickly to the external excitation. Recall that that |U1 |/U1static becomes unity at 1 and 2 .To determine 1 , we set U1 /U1static = −1 (refer to Fig. (2.11)), U1 1 − 2 = −1, = 4 static − (2 + rm ) 2 + 1 U1 which yields an equation for : 4 − (3 + rm ) 2 + 2 = 0.

(2.80)

The solution to Eq. (2.80) is given by 2 =

  1 3 + rm ± rm2 + 6rm + 1 . 2

(2.81)

For rm = 0.3, Eq. (2.81) yields 2 = 0.8, 2.5, i.e., =

√

0.8 = 0.894,

√

2.5 = 1.581.

(2.82)

As we will consider the behavior of U1 primarily near = 1 (refer to Fig. (2.11)), we choose 1 as 1 = To determine 2 , we set

√ 0.8 = 0.894.

(2.83)

2.6 Dynamic Vibration Absorber Interpreted by Effective Mass

47

U1 1 − 2 = 1, = 4 static − (2 + rm ) 2 + 1 U1 which yields   2 2 − (1 + rm ) = 0.

(2.84)

2 = 0, 2 = (1 + rm ).

(2.85)

The solution to Eq. (2.84) is

Upon selecting the frequency near = 1 and considering rm = 0.3, we obtain the following: 2 =

√

1 + 0.3 = 1.14.

(2.86)

Therefore, |U1 ( )| < U1static for 1 (= 0.894) < < 2 (= 1.14).

(2.87)

As suggested by Eq. (2.87), the magnitude of m 1 becomes smaller than U1static for all instances as long as the excitation frequency is between 1 = 0.894 and 2 = 1.14. The consequence of this phenomenon was discussed in the paragraph below Eq. (2.75). Interestingly, we can observe from Fig. 2.10 that −∞ < m eff 1 ( )< 0 for 1 < < 2 .

(2.88)

Equation (2.88) implies that the effective mass is negative for frequencies between = 1 and 2 . In this frequency range, |U1 ( )|/U1static is smaller than unity because the effective mass is negative. Negativity in the effective mass will be further discussed in Chap. 5. In addition, we evaluate m eff 1 at = 1 : m eff 1 ( 1 )

= m1 1 +

rm 1 − 21



= m1 1 +

0.3 1 − 0.8

 = 2.5m 1 .

(2.89)

Using the result in Eq. (2.89), one can conclude that if 2.5m 1 < m eff 1 ( ) 0), the general solution can be derived as u n = A1 ei(ωt−βn) + A2 ei(ωt+βn) (β > 0),

(3.20)

where exp[i(ωt − βn)] and exp[i(ωt + βn)] represent waves propagating rightward (i.e., along the positive x-axis) and leftward, respectively. For instance, we consider the real part of u n in Eq. (3.20) for A1 = 1 (A2 = 0), ω = 1, and β = π/2:  π  u n = cos t − n (n = 0, 1, 2, . . .), 2 and plot it in Fig. 3.5. At t = 0, u n = cos(π n/2), and at t = π/2, u n = cos(π(n − 1)/2). As the snapshot at t = π/2 is equal to that at t = 0, excluding its rightward shift by n = 1, exp[i(ωt − βn)] represents a wave moving rightward. Likewise, exp[i(ωt + βn)] indicates a leftward wave propagation (i.e., along the negative x-axis).

Fig. 3.5 Rightward propagating wave (u n = cos(t − π n/2)). In the plot, n refers to the discrete axial coordinate. Two snapshots are illustrated at t = 0 (black line) and t = π/2 (red lines)

58

3 Longitudinal Waves in 1D Monatomic Lattices

Fig. 3.6 Dispersion relation for a 1D lattice (for ω ≤ ωcutoff ). The zone of β between −π and π is called the first Brillouin zone

Herein, we investigate the dispersion relation stated in Eq. (3.14) and plotted in Fig. 3.6. Based on the dispersion relation, the following observations can be stated: 1) The frequency ω() is an even function of β. 2) The frequency ω() is 2π -periodic in β, implying ω(β + 2π ) = ω(β). 3) There is no real β for ω > ωcutoff ( > cutoff ), where  ωcutoff = 2

s , i.e.,cutoff ≡ 2. m

(3.21)

This indicates that no wave can propagate if the excitation frequency is larger than ωcutoff . More discussions for ω > ωcutoff are provided later. 4) The displacement u n or Un is 2π -periodic in β. Therefore, we need to consider the dispersion curve and system response for only −π ≤ β ≤ π , corresponding to the zone of the Bloch phase (wavenumber) called the first Brillouin zone. 5) As β → 0 (for long wavelengths), the relationship between frequency (ω) and wavenumber (β) becomes linear, yielding  ω≈

s β. m

(3.22)

6) The frequency corresponding to β = π (k = π/d) is called the cutoff frequency (ωcutoff ) above which there is no real-valued β. Thus, a complex-valued β satisfying Eqs. (3.13) or (3.14) can be expected for ω ≥ ωcutoff . In this case, no wave can propagate, as discussed later in more details. To prove the last observation 4), we replace β in the first expression in Eq. (3.17) by β + 2mπ (m: any integer): Un |β+2mπ = Ae−i(β+2mπ)n = Ae−βn e−i2mnπ = Ae−iβn = Un |β . In terms of the wavenumber k = β/d, Eq. (3.23a) becomes

(3.23a)

3.1 Governing Equation and General Solution

59

Fig. 3.7 Two waves of distinct wavelengths measured at the locations of discrete masses apart by d. A wave with a shorter wavelength λ1 is recognized as a wave with a longer wavelength λ0 because measurements occur only at finite locations

Un |k+ 2πd m = Un |k .

(3.23b)

Equation (3.23) implies that we cannot distinguish the system response for β = β0 (k0 ), where 0 ≤ β0 ≤ π (0 ≤ k0 ≤ π/d) and those for β1 = β0 + 2π (k1 = k0 + 2π/d), β2 = β0 +4π (k2 = k0 +4π/d),···. (Here, we simply consider a positive β0 , but the same argument applies to a negative β0 .) To understand the meaning of Eq. (3.23), we consider two Bloch phases, namely β0 = π/2 and β1 = β0 + 2π = 5π/2 at √  = 2 with the corresponding wavelengths λ1 and λ2 : λ0 =

2π 2π d 2π 2π d 4d . = = 4d and λ1 = = = k0 β0 k1 β1 5

The harmonic waves with λ0 and λ1 are displayed in Fig. 3.7. In particular, the two waves are indistinguishable as the displacements are only “sensed” at discretely located masses. (Refer to the aliasing phenomenon (e.g., Oppenheim and Willsky (1997).) Thus, the wave analysis will be performed only within the first Brillouin zone, |β| ≤ π . To consider β for |β| ≤ π implies that the shortest wavelength λshortest “expressed” by the system is λshortest = 2d, i.e.,    2π   |β| ≤ π → |kd| ≤ π →  · d  ≤ π → λ ≥ 2d. λ In fact, the shortest wavelength λshortest can be observed at β= ± π : λshortest = 2d, at β= ± π. Therefore, any wave with a wavelength shorter than λshortest = 2d cannot be considered in this model as the sensing occurs only at masses located discretely. To clarify this, the displacement field is considered at a specific time t = t0 for β = π :

60

3 Longitudinal Waves in 1D Monatomic Lattices

  u n = Re A0 ei(ωt0 −πn) = A0 cos(ωt0 − π n)

(3.24)

For n = 0, n = 1, and n = 2 in Eq. (3.24), u 0 = A0 cos(ωt0 ), u 1 = −A0 cos(ωt0 ), u 2 = A0 cos(ωt0 ). As the mass at n = 0 and the mass at n = 2 are 2d apart and the sign of u n alters at every distinct n, the shortest wavelength that the 1D system represents should be 2d. As will be discussed in Chap. 7, the waves propagating in a nonperiodic continuum body (such as a bar) can carry any wavenumber (wavelength).

3.2 Phase, Energy and Group Velocities To analyze wave velocities, we primarily consider the positive branch of the dispersion relation expressed in Eq. (3.14a) without the loss of generality. Referring to Eq. (3.15), φ = ωt − βn is called the phase, which depends on both the time t and location n, influencing the (complex-valued) magnitude of the displacement u n . However, u n can have the same magnitude if φ is maintained constant for varying t and n. Let us select a specific phase value of φ0 and examine two instances with the same phase φ0 : φ0 = ωt − kdn

for time: t; location: n, for time: t + t; location: n + 1,

φ0 = ω(t + t) − kd(n + 1)

(3.25a) (3.25b)

By setting φ0 in Eq. (3.25a) equal to φ0 in (3.25b), the following relationship can be obtained as: ωt = kd.

(3.26)

Note that d/t represents the velocity where d represents the distance between the n-th mass and (n + 1)-th mass, and t denotes the traveling period between the n-th mass and (n + 1)-th mass. As Eq. (3.26) represents the maintenance of the same phase at various times and locations, the phase velocity v p can be defined as vp ≡

ω d = . t k

(3.27)

The phase velocity in a continuum body can be defined as v p = d x/dt by replacing d (finite distance) with d x and t with dt. We will derive this result in details in Chap. 7. The significance of the phase velocity is that if we travel at the phase velocity of a wave, the displacement of the same magnitude and sign is observed. For instance, we consider a displacement field u n (t) depicted in Fig. 3.5:

3.2 Phase, Energy and Group Velocities

61

u n (t) = cos(t − π n/2) (ω = 1, β = π/2) In Fig. 3.5, the snapshots of the displacement field at t = 0 and t = π/2 are plotted. The phase velocity v p for this wave field (corresponding to the positive branch in Eq. (3.14a)) can be stated as vp =

ωd 1·d 2d ω = = = . k β π/2 π

(3.28)

If the displacement (marked by A) observed at the n = 0 location at time t = 0 becomes equal to the displacement (marked by A ) observed at the n = 1 location, it should be observed at a later instant t = d/v p = d/(2d/π ) = π/2, where d indicates the distance between the two masses. To explicitly express the phase velocity for the 1D lattice system, Eqs. (3.27) and (3.14b) are used 2 ω vp = = k

s m

sin kd 2 =d k



s sin kd 2 . m kd 2

(3.29)

As ω in Eq. (3.14) is an even function of β = kd, only the positive values of β are considered in Eq. (3.29). This indicates that for the negative value of β, the sign in Eq. (3.29) should be negative, implying that the wave propagates leftward. A vital observation one can make from Eq. (3.29) is that the phase velocity varies as a function of k. As the wavenumber is related to frequency ω by the dispersion relation in Eq. (3.14), v p is certainly a function of ω. The consequence of this observation is stated as follows: If an arbitrarily shaped pulse is used as an excitation pulse, it can be decomposed into harmonic waves with distinct frequency components (more precisely, represented by its Fourier transform (e.g., Mallat (1999)). The pulse signal measured at location A away from the excitation point can be reconstructed by adding all the harmonic waves. As the waves with different frequencies travel at different wave speeds, their arrival times differ as well. Therefore, the shape of the reconstructed pulse at A differs from that of the original pulse used for excitation. As the pulse appears to be dispersed at A caused by the frequency–wavenumber relation in Eq. (3.29), the relation is called the dispersion relation. We will now examine the phase velocities in limiting cases. In the long-wavelength limit corresponding to k → 0 (i.e., β → 0), we obtain  lim v p = d

k→0

sin kd s lim kd 2 = d m kd→0 2



s . m

(3.30)

In contrast, the phase velocity in the short-wavelength limit corresponding to k → π/d (i.e.,β → π ) becomes

62

3 Longitudinal Waves in 1D Monatomic Lattices

 lim v p = d

k→π/2

  sin β2 s s 2 lim β = d β→π m π m 2

= 0.637 v p |k=0 .

(3.31)

As expressed in Eq. (3.31), the phase velocity at the short-wavelength limit is slower than that at the long-wavelength limit. Certainly, the phase velocity at the shortwavelength limit can be faster than that at the long-wavelength limit in other wave systems. Another wave velocity useful for wave analysis is the energy velocity ve defined as

i T ≡ ve E iT O T T ,

(3.32)



where i T and E iT O T T denote the time-averaged power and total energy density, respectively. The subscript i stands for instantaneous. As defined in Eq. (2.42), the time-averaging operator • T is defined as

 • T =

T

 (•)dt /T.

(3.33)

0

To calculate i T , we require an expression for the instantaneous power i , representing the power flowing out of the system of interest to its adjacent system at time t. In the lattice system currently in consideration, we select the n-th unit cell comprising the n-th mass as the system of interest and the (n + 1)-th cell comprising the (n + 1)th mass as its adjacent system (refer to Fig. (3.8)). Therefore, the power is expressed as a product of the velocity of the n-th mass and the force acting on the (n + 1)-th cell by the n-th cell. Referring to Fig. 3.8, f n,n+1 denotes the force applied by the (n + 1)-th cell to the n-th cell such that the force applied by the n-th cell to the (n + 1)-th cell is “− f n,n+1 .” Therefore, the instantaneous power i absorbed by the (n + 1)-th cell can be expressed as     i = Re − f n,n+1 Re(vn ) = Re − f n,n+1 Re(u˙ n ).

Fig. 3.8 Free-body diagram for a spring lying between n-th and (n + 1)-th mass

(3.34)

3.2 Phase, Energy and Group Velocities

63

Upon considering the free-body diagram in Fig. 3.8, f n,n+1 acting on the spring along the indicated direction can be written as f n,n+1 = s(u n+1 − u n ).

(3.35)

To facilitate the calculation of Eq. (3.34), the following relation will be shown: (ReG(t))(ReF(t)) T =

 1  Re G(t)F(t)∗ , 2

(3.36)

where (·)∗ denotes the complex conjugate of (·). In Eq. (3.36), G and F are assumed to vary harmonically with respect to time as G(t) = Ae jωt = Aei(ωt+φ A ) ,

F(t) = Beiωt = Bei(ωt+φ B ) .

(3.37)

The validity of the relation in Eq. (3.36) can be simply confirmed through the following analysis: (ReG(t))(ReF(t)) T = A cos(ωt + φ A )B cos(ωt + φ B ) T 1 = AB 2π/ω

2π/ω 

cos(ωt + φ A ) cos(ωt + φ B )dt 0

ω 2π

2π/ω 

1 (cos(2ωt + (φ A + φ B )) + cos(φ A − φ B ))dt 2 0

 ω 1 1 2π =AB · cos(φ A − φ B ) · = AB cos(φ A − φ B ) 2π 2 ω 2  1   1  = Re AB∗ = Re G(t)F(t)∗ 2 2 =AB

To calculate i T , we use Eqs. (3.34) and (3.36):   

1  i T = Re − f n,n+1 Re(u˙ n ) T = − Re f n,n+1 u˙ ∗n 2  1  ∗ = − Re s(u n+1 − u n )u˙ n . 2

(3.38)

The substitution of Eq. (3.15) into Eq. (3.38) yields  ∗  s  i T = − Re A(e−βi − 1)ei(ωt−βn) Aiωei(ωt−βn) 2 sω 2 = |A| Re{[(cos β − 1) − i sin β]i} 2 sω 2 = |A| sin β. 2

(3.39)

64

3 Longitudinal Waves in 1D Monatomic Lattices

If the positive branch of the dispersion relation, i.e., Eq. (3.14a), is substituted into Eq. (3.39), the following result can be obtained:  i T = s|A|

2

β s sin sin β = 2s|A|2 m 2



β s β sin2 cos . m 2 2

(3.40)

Notably, β in Eq. (3.40) is assumed to be positive as the wave propagating from n-th cell to the (n + 1)-th cell is considered for calculating the power.

To calculate the time-averaged total energy density E iT O T T in the n-th cell, it is decomposed into the time-averaged potential energy density E iP T and kinetic energy density E iK T as

E iT O T

T

= E iP T + E iK T .

(3.41)

Note that we evaluate the stored energy in the n-th cell because the power flow is considered from the n-th cell to the (n + 1)-th cell. In Eq. (3.41), the instantaneous potential energy density E iP and kinetic energy density E iK can be defined as E iP =

2 1  1 s Re(u n − u n−1 ) , E iK = m[Re(u˙ n )]2 , 2d 2d

(3.42a, b)

where d in the denominator is required because E iP and E iK represent the energy densities. The expressions for E iP T and E iK T are can be now stated as

E iP

 s   2  s 1  Re(u · Re (u n − u n−1 )(u n − u n−1 )∗ − u = ) n n−1 T 2d 2d 2 T   s|A|2   s |A|2  Re 1 − e jβ 1 − e− jβ = 1 − 2 cos β + cos2 β + sin2 β = 2d 2 4d s|A|2 s|A|2 2 β = · 2(1 − cos β) = · 4 sin 4d 4d 2 2 s|A| β sin2 , = d 2  m K  m 1  Ei T = · Re u˙ n u˙ ∗n [Re(u˙ n )]2 = 2d 2d 2 T   m 1  · Re (iωA) −iωA∗ = 2d 2 β m|A|2 s m|A|2 2 ω = 4 sin2 = 4d 4d m 2

s|A|2 β sin2 = E iP T . = d 2 =

Therefore,

3.2 Phase, Energy and Group Velocities



E iT O T

65

T

=2

β s|A|2 sin2 . d 2

(3.43)

Ultimately, the energy velocity ve can be calculated using Eqs. (3.32), (3.39), and (3.43) as i

ve = T O TT = d Ei T



β s cos . m 2

(3.44)

Unless the ω − β (ω − k) relationship is linear, the energy velocity ve is not equal to the phase velocity v p . To comprehend the physical significance of energy velocity, let us consider an experiment such that a 1D lattice is excited at a certain location by a pulse with its center frequency at ω0 . The arrival time of the ω0 -frequency component of the pulse is calculated by dividing the distance by the energy velocity at ω0 and not by the phase velocity because the wave traveling in space carries mechanical energy. In addition to the phase and energy velocities, a velocity called the group velocity exists, which refers to the velocity of a certain frequency component ω, when it travels as a wave packet. To explain the group velocity, we consider a nominal wave such that u = A cos(ωt − kdn),

(3.45)

where A denotes a real-valued constant. Currently, we consider a group containing two waves with slightly perturbed wavenumbers and frequencies from the wave expressed in Eq. (3.45): u − = A cos[(ω − ω)t − (k − k)dn],

(3.46a)

u + = A cos[(ω + ω)t − (k + k)dn],

(3.46b)

A wave group u total formed by the two waves in Eqs. (3.46) can be expressed as u total = u − + u + = 2 A cos(ωt − kdn) · cos(ωt − kdn).

(3.47)

To facilitate subsequent analysis, we replace dn (discrete axial coordinate) by x (continuous axial coordinate) in Eq. (3.47): u total = u − + u + = 2 A cos(ωt − kx) · cos(ωt − kx).

(3.48)

For small ω and k, the illustration of u total following Eq. (3.48) is presented in Fig. 3.9, which depicts that φg = ωt − kx governs the variations in the envelop of u total , whereas φ = ωt − kx governs the variations of the wave inside the envelop. If φg is maintained constant, one can write

66

3 Longitudinal Waves in 1D Monatomic Lattices

Fig. 3.9 Snapshot of u total in Eq. (3.47) at a certain time

φg = ωt − kx = const.

(3.49)

The differentiation of Eq. (3.49) with respect to time t yields ωdt − kdx = 0.

(3.50)

Based on Eq. (3.50), we can define the group velocity vg as  dx  dω . = vg =  dt φg =const dk

(3.51)

Notably, by setting φ = const, one can obtain the phase velocity v p = dx/dt|φ=const = ω/k, which has been expressed in Eq. (3.27). Based on the definition of the group velocity in Eq. (3.51), vg for the 1D lattice system can be derived as  dβ dω β s dω = · =d cos . (3.52) vg = dk dk dβ m 2 Upon comparing Eqs. (3.44) and (3.52), we obtain ve = vg .

(3.53)

Unless the system exhibits dissipation, the group velocity and energy velocity are identical. (Notably, the energy velocity can be defined even in a dissipative system.) As we primarily consider the nondissipative systems in the remaining portions of the book, we interchangeably use the group and energy velocities. Based on the expressions stated in Eqs. (3.29) and (3.44), the phase and energy velocities are compared for 0 ≤ β ≤ π in Fig. 3.10. Note that ve becomes zero as β approaches π for which ω attains ωcutoff , implying that no wave can propagate for ω ≥ ωcutoff .

3.3 Characteristic Impedance

67

Fig. 3.10 Comparison of phase (v p ) and energy (ve ) velocities for a 1D mass–spring periodic system

As the group velocity is equal to the energy velocity (in a medium without any damping), the arrival time of a pulse centered at a certain frequency should be calculated based on the group velocity. Note that it is impossible to transmit a single pure tone signal (such as sin ω0 t) because a pure tone signal can be identified only for an infinite time duration. For instance, consider an excitation force f T (t) = sin ω0 t( − T < t < T ). Unless T → ∞, f T (t) contains not only the sin ω0 t component but also other nearby frequency components. (This fact can  ∞be verified by examining the Fourier transform F(ω) of f T (t) where F(ω)= −∞ f T (t) · e−2πitω dt= T −2πitω dt = δ(ω − ω0 ) (δ: the Kronecker delta).) This indicates that any −T sin ω0 t · e sinusoidal pulse of a finite duration should be decomposed into multiple sinusoids, where its center frequency is located at ω0 . Accordingly, a harmonic wave component forming a pulse propagates at its group velocity at all instances because it is always accompanied by waves of other harmonic components.

3.3 Characteristic Impedance In Chap. 2, the impedance has been defined as the ratio of the complex representation of a sinusoidal (or time-harmonic) force to that of the corresponding sinusoidal (or time-harmonic) velocity. For periodic mechanical 1D lattice systems, the “characteristic impedance” is defined as the ratio of the complex representation of a sinusoidal (or time-harmonic) force exerted by the left cell on the right cell with respect to the complex representation of the corresponding sinusoidal (or time-harmonic) velocity at the left cell for a wave propagating from the left to the right. As depicted in Fig. 3.11, we can use either an asymmetric or a symmetric unit cell to define the characteristic impedance. The characteristic impedance Z A of the asymmetric unit cell shown in Fig. 3.11a can be defined as

68

3 Longitudinal Waves in 1D Monatomic Lattices

...

...

...

...

Fig. 3.11 Asymmetric and symmetric unit cells considered for calculation of characteristic impedance. a Asymmetric; b symmetric cases

−Fn,n+1 = Z A Vn ,

(3.54)

where Fn,n+1 is defined in terms of f n,n+1 expressed in Eq. (3.35): f n,n+1 = Fn,n+1 eiωt ,

(3.55)

and Vn represents the complex-valued magnitude of the velocity vn of the n-th mass, such as vn = Vn eiωt = (iω)Un eiωt ,

(3.56)

The negative sign of Fn,n+1 in Eq. (3.54) is required because “− f n,n+1 ” denotes the force acting in the positive x-direction upon the (n + 1)-th unit cell by the n-th unit cell. Using Eqs. (3.35), (3.55), and (3.17),   Fn,n+1 = s(Un+1 − Un ) = sUn e− jβ − 1 .

(3.57)

Based on Eqs. (3.56) and (3.57), the impedance Z A can be derived as   sUn 1 − e− jβ −Fn,n+1 = ZA = Vn iωUn   − jβ s 1−e s s = sin β + i (cos β − 1). = iω ω ω Note that Z A can be directly expressed in terms of − f n,n+1 and vn :

(3.58)

3.3 Characteristic Impedance

69

ZA =

− f n,n+1 −Fn,n+1 = . vn Vn

(3.59)

If Z A is decomposed into its real (Z rA ) and imaginary (Z iA ) components as Z A ≡ Z rA + i Z iA , they can be explicitly written as Z rA =

s s sin β, Z iA = (cos β − 1). ω ω

(3.60a, b)

If the dispersion relation given by Eq. (3.14) is used (with positive β’s), Eq. (3.60) can be simplified as

 

s s β β 2 sin cos √ (sin β) = ω 2 2 2 s/m sin β/2 √ m β = ms cos = ve , 2 d Z rA =

(3.61a)

and s Z iA = (cos β − 1) ω

 

s β = −2 sin2 √ 2 2 s/m sin β/2 √ β = − sm sin . 2

(3.61b)

The results in Eq. (3.61) reveal that the real component of the characteristic impedance contributes toward power transport, whereas the imaginary component does not. This can be confirmed by evaluating the power using the characteristic impedance according to Eqs. (3.33), (3.36), and (3.34):   

1  i T = Re − f n,n+1 Re(vn ) T = Re −Fn,n+1 Vn∗ 2  1  2 1  1 ∗   = Re Z A Vn Vn = Vn Re(Z A ) = Z rA |u˙ n |2 . 2 2 2

(3.62)

Equation (3.62) clearly indicates that only the real component of the characteristic impedance affects the power transport. To derive the characteristic impedance Z S for the symmetric unit cell model depicted in Fig. 3.11b, we introduce symbol f n − ,n + denoting the force acting on the left-half side of the n-th mass by the right-half side of the n-th mass attached to a semi-infinite lattice system. Thus, “− f n − ,n + ” denotes the force exerted by the left-half side of the n-th mass on the right-half side of the n-th mass attached to a semi-infinite lattice system. Therefore, Z S can be defined as −Fn − ,n + = Z S Vn ,

(3.63)

70

3 Longitudinal Waves in 1D Monatomic Lattices

Fig. 3.12 Variation of real (Z rA ) and imaginary (Z iA ) components of characteristic impedance as function of β for asymmetric unit cell model

Fig. 3.13 Free-body diagram of the left half of the n-th mass in a symmetric unit cell model shown in Fig. 3.11b

Fig. 3.14 Dispersion curve plotted for 0 ≤ Reβ ≤ π and Imβ ≥ 0. a 2D plot and b 3D plot

where Fn − ,n + is defined in the following expression: f n − ,n + = Fn − ,n + eiωt .

(3.64)

Upon considering the free-body diagram of the left half of the n-th mass illustrated in Fig. 3.13, the equation of motion for the left half of the mass becomes m u¨ n = f n − ,n + − s(u n − u n−1 ). 2 The substitution of Eqs. (3.15), (3.64), and (3.63) into Eq. (3.65) yields

(3.65)

3.4 Dispersion Relation for ω ≥ ωcutoff

−

71

m 2 ω Un = −Z S Vn − s(Un − Un−1 ). 2

(3.66)

Using Vn = iωUn and substituting Eq. (3.17) into Eq. (3.66), the following expression can be obtained as:   m 2 + s eiβ − 1 ω + s(cos β + i sin β − 1) ZS = = 2 iω    iω m s 2 β 2 β 4 m sin 2 + s −2 sin 2 + i sin β 2 = iω s r = sin β = Z A . ω m 2 ω 2

(3.67)

Equation (3.67) shows that the characteristic impedance Z S of the symmetric unit cell model is real-valued without any imaginary component. In addition, Z S is identical to the real component Z rA of the impedance Z A of the asymmetric unit cell model.

3.4 Dispersion Relation for ω ≥ ωcutoff Till the previous section, we primarily considered the wave phenomena for ω ≤ ωcutoff , where ωcutoff is the cutoff frequency. The corresponding dispersion relation was plotted in Fig. 3.6. In this section, we consider the dispersion relation for ω ≥ ωcutoff . Here, we √ consider the positive branch expressed in Eq. (3.13b). Although ω > ωcutoff = 2 s/m, sin2 (β/2) in Eq. (3.14a) must be real-valued and larger than unity to satisfy Eq. (3.14a). This can be possible if β becomes complex-valued as β = β R + iβ I .

(3.68)

If Eq. (3.68) is used,sin(β/2) in Eq. (3.14a) can be expressed as sin

βR βI βR βI β = sin cosh + i cos sinh . 2 2 2 2 2

(3.69)

As sin(β/2) should be real under Eq. (3.14a), its imaginary part should vanish: cos

βI βR sinh = 0. 2 2

(3.70)

Equation (3.70) yields β R = π, 3π, 5π, . . .. Because the dispersion curve for ω ≤ ωcutoff yields β = π at ω = ωcutoff and β for ω ≥ ωcutoff in Eq. (3.68) should be continuous at ω = ωcutoff , β R must be

72

3 Longitudinal Waves in 1D Monatomic Lattices

β R = π.

(3.71)

With Eq. (3.71), sin

βI β = cosh . 2 2

(3.72)

Substituting Eq. (3.72) into Eq. (3.13b) yields, ω2 = 4

 βI s cosh2 with (β I ≥ 0) m 2

(3.73a)

or  ω=2

 βI s cosh with (β I ≥ 0). m 2

(3.73b)

Because Eq. (3.70) also holds for β R = −π and Eq. (3.73) holds for –β I , four pairs of solutions are possible for ω > ωcuto f f , namely β = (β R + iβ I ), (β R − iβ I ), (−β R + iβ I ), (−β R − iβ I )(with β R , β I ≥ 0). (3.74) Thereafter, the substitution of Eq. (3.74) into (3.15) yields a general solution: u n = Un eiωt =A1 eiωt e−iπn eβ I n + A2 eiωt e−iπn e−β I n + A3 eiωt e+iπn eβ I n + A4 eiωt e+iπn e−β I n   = ( − 1)n eiωt A1 eβ I n + A2 e−β I n + A3 eβ I n + A4 e−β I n .   = ( − 1)n eiωt C1 eβ I n + C2 e−β I n (3.75) Owing to the existence of exp(±β I n), the waves for ω > ωcutoff cannot propagate but decay exponentially. Note that for n > 0 (or x > 0), we should select only the C1 exp(−β I n) term because exp(β I n) grows indefinitely as n → ∞. (Any disturbance of finite strength cannot be indefinitely amplified.) Similarly, we should select only the C2 exp(β I n) term for n < 0 (or x < 0). Either for u n = ( − 1)n eiωt C1 exp(−β I n) or for u n = ( − 1)n eiωt C2 exp(β I n), one can show that the time-averaged power i T (T = 2π/ω) is identically zero according to Eq. (3.38), implying that no energy transport occurs at frequencies greater than ωcutoff . Using Eqs. (3.71) and (3.73b), the dispersion curve for ω > ωcutoff is plotted in Fig. 3.14. The curve is plotted only for β R ≥ 0 and β I ≥ 0 without loss of generality. As waves cannot propagate at ω > ωcutoff , the corresponding frequency band is called a stopband. In principle, the origin of the stopband is caused by the periodicity of

3.5 Problem Set

73

the system, involving the physics known as the Bragg scattering (Bragg and Bragg 1913), which will be further explained in Chap. 7.

3.5 Problem Set Problem 3.1. Consider the following one-dimensional periodic system:

s2 d s1

s1 m

s2

s2 s1

m

s1 m

s2

s2 s1

m

s1 m

s1 m

s2

(a) Derive the equation of motion. (b) Derive the dispersion relation in terms of ω and β = kd, where k is the wavenumber and d is the distance between masses. (c) Plot the dispersion curve for −π ≤ β < π in the following systems with m = 1 kg: (i) s1 = s2 = 1, (ii) s1 = 0, s2 = 1, and (iii) s1 = 1, s2 = 0 (unit: N/m). Problem 3.2. (a) Referring to the periodic system shown in Problem 3.1, what relation must hold between s1 and s2 if the group velocity remains positive for 0 ≤ β < π ? (b) Plot the dispersion curves for (i) s1 = 1 and s2 = 0.5 and (ii) s1 = 1 and s2 = 0.2 and check if the relation derived in (a) is consistent with the dispersions plotted. Problem 3.3. Consider a symmetric unit cell model and derive the expression for the characteristic impedance Z S . Problem 3.4. Consider wave reflection and transmission across the joint of two semi-infinite one-dimensional lattice systems having different impedances (Z 1 and Z 2 ) shown below. Assume that a wave u inc n is incident toward the joint from −∞ in the lattice system 1: j(ωt−β1 n) u inc (A1 : given real-valued magnitude), n = A1 e where βi is the dimensionless wavenumber in the lattice system i. The reflected wave in the lattice system 1 and the transmitted wave in the lattice system 2 may be written as

74

3 Longitudinal Waves in 1D Monatomic Lattices j(ωt+β1 n) u ref , u trans = A2 e j(ωt−β2 n) n = A3 e n

Assume that A3 and A2 are also real-valued. (a) Express the displacement (or velocity) continuity at x = 0 (n = 0) in terms of Ai (i = 1, 2, 3), Z 1 , and Z 2 . (b) Express the power balance at x = 0 using the time-averaged power flow. (c) Find the transmission coefficient t = A2 /A1 and the reflection coefficient r = A3 /A1 . And establish the condition for t = 1 (r = 0) from your result.

Incident wave s1

s1 M1

x ( β1n or β 2 n ) s1

s2

M1

s2 M2

M1 / 2 Reflected wave Lattice system 1 having impedance Z1

s2 M2

M2 / 2 Transmitted wave Lattice system 1 having impedance Z2

References Banerjee B (2011) An introduction to metamaterials and waves in composites. CRC Press Bragg WH, Bragg WL (1913) The reflection of X-rays by crystals. In: Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character, vol 88, pp 428–438 Brillouin L (1953) Wave propagation in periodic structures: electric filters and crystal lattices. Dover Hussein MI, Leamy MJ, Ruzzene M (2014) Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook. Appl Mech Rev 66:040802 Mallat S (1999) A wavelet tour of signal processing. Academic Oppenheim AV, Willsky AS (1997) Signals and systems. Prentice Hall

Chapter 4

Longitudinal Waves in 1D Diatomic Lattices

This chapter discusses wave phenomena in a diatomic lattice, whereas Chap. 3 focuses on monatomic lattices. Figure 4.1 illustrates a diatomic lattice system. As more than one mass is considered in the lattice, the resulting dispersion curve will be more complicated than that of a system with a monatomic lattice. Brillouin (1953) provided a comprehensive treatment of the subject (see also Banerjee 2011; Hussein et al. 2014).

4.1 Governing Equation In a diatomic lattice in Fig. 4.1, the distance between the two masses M and m (M > m) is given by d/2, and the masses m and M are assumed to be located at even (2n) and odd (2n+1) locations (where n is an integer), respectively. The displacement of mass m at the (2n)-th location is denoted by u 2n , whereas the displacement of mass M at the (2n + 1)-th location is denoted by u 2n+1 . The unit cell is shown in Fig. 4.2. To derive the equations of motion, let us first consider the force acting on the (2n + 1)-th mass M. Referring to Fig. 4.3, the equation of motion for the mass M can be expressed as M

d2 u 2n+1 = −s(u 2n+1 − u 2n+2 ) − s(u 2n+1 − u 2n ) = s(u 2n + u 2n+2 − 2u 2n+1 ). dt 2 (4.1)

Similarly, the equation of motion for the (2n)-th mass m can be expressed as m

d2 u 2n = s(u 2n−1 + u 2n+1 − 2u 2n ). dt 2

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_4

(4.2)

75

76

4 Longitudinal Waves in 1D Diatomic Lattices

Fig. 4.1 One-dimensional diatomic lattice with two masses (M > m) connected by springs of stiffness s

Fig. 4.2 Unit cell of a diatomic lattice in Fig. 4.1. (Alternatively, one can select a unit cell containing whole mass M at center and two m/2 masses on boundaries)

Fig. 4.3 Free-body diagram showing the forces acting on the (2n + 1)-th mass M

Upon assuming harmonic motion and referring to Eq. (3.18), u 2n+1 and u 2n representing the displacements sampled at x = (2n + 1)(d/2) and x = 2n(d/2) can be expressed as u 2n+1 = A M ei(ωt−kx) |x=(2n+1)(d/2) = A M ei (ωt−(2n+1) 2 ) = A M ei(ωt−(2n+1)β1 ) , (4.3) kd

and u 2n = Am ei(ωt−kx) |x=2n(d/2) = Am ei (ωt−(2n) 2 ) = Am ei(ωt−2nβ1 ) , kd

(4.4)

where A M and Am are the magnitude of u 2n+1 and u 2n , respectively, and β1 is defined as β1 =

kd β = . 2 2

(4.5)

Regarding the periodicity of the dispersion curve, we are primarily concerned with the curve for β in the range of −π ≤ β = kd ≤ π . This range of β corresponds to

4.2 Dispersion Relation in Passband

77

the first Brillouin zone. In terms of β1 , the first Brillouin zone is given by −π/2 ≤ β1 = kd/2 ≤ π/2.

4.2 Dispersion Relation in Passband The substitution of u 2n+1 and u 2n in Eqs. (4.3) and (4.4) into Eqs. (4.1) and (4.2) yields 

Mω2 − 2s 2s cos β1 2s cos β mω2 − 2s



AM Am



  0 = . 0

(4.6)

To obtain a nontrivial solution from Eq. (4.6), the determinant of the 2 × 2 matrix in Eq. (4.6) should be zero. Accordingly, the following dispersion relation can be obtained:   1 4s 2 1 4 ω2 + (4.7) + sin2 β1 = 0. ω − 2s M m Mm Solving Eq. (4.7) for ω2 yields ⎡

⎤   2 1 1 1 1 4 + ± + sin2 β1 ⎦ ω2 = s ⎣ − M m M m Mm   s = (M + m) ± M 2 + m 2 + 2Mm cos 2β1 . Mm 

(4.8)

From the second equation in the matrix Eq. (4.6), the relation between A M and Am can be determined as AM −mω2 + 2s = . Am 2s cos β1

(4.9)

The substitution of Eq. (4.8) into Eq. (4.9) yields AM M −m∓ = Am



M 2 + m 2 + 2Mm cos 2β1 . 2M cos β1

(4.10)

The minus sign in Eq. (4.10) corresponds to the plus sign in Eq. (4.8). Because A M / Am in Eq. (4.10) is always real (for real β1 ), the phase difference between M and m is 0 if A M / Am > 0 and π if A M / Am < 0. For real-valued β’s, the following condition holds:

78

4 Longitudinal Waves in 1D Diatomic Lattices

√ √ Fig. 2s/M, ω2 = 2s/m, ω3 = √ 4.4 Dispersion curve for a diatomic lattice (ω1 = 2s(1/M + 1/m). The upper and lower branches correspond to plus and negative signs in Eq. (4.8), respectively



M 2 + m 2 − 2Mm ≤



M 2 + m 2 + 2Mm cos 2β1 ≤



M 2 + m 2 + 2Mm,

i.e., M −m ≤



M 2 + m 2 + 2Mm cos 2β1 ≤ M + m.

Therefore, ω2 in Eq. (4.8) is nonnegative in all instances, which implies the existence of a propagating wave for a given real wavenumber β. (This does not imply the existence of a real-valued β for any ω). The dispersion curve (denoting the relation between ω−β) obtained from Eq. (4.8) is plotted in Fig. 4.4, and the ratio of A M / Am is plotted in Fig. 4.5. Because M = m is assumed, there appear two branches in the dispersion curve. (Recall that there is only a single branch for a monatomic system considered in Chap. 3). The dispersion curve corresponding to the lower branch looks similar to that found in a monatomic lattice system considered in Chap. 3. Note that in the upper branch connecting ➅ and ➃ (for 0 ≤ β ≤ π ), the group velocity is negative, whereas the phase velocity is positive. Therefore, the upper branch for 0 ≤ β ≤ π describes the wave propagating leftward,1 whereas the upper branch for −π ≤ β ≤ 0 describes the wave propagating rightward. (The lower branch for 0 ≤ β ≤ π corresponds to the rightward propagating wave.) The lower branch starting from ω = 0 corresponds to the minus sign in Eq. (4.8) (and to the plus sign in Eq. (4.10)); the higher branch starting from ω = ω2 corresponds to the plus sign in Eq. (4.8) (and to the minus sign in Eq. (4.10)). Note that that the lower branch is referred to as the acoustic branch, because the frequencies belonging to this branch for certain lattices pertain to the same order of magnitude 1

If the signs of the group and phase velocities are different, the group velocity determines the direction of wave propagation. See Sect. 3.2.

4.2 Dispersion Relation in Passband

79

Fig. 4.5 Ratio of A M / Am as a function of β for M > m

of acoustic or supersonic vibrations. In contrast, the frequencies associated with the higher branch connecting ω2 and ω3 (marked in Fig. 4.4) are in the order of magnitude of infrared frequencies. Thus, the upper branch is referred to as the optical branch. To determine the cutoff frequencies, ω1 and ω2 , β = π (2β1 = π ) is substituted into Eq. (4.8): ω2 =

2s s 2s or , [(M + m) ± (M − m)] = Mm M m

thereby yielding  ω1 =

2s and ω2 = M



2s . m

(4.11a, b)

To determine ω3 , β = 0 is substituted into Eq. (4.8) to obtain ω3 =



  1 1 2 2 + . ω1 + ω2 = 2s M m

(4.12)

To examine the dispersion curve for small wavenumbers β = 2β1 → 0 (i.e., for long wavelengths), cos 2β1 can be approximated as cos 2β1 = cos β ≈ 1 − β 2 /2. With this approximation, the radical (square root) in Eqs. (4.8) and (4.10) becomes 

 M 2 + m 2 + 2Mm cos 2β1 ≈ M 2 + m 2 + 2Mm(1 − β 2 /2)

  Mmβ 2 Mmβ 2 1 − . ≈ + m) = (M + m) 1 − (M (M + m)2 2(M + m)2 The substitution of Eq. (4.13) into Eq. (4.8) yields.

(4.13)

80

4 Longitudinal Waves in 1D Diatomic Lattices

• for lower branch (minus sign in Eq. 4.8):    s Mmβ 2 s = (M + m) − (M + m) 1 − β2, ω ≈ 2 Mm 2(M + m) 2(M + m) (4.14) 2

• for upper branch (plus sign in Eq. 4.8):    Mmβ 2 s (M + m) + (M + m) 1 − ω ≈ Mm 2(M + m)2    2 1 β 1 + − . = 2s M m 4(M + m) 2

(4.15)

Equation √ (4.14) shows that the lower branch exhibits a linear ω − β relation such that ω = s/2(M + m)β as β → 0. This resembles the long-wavelength behavior of the dispersion curve in the monatomic lattice (refer to Eq. (3.22)). On the other hand, the upper branch decreases in the form of a parabolic function of β from its maximum frequency √ value at β = 0, corresponding to point ➅ in Fig. 4.4 with the frequency ω3 = 2s/(1/M + 1/m). For β = 2β1 → 0, Eq. (4.10) reduces to • Lower branch (plus sign in Eq. 4.10):   Mmβ 2 M − m + (M + m) 1 − 2(M+m) 2 AM   ≈ 2 Am 2M 1 − β8   Mmβ 2   M − m + (M + m) 1 − 2(M+m) 2 β2 1+ ≈ 2M 8    2 2 β β M −m 1 m β2 1 + ≈1+ = 1− 4 M +m 8 8 M +m → 1 at β = 0,

(4.16)

• Upper branch (plus sign in Eq. 4.10): AM Am

  Mmβ 2 M − m − (M + m) 1 − 2(M+m) 2   ≈ 2 2M 1 − β8   Mmβ 2   M − m − (M + m) 1 − 2(M+m) 2 β2 1+ ≈ 2M 8     2 β 1 M m β2 M − m m 2 1+ 1− ≈− 1− =− β M 4 M +m 8 M 8 M +m

4.2 Dispersion Relation in Passband

81

Fig. 4.6 Snapshots of particle motions of a diatomic lattice at β values in the dispersion relation in Fig. 4.4

→−

m at β= 0. M

(4.17)

The limiting result for the lower branch at β = 0 (corresponding to point ➀) exhibits that particles M and m move with equal amplitudes bearing no phase difference (refer to Fig. (4.6)). Therefore, all the particles oscillate by the same amount in the same direction, implying that the lattice oscillates as a single entity. As the corresponding frequency is zero, there is no restoring force. For the higher branch at ◦ β = 0 (corresponding to point ➅), the displacements of particles M and m are 180 out-of-phase. Therefore, two adjacent particles M and m move exactly in opposite directions; (refer to Fig. (4.6)). As the corresponding frequency ω3 is not zero, there is a restoring force ensuring the movement of the particles. Next, we examine the limiting behavior at β = π . (Owing to symmetry, the wave behavior at β = −π is identical to that at β = π ). For this analysis, it is convenient to introduce a small variable ε such that 2β1 = β = π − ε. Therefore, cos 2β1 = cos β = cos(π − ε) = − cos ε ≈ −1 +

ε2 , 2

(4.18a)

82

4 Longitudinal Waves in 1D Diatomic Lattices

cos β1 = cos

π ε ε ε β = cos − = sin ≈ . 2 2 2 2 2

(4.18b)

The substitution of Eq. (4.18a) into the radical in Eq. (4.8) yields 

 M 2 + m 2 + 2Mm cos 2β1 ≈ M 2 + m 2 + 2Mm(−1 + ε2 /2)

  Mmε2 Mmε2 1 + . ≈ − m) = (M − m) 1 + (M (M − m)2 2(M − m)2

(4.19)

In obtaining the terminating expression in Eq. (4.19), ε/(M − m) is assumed to be adequately small.2 Using Eq. (4.19), Eq. (4.8) reduces to • for lower branch (minus sign in Eq. 4.8):    2s Mmε2 s s = (M + m) − (M − m) 1 + − ε2 , ω ≈ 2 Mm 2(M − m) M 2(M − m) (4.20) 2

• for upper branch (plus sign in Eq. 4.8):    s 2s Mmε2 s ε2 , = (M + m) + (M − m) 1 + + ω ≈ 2 Mm 2(M − m) m 2(M − m) (4.21) 2

As ε increases from zero, (i.e., √ β deviates from π ), the lower branch parabolically decreases from ω√= ω1 = 2s/M and the upper branch parabolically increases from ω = ω2 = 2s/m. We also substitute Eq. (4.18b) into Eq. (4.10) to obtain. • Lower branch (plus sign in Eq. 4.10): AM Am

 M − m + (M − m) 1 +   ≈ 2M 2ε ≈

2(M − m) + Mε

Mmε2 2(M−m)

Mmε2 2(M−m)2



→ ∞ at ε → 0,

• Upper branch (plus sign in Eq. 4.10):

2

We used

√

1 + x ≈ 1 + x/2 for small x’s with x = Mmε2 /(M − m)2 .

(4.22)

4.3 Dispersion Relation in Stopband

83

 M − m − (M − m) 1 + AM   ≈ Am 2M 2ε ≈−

Mmε2 2(M−m)

Mε

=−

Mmε2 2(M−m)2



mε → 0 as ε → 0. 2(M − m)

(4.23)

To interpret the results in Eq. (4.22) for the lower branch, we observe that the particles M and m oscillate in the same direction (as A M / Am > 0). As β approaches π toward ➂, only the heavy particles (M) oscillate and form a standing wave, whereas the light particles (m) remain stationary, as illustrated in Fig. 4.6. In contrast, the particles M and m oscillate in opposite directions (as A M / Am < 0) for the upper branch. At the limiting wavelength of β = π corresponding to ➃, only the light particles m oscillate and form a standing wave, whereas the heavy particles M remain stationary, as illustrated in Fig. 4.6. The points ➁ and ➄ in Fig. 4.4 correspond to propagating waves. They correspond to the same wavenumber, but the masses M and m oscillate in the same direction at ➁, while they oscillate in opposite directions at ➄.

4.3 Dispersion Relation in Stopband In Fig. 4.4, a stopband is formed in the region between ω = ω1 and ω = ω2 and the region above ω = ω3 . For analyzing the stopband, Eq. (4.7) is rewritten as sin2 β1 = sin2

    β M +m Mm = ω2 − 2 ω2 , 2 2s 4s

(4.24)

As ω increases from 0 to ω1 , sin2 β1 in Eq. (4.24) increases from 0 to 1. If ω increases further, Eq. (4.24) requires that sin2 β1 be larger than 1. Because sin2 β1 > 1 is only possible for complex-valued β’s, we set β as (see Sect. 3.4),3 β = β R + iβ I ,

(4.25)

where β R and β I stand for the real and imaginary parts of β. The real part β R should be equal to π because the branches belonging to the passbands yield β = π as ω → ω1 and ω → ω2 ; otherwise, β cannot vary continuously as ω varies. Therefore, we can set β = π + iβ I ( for ω1 ≤ ω ≤ ω2 ).

(4.26)

With Eqs. (4.26) and (4.24) can be rewritten as If β (or β 1 ) is allowed to be complex-valued, Eq. (4.24) can be numerically solved (e.g., using MATLAB) to find the dispersion curve as shown in Fig. 4.7.

3

84

4 Longitudinal Waves in 1D Diatomic Lattices

Fig. 4.7 Dispersion relation for a diatomic lattice system. If Im β = 0, the corresponding band is a stopband

 cosh

2

βI 2



 =ω

2

M +m Mm − 2 ω2 2s 4s

 (for ω1 ≤ ω ≤ ω2 ).

(4.27)

For ω ≥ ω3 , we assume β = β R + iβ I and impose the condition that β = 0 at ω = ω3 , resulting in β = iβ I (for ω ≥ ω3 ).

(4.28)

If Eq. (4.28) is substituted into Eq. (4.24),   β 2 iβ = − sin − sin 2 2     βI 2 M +m Mm = sinh + 2 ω2 = ω2 − (for ω ≥ ω3 ). 2 2s 4s 2

(4.29)

As the parenthetical quantity in the last expression is positive for ω ≥ ω3 , a nontrivial value of β I can be determined for all instances. The dispersion curve including the stopband is plotted in Fig. 4.7.

4.4 Transition from Diatomic Lattice to Monatomic Lattice In this section, the wave behavior was examined at the instant the diatomic lattice reduces to a monatomic lattice, either by m → 0 (period = d), by M → ∞ (period

4.4 Transition from Diatomic Lattice to Monatomic Lattice

85

= d), or by m = M (period = d/2). As the last scenario involves a reduced period, the dispersion curve originally set up for M = m (period = d) should be carefully interpreted when m → M. Let us examine the limiting behaviors for each of the three cases. Case 1: m → 0 (period = d). As m → 0, Eqs. (4.11) and (4.12) become  ω1 =

2s , ω2 = M



2s → ∞, ω3 = m



2s 2s + → ∞. m M

In this limit, the width ω23 = ω3 − ω2 of the passband of the diatomic lattice system reduces to zero:

 2 ω1 ω23 = ω3 − ω2 = ω12 + ω22 − ω2 = ω2 1 + − ω2 ω2     1 ω12 1 ω1 2 = ω2 1 + → 0 as ω2 → ∞. − ω2 = 2 ω2 2 ω2 

Therefore, no passband is formed in a monatomic lattice above ω > ω1 . The dispersion curve precisely resembles the one4 shown in Fig. 3.6. Case 2: M → ∞(period = d). As M → ∞, Eqs. (4.11) and (4.12) become  ω1 =

2s → 0, ω2 = M



2s , ω3 = m



2s 2s + → ω2 . m M

The dispersion curve corresponding to this limit is presented in Fig. 4.8a. Evidently, only a single frequency is possible, and no wave can propagate at this frequency (the group velocity is zero everywhere); only small particles oscillate separately from each other (without any interaction), as illustrated in Fig. 4.8b–c for β = 0 and β = π. Case 3: m → M (period → d/2). This limiting case requires careful analysis and interpretation because the period of the lattice system suddenly reduces to d/2 from its original period d. As β is defined as β = kd and the actual period is d/2, the first Brillouin zone should be 4

In comparing the dispersion relation given by Eq. (3.14) (or in Fig. 3.6) for the monatomic lattice system and the limiting result in Case 1, some care must be taken. In Case 1, the stiffness of the spring connecting two adjacent masses (M) becomes s/2 because two springs of stiffness s serially connect the two masses. Therefore, we put s/2 for s and M for m in Eq. (3.21) to obtain ω1 for Case 1.

86

4 Longitudinal Waves in 1D Diatomic Lattices

Fig. 4.8 Limiting case of M → ∞ from the diatomic lattice system. a Dispersion curve, b particle motions at β = 0, and c particle motions at β = π

−2π ≤ β ≤ 2π . In this case, the Bloch phase β1 = kd/2 is more suitable for analysis because −π ≤ β1 ≤ π . If m = M is inserted in the dispersion relation (4.8), we obtain. ⎡



2

⎤

2 2 4 ± − 2 sin2 β1 ⎦ M M M    2s 2s 2 1 ± 1 − sin β1 = = (1 ± cos β1 ) M M 2s 2s (1 − cos β1 ) or (1 + cos β1 ) = M M 4s 4s β β1 1 = sin2 or cos2 . M 2 M 2

ω2 = s ⎣

Therefore,  ω=2

   s  β1  s sin = 2 M 2 M

   β sin ,  4

(4.30a)

   β1  s  s cos = 2 M 2 M

    cos β .  4

(4.30b)

or  ω=2

The dispersion curves based on Eqs. (4.30a, b) are plotted in the 1st Brillouin zone (−π ≤ β ≤ π ) of the diatomic lattice, as depicted in Fig. 4.9.

4.4 Transition from Diatomic Lattice to Monatomic Lattice

87

Fig. 4.9 Dispersion curve for the limiting case of m → M plotted using the dispersion curve of a diatomic lattice for m = M. If a dispersion curve were plotted for m = M (period = d/2) in its own 1st Brillouin zone (−π ≤ β1 = k(d/2) ≤ π , the corresponding curve would be a single √ continuous line connecting two points (ω = 0, β1 = 0) and (ω = 2 S/M, β1 = π ) Table. 4.1 Variations in key parameters as M → m in a diatomic lattice Lattice distance

λmin

[βmin , βmax ]

Range of β of the 1st Brillouin zone

Range of β1 of the 1st Brillouin zone

M = m

d

2d

[−π, π ]

2π

π

M =m

d/2

d

[−2π, 2π ]

4π

2π

Before further discussing the dispersion curves plotted in Fig. 4.9, the key parameters between the diatomic and monatomic lattice systems are comparatively presented in Table 4.1. As the range of the first Brillouin zone varies depending on the lattice distance (d or d/2), the dispersion curve of the limiting case for m → M should be interpreted with careful consideration.√Indeed, the solid-dotted lines connecting the points (ω = 0, β1 = 0) and (ω = 2 s/M, β1 = π ) represent the dispersion curves for the monatomic lattice system with the “correct” period of d/2 plotted in its 1st Brillouin zone (−π ≤ β1 = k(d/2) ≤ π ). Furthermore, the monatomic lattice includes only a single branch for −π ≤ β1 = k(d/2) ≤ √ π , as portrayed in Fig. 3.6. In reality, the curve representing Eq. (4.30b) (ω = 2 s/M| cos β1 /2|) is formed as a result of the band-folding phenomena. More details on band-folding will be explained below. As observed in Fig. 4.9, the dispersion curve of the monatomic lattice (period = d/2) expressed in the dotted line for π/2 ≤ |β1 = k(d/2)| ≤ π is folded 5 within 5

See Ahn et al. (2016) for two-dimensional problems.

88

4 Longitudinal Waves in 1D Diatomic Lattices

0 ≤ |β1 = k(d/2)| ≤ π/2 or 0 ≤ |β = kd| ≤ π with respect to β1 = π/2 or β = π when the curve is forced to be presented only within √ the 1st Brillouin zone of the diatomic lattice. Thus, the curve marked by ω = 2 s/M| cos β1 /2| for 0 ≤ |β1 = k(d/2)| ≤ π/2 denotes the mirror image of the dotted line of the monatomic lattice originally plotted for π/2 ≤ |β1 = k(d/2)| ≤ π with respect to β1 = π/2. In this case, the two branches from Eqs. (4.30a) and (4.30b) √ coincide at β1 = π/2 or β = π . More importantly, the curve marked by ω = 2 s/M| cos β1 /2| should not be understood as a separate branch with v p · vg < 0, but it should be interpreted as the dotted line plotted between π/2 ≤ |β1 = k(d/2)| ≤ π with v p · vg > 0. This band-folding phenomenon can typically occur when investing the dispersion curves of diatomic lattice systems in the limit of m → M abruptly changing the period from d/2 to d. Therefore, care must be taken when interpreting the dispersion curve of diatomic lattice systems for this limit.

4.5 Problem Set Problem 4.1 Consider the following diatomic system:

(a) Write down the equations of motion for m 1 and m 2 . (b) Derive the dispersion relation in terms of ω and β1 = kd/2 where ω and k are the angular frequency and wavenumber, respectively. (c) Find the values of ω’s when β1 → 0 and β1 → π/2. Problem 4.2 (a) For the one-dimensional lattice system shown below, derive the dispersion relation in terms of ω (frequency) and β1 = kd/3 where k is the wavenumber.

References

89

(b) Find ω’s corresponding to k → 0 for m 1 = 1 kg, m 2 = 2 kg, m 3 = 3 kg, and s = 500 N/m. Also, find ω’s corresponding to kd = π . Sketch the dispersion curve using ω and β = kd for 0 ≤ β ≤ π . Consider only the real part of β in the plot. Mark the passband and stopband in your sketch. (c) Plot the dispersion curves (i) when m 1 = 1 kg, m 2 = 2 kg, m 3 = 2 kg, and (ii) when m 1 = 1 kg, m 2 = 1.01 kg, m 3 = 1.01 kg, (m 1 ≈ m 2 ≈ m 3 ). Use s = 500 N/m for all problems. Compare the two result curves and make comments if there are any interesting phenomena.

References Ahn YK, Oh JH, Ma PS, Kim YY (2016) Dispersion analysis with 45°-rotated augmented supercells and applications in phononic crystal design. Wave Motion 61:63–72 Banerjee B (2011) An introduction to metamaterials and waves in composites. CRC Press Brillouin L (1953) Wave propagation in periodic structures: electric filters and crystal lattices. Dover Hussein MI, Leamy MJ, Ruzzene M (2014) Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl Mech Rev 66(4):040802

Chapter 5

Effective Material Property Manipulation in 1D Lattice Systems

The previous chapters investigate wave phenomena in one-dimensional lattice systems. Even at extremely low frequencies, intriguing wave phenomena, such as bandgap formation, can be observed if additional masses and springs are elaborately added to the system. Such phenomena are not typically observed in conventional lattice systems. Recently, the concept of effective material properties has been introduced and found to be extremely useful for interpreting observed physical phenomena. In the effective material concept, mass m and rigidity s can be frequency-dependent quantities. At a particular frequency, they can be negative or infinitely large (positive or negative). This section aims to introduce particular 1D lattice systems in which the concept of effective material property aids in interpreting observed wave phenomena.

5.1 Frequency-Dependent Effective Mass We begin with a one-dimensional mass-in-mass lattice system shown in Fig. 5.1. Unlike the lattice system considered in Fig. 3.1, a mass-in-mass system contains a mass–spring system attached to the mass m 1 . Let u n denotes the displacement of the n-th mass m 1 and vn , the displacement of the n-th mass m 2 inside m 1 . The equations of motion for the n-th masses m 1 and m 2 can be stated as m 1 u¨ n = −s1 (u n − u n+1 ) − s1 (u n − u n−1 ) − s2 (u n − vn ),

(5.1a)

m 2 v¨n = −s2 (vn − u n ).

(5.1b)

As discussed in Sect. 3.1, the displacement u n and vn can be assumed as u n = Aei(ωt−βn) , vn = Bei(ωt−βn) , © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_5

(5.2) 91

92

5 Effective Material Property Manipulation in 1D Lattice Systems

Fig. 5.1 One-dimensional mass-in-mass lattice system

where A and B are complex-valued constants. The substitution of Eq. (5.2) into Eq. (5.1b) yields the relation between A and B as B=

s2 A . (s2 − m 2 ω2 )

(5.3)

If Eqs. (5.2) and (5.3) are substituted into Eq. (5.1a), one can find   −ω2 m 1 A = s1 Ae−iβ + Aeiβ − 2A + s2



 s2 A − A . (s2 − m 2 ω2 )

(5.4)

The simplification of Eq. (5.4) yields   A s2 1 −

s2 (s2 − m 2 ω2 )



 − m 1 ω + 2s1 (1 − cos β) = 0. 2

(5.5)

To obtain a nontrivial solution from Eq. (5.5), the term inside the square bracket should be zero. The result can be written as m 1 m 2 ω4 − [(m 1 + m 2 )s2 + 2m 2 s1 (1 − cos β)]ω2 + 2s1 s2 (1 − cos β) = 0. (5.6) Equation (5.6) represents the dispersion relation for the system illustrated in Fig. 5.1. Compared with the dispersion relation (3.13), Eq. (5.6) is a quartic equation for ω as it involves two sets of masses m 1 and m 2 . The dispersion relation given by Eq. (5.6) is plotted in Fig. 5.2. Unlike the dispersion curves plotted in the previous chapters, the horizontal axis was set as the frequency (ω) to conveniently reveal the mechanism that renders the imaginary component of β as nonzero. In Fig. 5.2, three stopbands, which have nonzero Im (β)’s, are marked as A, B, and C, but the mechanism yielding the stopbands are different. The stopband marked by C is similar to the stopband appearing for ω > ωcutoff , as depicted in Fig. 3.14; in fact, the stopband marked as C is formed by the same mechanism as that observed in the 1D lattice system, i.e., due to the periodic arrangement of the lattice. (The dispersion equation cannot be satisfied unless β becomes complex

5.1 Frequency-Dependent Effective Mass

93

for sufficiently large values of ω.) To interpret the physics involved in the stopbands marked as A and B, we introduce the concept of “effective” material properties. Thus, we now delve into the “effective” mass. The concept of the effective mass was introduced to explain the physics involved in the dynamic absorber in Chap. 2. To provide a more concrete concept for the effective mass, we consider the mass-in-mass model displayed in Fig. 5.3a. According to the discussion in Sect. 2.4, the equations of motion for the system presented in Fig. 5.3a can be expressed as ¨ = −s2 [u(t) − v(t)] + f (t), m 1 u(t)

(5.7)

¨ = −s2 [v(t) − u(t)], m 2 v(t)

(5.8)

Fig. 5.2 a Dispersion curve for a mass-in-mass lattice system shown in Fig. 5.1 and b effective mass m eff the following data are used: 1 () ( = ω/ω2 ). For the plot, √ √ m 1 = 0.5 kg, m 2 = 0.2 kg, s1 = 150 N/m and s2 = 300 N/m. (ω2 = s2 /m 2 = 38.73, ω12 = s2 /m 1 + s2 /m 2 = 45.83)

94

5 Effective Material Property Manipulation in 1D Lattice Systems

Fig. 5.3 Mass-in-mass model used for explaining the effective mass concept

where the displacements of mass m 1 and m 2 are denoted as u(t) and v(t), respectively. If the applied force f (t) is assumed as a harmonic force, f (t) = F exp(iωt),

(5.9)

the displacements can be also assumed to harmonic as u(t) = Aeiωt , v(t) = Beiωt ,

(5.10a, b)

where F, A, and B denote the constants. The substitution of Eq. (5.10) into Eq. (5.8) yields B=

ω2 A s2 A = 2 2 2. 2 s2 − m 2 ω ω2 − ω

(5.11)

s2 . m2

(5.12a)

In Eq. (5.11), ω2 is defined as ω22 =

For a later use, the following symbol is also introduced 2 ω12 =

s2 s2 + = ω22 (1 + rm ), m2 m1

(5.12b)

where rm is defined as

m2 . m1 Using Eq. (5.11), v(t) may be expressed as rm =

v(t) =

ω22

ω22 u(t). − ω2

The substitution of Eq. (5.13) into Eq. (5.7) yields

(5.13)

5.1 Frequency-Dependent Effective Mass

95



 ω22 f (t) = m 1 u(t) ¨ − s2 − 1 u(t) ω22 − ω2   −ω2 u(t) =m 1 u(t) ¨ + s2 ω22 − ω2 .   s2 = u(t) ¨ m1 + 2 ω2 − ω2  2  ω12 − ω2 . = u(t)m ¨ 1 ω22 − ω2

(5.14)

If the system in Fig. 5.3a is interpreted as an equivalent system in Fig. 5.3b, Eq. (5.14) can be expressed as f (t) = m eff ¨ 1 (ω)u(t),

(5.15)

where the effective mass m eff 1 (ω) can be defined as  m eff 1 (ω)

= m1

 2 ω12 − ω2 . ω22 − ω2

(5.16)

As Eq. (5.16) suggests, the effective mass should be regarded as a frequencydependent quantity—this is a nontraditional interpretation of mass, expanding our dimension of thinking. As stated in the subsequent discussion, this new concept facilitates the interpretation of wave phenomena that is otherwise difficult to interpret. Let us introduce the following dimensionless frequencies to facilitate the subsequent analysis: =

ω ω2 and 212 = 12 = 1 + rm . ω2 ω22

(5.17a, b)

Using the symbols in Eq. (5.17), Eq. (5.16) can be rewritten as  m eff 1

= m1

 212 − 2 . 1 − 2

(5.18)

Herein, we consider the mass-in-mass system in Fig. 5.1 as if it were a monatomic lattice system (Fig. 3.1), where m 1 is replaced with m eff 1 (ω) (Eqs. (5.16) or (5.18)). After substituting m eff (ω) in Eq. (5.16) into Eq. (3.13b), the dispersion relation of 1 the monatomic lattice system with the effective mass becomes 2 2 m eff 1 (ω)ω = 2s1 (1 − cos β)= 4s1 sin

  β . 2

(5.19)

96

5 Effective Material Property Manipulation in 1D Lattice Systems

After certain algebraic manipulations, the substitution of Eq. (5.16) into Eq. (5.19) identically yields the same equation as the dispersion relation Eq. (5.6). Accordingly, the behavior of the dispersion relation of the mass-in-mass system (Fig. 5.1) can be interpreted using the dispersion relation of a monatomic lattice system (Fig. 3.1) based on the effective mass given in Eqs. (5.16) or (5.18). Let us now interpret certain behaviors in the dispersion in Fig. 5.2a using the effective mass plotted in Fig. 5.2b. First, the stopband marked by A is formed because 2 m eff 1 (ω) becomes (positively) sufficiently large to render sin (β/2) larger than unity. Therefore, β becomes complex-valued in this stopband. If we introduce ω( = 2 ω/ω2 ) such that ω2 m eff 1 (ω) = 4s1 , sin (β/2) > 1 for  <  < 1. (Note that  = 0.907 for the data illustrated in Fig. 5.2a.) In contrast, the stopband marked as B is formed for 1 <  < 12 , because m eff 1 (ω) is negative for which β must be complex-valued to satisfy Eq. (5.19). (The value of √ 12 is 1.4 for the case presented in Fig. 5.2). As discussed earlier, the stopband marked by C is formed because of large ω values, as observed in a monatomic lattice system considered in Sect. 3.1.

5.2 Frequency-Dependent Effective Stiffness Consider the two-mass lattice system presented in Fig. 5.4. Unlike the two-mass lattice system in the previous section, mass m 3 is supposed to move in the vertical direction, and the m 3 − s3 system has a resonance in the vertical motion. To analyze the wave behavior of the system in Fig. 5.4, our strategy is to treat it as an equivalent 1D lattice system in which the effects of the m 3 − s3 system on the m 1 −s1 lattice system are treated as an effective stiffness of the 1D lattice system. To analyze wave propagation phenomena in the system in Fig. 5.4, the horizontal displacement of the n-th mass m 1 is denoted as u n and the vertical displacement of the n-th mass m 3 as wn . Notably, if the m 3 − s3 system is not used, the total stiffness s1T connecting two adjacent m 1 ’s is simply expressed as s1T = s1 , because

Fig. 5.4 Two-mass lattice system which can be modeled as a monatomic lattice with effective stiffness. Masses denoted by m 1 are roller-supported that are allowed to travel only horizontally

5.2 Frequency-Dependent Effective Stiffness

97

Fig. 5.5 Free-body diagram used to calculate effective stiffness. The system under consideration consists of a set of springs and mass m 3 . For stiffness analysis, force f is assumed to be applied to both ends of the system. Points B and C denote pinned joints; A and B are assumed to travel horizontally

1 1 1 = + T 2s1 2s1 s1 To identify the effective stiffness when the lattice system in Fig. 5.4 is modeled as a monatomic system, we focus on the system involving mass m 3 in Fig. 5.5. For stiffness analysis, f is assumed to be applied to both sides of the system, and by symmetry, the horizontal displacement along the line passing point C can be assumed as zero. The consideration of the force equilibrium around point A yields f = 2s1 (u n − u˜ n ),

(5.20)

where u˜ n denotes the horizontal displacement of the revolute joint B indicated in Fig. 5.5. The consideration of the equilibrium condition around B in the horizontal direction (refer to Fig. (5.6a)) yields 2s1 (u n − u˜ n ) + T cos α − 2s3 u˜ n = 0,

(5.21)

where T denotes the force acting along the link BC, and α indicates the angle between the link BC and horizontal axis. A vertical reaction force is marked by the vertical arrow in Fig. 5.6a, but we do not need to consider it for the horizontal equilibrium analysis. Using the free-body diagram for mass m 2 in Fig. 5.6b, the equation of motion in the vertical direction can be expressed as m3

d2 wn = −2T sin α. dt 2

(5.22)

Solving Eq. (5.22) for T and inserting T into Eq. (5.21) yields 1 d2 wn 2s1 (u n − u˜ n ) − m 3 2 cot α − 2s3 u˜ n = 0. 2 dt

(5.23)

98

5 Effective Material Property Manipulation in 1D Lattice Systems

Fig. 5.6 a Forces acting around B, b forces acting around mass m 3 , and c geometric constraints on link BC

As the length (l) of link BC should remain unaltered even though points B and C move by and wn , respectively, the following relation must hold (see Fig. 5.6c):u˜ n l 2 = (a − u˜ n )2 + (b + wn )2 = a 2 + b2 − 2a u˜ n + 2bwn + u˜ 2n + wn2

.

(5.24)

Using l 2 = a 2 + b2 and assuming small displacements (i.e., wn2  |wn | and u˜ 2n  |u˜ n |), Eq. (5.24) becomes

Therefore, the relationship between wn and u˜ n can be given by wn ≈

a u˜ n u˜ n = . b tan α

(5.25)

If wn in Eq. (5.23) is eliminated using the expression stated in Eq. (5.25), Eq. (5.23) becomes 1 d2 u˜ n 2s1 (u n − u˜ n ) − m 3 2 cot 2 α − 2s3 u˜ n = 0. 2 dt

(5.26)

˜ n eiωt , Eq. (5.26) becomes Upon assuming u n = Un eiωt and u˜ n = U   m 3 ω2 ˜ Un . 2s1 Un = 2s1 + 2s3 − 2 tan2 α

(5.27)

Assuming that f is harmonic in Eq. (5.20) as f = Feiωt , the combination of Eqs. (5.20) and (5.27) yields   ˜ F = 2s1 Un − Un = 2s1 1 −

 2s1 Un 2s1 + 2s3 − m 3 ω2 /2 tan2 α

5.3 Doubly Negative Effective Material Properties

99



 4s1 tan2 α/m 3 Un =2s1 1 + 2 ω − 4(s1 + s3 ) tan2 α/m 3   ω2 − 4s3 tan2 α/m 3 Un . = 2s1 ω2 − 4(s1 + s3 ) tan2 α/m 3

(5.28)

If we write Eq. (5.28) as F  2s1eff (ω)Un ,

(5.29)

s1eff (ω) can be defined as  s1eff (ω)

= s1

 ω˜ 32 − ω2 , 2 ω˜ 13 − ω2

(5.30)

where

ω˜ 3 = 2 tan α

s3 s3 + s1 , ω˜ 13 = 2 tan α . m3 m3

(5.31a, b)

The symbol s1eff (ω) is referred to as the effective stiffness. Using the effective stiffness seff (ω), the dispersion relation of the system in Fig. 5.4 can be expressed as that of an equivalent monatomic system: m 1 ω2 = 4s1eff (ω) sin2

  β . 2

(5.32)

The behavior of s1eff as a function of ω is plotted in Fig. 5.7. As depicted in Fig. 5.7b, the effective stiffness exhibits a range of values from the negative infinity to the positive infinity owing to the presence of resonance. The stopbands are formed for the nonzero imaginary component of the wavenumber β, marked by A, B, and C in Fig. 5.7a. Note that stopband A is formed because the effective stiffness is inadequately small, whereas stopband B is formed because the effective stiffness is negative. These two bands are formed because of the resonance of the m 3 − s3 system. Stopband C is formed due to periodicity.

5.3 Doubly Negative Effective Material Properties In Sects. 5.1 and 5.2, the concept of the effective material properties has been introduced. In particular, the negative effective properties were introduced, and it was argued that the formation of certain stopbands in the dispersion curve can be explained by the negativity in the material properties. Herein, we investigate the wave phenomena for more general cases, especially for the case in which the effective mass

100

5 Effective Material Property Manipulation in 1D Lattice Systems

Fig. 5.7 a Dispersion curve and b effective stiffness s1eff (ω) when the system in Fig. 5.4 is regarded as a monatomic system. (m1 = 0.5 kg, m3 = 0.6 kg, s1 = 150 N/m, s3 = 200 N/m, α = π/4, ω˜ 3 ≈ 36.51, ω˜ 13 ≈ 48.30)

and stiffness are simultaneously negative. To this end, we combine a mechanical system possibly yielding negative effective mass (Sect. 5.1) with another possibly yielding negative effective stiffness (Sect. 5.2) to form the compound system shown in Fig. 5.8. Based on the results discussed in Sects. 5.1 and 5.2, the dispersion relation for the system shown in Fig. 5.8 can be directly stated as 2 m eff 1 (ω)ω

=

4s1eff (ω) sin2

  β . 2

(5.33)

eff The effective properties m eff 1 (ω) and s1 (ω) are explicitly rewritten below (see Eqs. (5.16) and (5.30)):

5.3 Doubly Negative Effective Material Properties

101

Fig. 5.8 Periodic lattice system with a horizontal and vertical resonator. It simultaneously realizes negativity in effective mass and stiffness

 m eff 1 (ω)

= m1

2 ω12 − ω2 ω22 − ω2



 and

s1eff (ω)

= s1

ω˜ 32 − ω2 2 ω˜ 13 − ω2

 (5.34)

where the parameters ω2 , ω12 , ω˜ 3 , and ω˜ 13 are defined as

ω2 =

s2 ,ω = m 2 12





s2 s3 s3 + s1 s2 + , ω˜ 3 = 2 tan α , ω˜ 13 = 2 tan α m2 m1 m3 m3

For the system shown in Fig. 5.8, the effective material properties are plotted in Fig. 5.9a, b, and its dispersion curve is plotted as the function of frequency in Fig. 5.9c. The stopbands were formed by the same mechanism as in the cases considered in Sects. 5.1 and 5.2, i.e., when only one of the effective mass and stiffness is negative. However, a peculiar behavior is observed in the dispersion curve in Fig. 5.9c, wherein a wave passing band is formed in some frequency zones of the simultaneous negativity in the effective mass and stiffness. The corresponding frequency zone is ωˆ < ω < ω12 (when ωˆ > ω2 as in this case), where ωˆ can be found solving Eq. (5.33) with β = π .1 Thus, the explicit equation to determine ωˆ is given by.  m1

2 ω12 − ωˆ 2 ω22 − ωˆ 2



 = 4s1

 ω˜ 32 − ωˆ 2 . 2 ω˜ 13 − ωˆ 2

eff Note that for ωˆ < ω < ω12 , both m eff 1 (ω) and s1 (ω) are simultaneously negative. Figure 5.9c shows that the group or energy velocity (vg ) is negative, but the phase velocity (v p ) is positive in the frequency zone of ωˆ < ω < ω12 . Although not 1 For ω < ω < ω, ˆ both effective mass and stiffness are simultaneously negative, too. However, β 2 eff is complex-valued because |m eff 1 (ω)| in this frequency range is larger than |4s1 (ω)|. Therefore, a stopband is formed in this frequency zone. Based on this observation, it is noted that the pass band is not always formed over the whole frequency zone of double negativity. In the case considered in Fig. 5.9, the passband is formed in a narrower band described by ωˆ < ω < ω12 than the double negative frequency band described by ω2 < ω < ω12 .

102

5 Effective Material Property Manipulation in 1D Lattice Systems

Fig. 5.9 a Effective mass, b effective stiffness, and c dispersion curve for the system depicted in Fig. 5.8. (m 1 = 0.5 kg, m 2 = 0.2 kg, m 3 = 0.6 kg, s1 = 150 N/m, s2 = 300 N/m, s3 = 200 N/m, α = π/4, ω˜ 3 ≈ 36.51, ω2 ≈ 38.73, ωˆ ≈ 43.05, ω12 ≈ 45.83, ω˜ 13 ≈ 48.30)

5.3 Doubly Negative Effective Material Properties

103

explicitly plotted in Fig. 5.9c, the group velocity for −π ≤ kd ≤ 0 in the same frequency band will be positive, whereas the phase velocity is negative. Therefore, we must consider the wave passing branch corresponding to the negative wavenumbers (−π ≤ kd ≤ 0), when we considered the waves propagating to the +x-axis, because the wave power or energy should be transported along the x-axis with a positive group velocity. In two- or higher-dimensional problems, the negative velocity provides an interesting consequence in the wave field. Snell’s law, the law of refraction between two dissimilar media, can be expressed as sin θ2 sin θ1 = , c1 c2

(5.35)

where θ1 and θ2 denote the incident and refracted angles, and c1 and c2 represent the phase velocities in media 1 and 2, respectively. In Fig. 5.10a, the refracted angle θ2 becomes negative if c2 of medium 2 is negative. The phenomenon of negative refraction can be realized if a metamaterial with doubly negative effective material properties at a selected frequency is used in constituting medium 2. A material with doubly negative properties is often referred to as the left-handed material with negative refractive index, which is proportional to the inverse of the phase velocity. If such a material (with c2 < 0) is inserted in a normal medium, a flat lens can be realized, as presented in Fig. 5.10b. Historically, the notion of the negative refraction in electromagnetic waves was first conceived by Veselago (1967). For elastic waves, the realization and experiment of the negative refraction left-handed metamaterials were reported by Zhu et al. (2014) and Oh et al. (2017). Note that the negative refraction can be realized using a phononic crystal, but in this case, its geometric feature size should be in the same order of magnitude as the wavelength. If a metamaterial is used, the negative refraction can be realized with its feature size in the subwavelength scale. A typical wave behavior at the frequency of negative phase velocity will be discussed in Chap. 6.2 A sketch of the left-handed metamaterial embedded in an aluminum plate (thickness: 1 mm) with the target frequency of 35 kHz is presented in Fig. 5.11. The detailed geometry of the 25 mm × 25 mm unit cell of the metamaterial is given in Oh et al. (2017). The wave used for simulations and experiments is the S0 wave (the lowest symmetric Lamb wave in a plate) with appropriate correspondence to a longitudinal wave. An omnidirectional wave source can be focused through a flat lens composed of the left-handed metamaterial. Although it is beyond the scope of this book, it is remarked that the left-handed metamaterial in Fig. 5.11 is designed as “isotropic” at 35 kHz. If the medium is anisotropic, the metamaterial cannot function as a flat lens. The wave simulation result using the finite element analysis is presented in Fig. 5.12a. The time-harmonic finite element calculations were performed using In Fig. 6.10c, the wave is supposed to propagate with positive group velocity along the +x-axis. The comparison of the wave behavior for the negative phase case with that for the positive phase velocity in Fig. 6.10b, c demonstrates the peculiar wave phenomenon occurring when the phase velocity is negative.

2

104

5 Effective Material Property Manipulation in 1D Lattice Systems

Fig. 5.10 Negative refraction for a medium having a negative phase velocity at a particular frequency. A metamaterial having doubly negative effective material properties can realize a negative phase velocity medium. a Comparison of positive and negative refractions and b an application of negative refraction to fabricate a flat lens (Oh et al. 2017)

COMSOL Multiphysics. The flat lens is composed of 6 × 12 metamaterial unit cells. The distance d from the center of the circular source to the left-hand side of the metamaterial lens is 20 mm. The time-harmonic omnidirectional forces centered at 35 kHz, which are distributed along the 10-mm-radius circle, were exerted in the homogeneous aluminum plate. To avoid any undesired reflected waves from the plate boundaries, perfectly-matched layers (PML) were placed adjacent to the boundaries. (Refer to Oh et al. (2017) for more detailed modeling techniques.) The color plot in Fig. 5.12a represents the magnitude of pressure (corresponding to longitudinal wave field) of the S0 wave in the plate (which is equivalent to the longitudinal wave in an elastic medium). The results confirmed that the actuated

Fig. 5.11 Left-handed metamaterial fabricated in a 1-mm-thick plate at 35 kHz. The S0 wave (lowest symmetric Lamb wave in a plate) is a symmetric guided wave propagating in a plate that corresponds well with a longitudinal wave. (Oh et al. 2017)

5.4 Problem Set

105

Fig. 5.12 Subwavelength focusing with flat lens composed of an isotropic left-handed metamaterial at 35 kHz. a Finite element simulation result (pressure field), b experimental result, and c simulation and experimental results compared along the vertical line connecting Pa and Pb (λ: wavelength of 35 kHz S0 wave mode in a 1-mm-thick aluminum plate). (Oh et al. 2017)

wave from the circular wave source is focused after passing through the left-handed metamaterial. The exact focal point is situated at (x, y) = (67, 0) mm. Based on the experimental result representing the distribution of the S0 wave amplitude in Fig. 5.12b, we can identify the high-strength point, and equivalently, the focal point is located in the box surrounded by the (50, 70) × (−10, 10) [unit: mm] region; this result is consistent with the theoretical prediction. The amplitude variations along the y-direction across the focal point, i.e., along line Pa − Pb , obtained by numerical simulation and experiment are presented in Fig. 5.12c. The estimated full-width at half maximum (FWHM) from the experimental result is 0.486λ, which favorably agrees with that estimated from the simulation result—0.427λ. In particular, the wavelength λ of the 35 kHz S0 wave mode at 1-mm-thick aluminum plate was λ = 152.94 mm. Notably, as FWHM < 0.5λ by the designed flat lens, it can surpass the diffraction limit3 (= 0.5λ) and achieve “sub-wavelength” focusing, which is otherwise difficult.

5.4 Problem Set Problem 5.1.

3

Abbe (1873) discovered a fundamental “diffraction limit” in optics, which states that any feature smaller than half the wavelength of the light used for imaging by a conventional optical system is permanently lost in the image. Metamaterials have offered new possibilities to overcome the limit, as provided in this example.

106

5 Effective Material Property Manipulation in 1D Lattice Systems

Consider the following one-dimensional infinitely-long periodic system. (a) The diatomic system shown above may be treated as a monatomic system having ef f ef f effective material properties m 1 (ω) and s1 (ω) where u n is treated as the primary variable. Derive its effective mass and stiffness. Assume that all particles move harmonically at the angular frequency of ω. (b) Plot the effective properties as the function of angular frequency. Also, plot the dispersion curve. In the plot, consider both the real and imaginary parts of the Bloch phase. Use m 1 = 0.5 kg, m 2 = 0.2 kg, and s1 = 500 N/m. Problem 5.2.

ef f

(a) Derive the effective mass m 1 in the system shown above by smearing out the effects of m 3 and s3 in m 1 . Treat the system as a monatomic system as in Problem 5.1. Assume harmonic motion at an angular frequency of ω. ef f (b) Plot m 1 (ω) as the function of frequency. Also plot the dispersion curve. In the plot, consider both the real and imaginary parts of the Bloch phase. Use m 1 = 0.5 kg, m 2 = 0.2 kg, m 3 = 0.1 kg, s1 = 500 N/m, and s3 = 120 N/m. Problem 5.3. Consider an infinitely-long one-dimensional periodic system shown below. Assume harmonic motion at an angular frequency of ω.

(a) Derive the effective parameters (mass and stiffness) of the one-dimensional periodic system by treating the system as a monatomic system with primary displacement u n . (b) Plot the effective properties of the equivalent monatomic system as the function of angular frequency ω. Also plot the dispersion curve for both the real and imaginary part of the Bloch phase. Use m 1 = 0.5 kg, m 2 = 0.2 kg, m y = 0.5 kg, s1 = 500 N/m, s = 200 N/m, and α = π/5.

References

107

Problem 5.4. Consider an infinitely-long one-dimensional periodic system shown below. Find the range of angular frequencies where the effective mass and stiffness are simultaneously negative. Assume harmonic motion at an angular frequency of ω. Use m 1 = 0.2 kg, m 2 = 0.1 kg, m y = 0.1 kg, s1 = 400 N/m, s2 = 300 N/m, s3 = 100 N/m, and α = π6 .

Problem 5.5. Consider the following infinitely-long one-dimensional periodic system.

ef f

(a) Derive and plot the effective mass m 1 by treating the system as a monatomic system where the primary displacement is u n . Use m 1 = 0.8 kg, m 2 = 0.4 kg, m y = 0.6 kg, s1 = 100 N/m, s2 = 600 N/m, s3 = 100 N/m, α = π/4. (b) Plot the effective mass as the function of ω as α → π/2. Explain why the resulting plot is much different from that obtained in part (a).

References Abbe E (1873) Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung. Arch Mikrosk Anat 9:413–468 Oh JH, Seung HM, Kim YY (2017) Doubly negative isotropic elastic metamaterial for subwavelength focusing: design and realization. J Sound Vib 410:169–186

108

5 Effective Material Property Manipulation in 1D Lattice Systems

Veselago VG (1967) Electrodynamics of substances with simultaneously negative values of ε and μ. Usp Fiz Nauk 92:517 Zhu R, Liu XN, Hu GK, Sun CT, Huang GL (2014) Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial. Nat Commun 5:1–8

Chapter 6

Metamaterials: Effective Property Realization

In the previous chapter, a one-dimensional model consisting of simple mass and stiffness components was introduced and used to examine the corresponding dispersion relations. However, these systems are difficult to realize (or fabricate) if their mass and stiffness properties cannot be independently tuned to the desired values. This chapter examines a continuum system whose effective mass and stiffness are either or both negative. Using a discrete model discussed in the previous chapter, we explain the wave behavior of the continuous system. Figure 6.1 depicts such an elastic metamaterial embedded in a base aluminum plate. Actual wave mode propagating in the plate is the lowest symmetric Lamb wave mode (S0 mode) (see, e.g., Achenbach (1976), Miklowitz (1978), Rose (2014)), which corresponds well to the longitudinal wave observed in a one-dimensional discrete or continuous system at low frequencies. Using the system depicted in Fig. 6.1, we investigate the longitudinal wave motion propagating along the x-axis. A metamaterial is an artificially engineered material with effective material properties not observed in natural materials. This material is engineered by arranging repetitive patterns in a base material. Typically, the pattern scale is smaller than the wavelengths of interest. If the wavelength of a considered wave is much longer than the unit cell size (say, 10 times longer) of the engineered material, it can be a metamaterial. An example is a holey structure shown in Fig. 6.1. Even though the system displayed in Fig. 6.1 is a continuum system, its wave behavior can be more conveniently analyzed using its equivalent 1D lattice model. Following the work conducted by Oh et al. (2016), we demonstrate the actual realization of effective material properties with values ranging from negative to extremely positive.

6.1 Metamaterial Modeling via Spring–Mass System The square unit cell of the metamaterial (Fig. 6.1) is presented in Fig. 6.2a, as denoted by C mk . The subscripts m and k in C mk indicate that both the effective mass (“m”) © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_6

109

110

6 Metamaterials: Effective Property Realization

Fig. 6.1 Image of a 2 mm-thick aluminum plate on which a metamaterial is embedded (black background: void). Its unit cell (2 cm × 2 cm) is designed to exhibit frequency-dependent effective mass and stiffness for longitudinal waves propagating along the x-axis. Arrows indicate longitudinal waves propagating along the x-axis. (Oh et al. 2016)

and stiffness (“k”) of the unit cell can be adjusted. On the contrary, the unit cells C m and C k presented in Fig. 6.2b, c allow only the effective mass and stiffness adjusted, respectively. As will be shown, the longitudinal wave behavior of C mk can be explained by the superposition of the wave behaviors of C m and C k using equivalent discrete mass–spring models. Upon examining the mechanical components of the unit cell C mk , we observe that mass m 1 serves as the fundamental mass element for the longitudinal wave in the x-direction, whereas the masses m 2 and m 3 serve to induce the resonating motions in the y- and x-directions, respectively, thereby possibly varying effective material properties. In particular, the thin members connecting these masses function as spring elements. Referring to the analyses presented in Chap. 5, one can see that the vibrational motions in the x- and y-directions influence the effective mass and stiffness, respectively; masses m 3 and m 2 primarily vibrate in the xand y-directions. The longitudinal wave motion in the x-direction in the metamaterial containing the unit cell C mk (see Fig. 6.1) will be analyzed using a periodic mass–spring system depicted in Fig. 6.3. To estimate two frequency-dependent parameters, i.e., the effective mass m eff (ω) and stiffness αeff (ω), the dispersion relation and expression for impedance Z A , earlier derived in Chap. 3, will be utilized (see Eqs. (3.13b) and (3.59)): ω2 m eff (ω) = 4αeff (ω) sin2 ZA =

kd , 2

 αeff (ω)  −Fn,n+1 − f n,n+1 = 1 − exp(−ikd) . = Vn ∂u n /∂t iω

(6.1) (6.2)

In Eq. (6.1), k denotes the wave number and d is the unit cell size in the x-direction, and in Eq. (6.2), −Fn,n+1 and Vn denote the force acting upon the (n + 1)-th unit cell by the n-th unit cell and the velocity of the n-th mass. If the characteristic impedance

6.2 Evaluation of Frequency-Dependent Effective Mass

111

Fig. 6.2 a Unit cell C mk of a metamaterial designed for independent tuning of effective mass and stiffness, b unit cell C m with an x-resonating component, and c unit cell C k with two y-resonating components. (Oh et al. 2016)

Fig. 6.3 Equivalent periodic mass–spring system with effective mass meff (ω) and seff (ω)

is known or derived for an unknown system, we can identify the effective stiffness αeff (ω) from Eq. (6.2) because it involves only the effective stiffness term. Once αeff (ω) is determined, the effective mass m eff (ω) can be determined using Eq. (6.1).

6.2 Evaluation of Frequency-Dependent Effective Mass The equivalent mass–spring system of the metamaterial containing the unit cell C m described in Fig. 6.2b is elaboratively illustrated in Fig. 6.4. Although the square unit cell is slightly off-centered to facilitate analysis, the resulting dispersion curve is not influenced because of the periodicity of the unit cells. Referring to the off-centered unit cell in Fig. 6.4, the x-directional displacements of m 1 and m 3 in the (n, j)-th unit cell are denoted as u 1n, j and u 3n, j , respectively. The indices n and j denote the unit cell location in the x- and y-coordinates, respectively. Considering the x-directional

112

6 Metamaterials: Effective Property Realization

Fig. 6.4 Discrete mass–spring system corresponding to a metamaterial composed of Cm . Symbol δ denotes the x-directional stiffness of a thin member connecting lumped masses m 1 and m 3

displacements of m 1 and m 3 for analyzing the longitudinal waves propagating along the x-direction, one can set up the following equations of motion: m1

∂ 2 u 1n, j ∂t 2

= α(u 1n+1, j + u 1n−1, j − 2u 1n, j ) + δ(u 3n, j+1 + u 3n, j − 2u 1n, j ), m3

∂ 2 u 3n, j ∂t 2

= δ(u 1n, j + u 1n, j−1 − 2u 3n, j ).

(6.3)

(6.4)

Note that the displacements in the y-axis need not be considered because they do not interact with the S0 wave mode (dominated by x-displacements of m 1 ). In addition, the displacements u 1n, j and u 3n, j can be assumed to not vary along the y-axis. Therefore, we can set u 3n, j+1 = u 3n, j and u 1n, j+1 = u 1n, j . If time-harmonic wave motion at an angular frequency of ω is considered, Eqs. (6.3) and (6.4) become −ω2 m 1 u 1n, j = α(exp(ikd) + exp(−ikd) − 2)u 1n, j + 2δ(u 3n, j − u 1n, j ), −ω2 m 3 u 3n, j = 2δ(u 1n, j − u 3n, j ). The solution of Eq. (6.6) for u 3n, j is given by

(6.5) (6.6)

6.2 Evaluation of Frequency-Dependent Effective Mass

u 3n, j =

113

2δ u1 . 2δ − ω2 m 3 n, j

(6.7)

Substituting Eq. (6.7) yields   2δm 3 ω2 1 u . −ω2 m 1 u 1n, j = α exp(ikd) + exp(−ikd) − 2 u 1n, j + 2δ − ω2 m 3 n, j

(6.8)

Equation (6.8) can be simplified as the following dispersion relation for the periodic system consisting of C m :  ω2 m 1 +

2δm 3 2δ − ω2 m 3

 = 4α sin2

kd . 2

(6.9)

To determine an expression for the characteristic impedance, the force exerted on the (n + 1)-th mass by the n-th mass can be expressed as   − f n,n+1 = α(u 1n, j − u 1n+1, j ) = α 1 − exp(−ikd) u 1n, j .

(6.10)

Thus, the expression for the characteristic impedance Z A can be derived based on Eq. (6.10) as1 ZA =

 α − f n,n+1 1 − exp(−ikd) . = 1 iω ∂u n, j /∂t

(6.11)

Upon comparing Eqs. (6.9) and (6.11) with Eqs. (6.1) and (6.2), one can identify the effective properties for the metamaterial comprising C m . The results are as follows: for the metamaterial made of C m : effective mass : m eff (ω) = m 1 +

2δm 3 ω2 m 3 = m1 + 2 x 2 , 2 2δ − ω m 3 ωx − ω

effective stiffness αeff (ω) = α,

(6.12) (6.13)

where ωx corresponds to the resonance frequency of the x-resonating component of the m 3 − δ system:  ωx =

1

2δ . m3

(6.14)

The subscript A is used to denote that the considered unit cell model is an asymmetric model.

114

6 Metamaterials: Effective Property Realization

Fig. 6.5 Discrete mass–spring system corresponding to a metamaterial composed of Ck . In figure, α and β denote the x- and y-directional stiffnesses and γ denotes the x–y coupling stiffness of a thin member connecting lumped masses, m 1 and m 2

The examination of Eqs. (6.12) and (6.13) shows that the effective mass can become negative if ω is slightly larger than ωx while the effective stiffness remains frequencyindependent. (One can find the frequency range for the negative effective mass using Eq. (6.12) by setting m eff (ω) < 0.)

6.3 Evaluation of Frequency-Dependent Effective Stiffness The equivalent mass–spring system of the metamaterial comprising the unit cell C k is illustrated in Fig. 6.5. Note that the two slender obliquely-oriented members connecting m 1 and m 2 produce stiffness (α and β) in the x- and y-directions as well as the coupling stiffness (γ ) between the two directions. Using α, β, and γ , the force–displacement relations for the two oblique members can be expressed as 

Fx Fy



=

αγ γ β

 

    α −γ u u Fx = or , Fy −γ β v v

(6.15)

where Fx and Fy denote the forces acting along the x- and y-axes; u and v denote the x- and y-directional displacements.2 Depending on the relative orientation angle of the slender member with respect to the x-axis, the signs of the off-diagonal terms Here, symbol v denotes the y-directional displacement. In other chapters, the same symbol is also used to represent the velocity given by v = ∂u/∂t (u: displacement).

2

6.3 Evaluation of Frequency-Dependent Effective Stiffness

115

vary in the 2 × 2 stiffness matrix in Eq. (6.15). As the motion of m 2 in the y-direction influences the wave motion in the x-direction, we should additionally consider the equation of motion for m 2 in the y-direction. Moreover, the coupling of u and v should be considered because of the coupling stiffness (γ ) term. Now, all necessary equations of motion for m 1 and m 2 can be combined to obtain the following relations3 : m1

∂ 2 u 1n, j ∂t 2





= −4αu 1n, j + αu 2n, j + αu 2n−1, j + αu 2n, j + αu 2n−1, j 2 2 2 2 +γ vn, j − γ vn−1, j − γ vn, j + γ vn−1, j , m2

∂ 2 u 2n, j ∂t 2

(6.16)

= −2αu 2n, j + αu 1n+1, j + αu 1n, j ,

(6.17)



m2

m2

∂ 2 u 2n, j ∂t 2 2 ∂ 2 vn, j

∂t 2



= −2αu 2n, j + αu 1n+1, j + αu 1n, j ,

(6.18)

2 = −γ u 1n+1, j + γ u 1n, j − 2βvn, j,

(6.19)



m2

2 ∂ 2 vn, j

∂t 2



2 = γ u 1n+1, j − γ u 1n, j − 2βvn, j.

(6.20)

Note that the quantities with superscripts 2 and 2 are related to the upper and the lower mass m 2 in the above equations, respectively. Thus, assuming time-harmonic wave motion, Eqs. (6.16)–(6.20) reduce to 

−ω2 m 1 u 1n, j = −4αu 1n, j + α(1 + exp(ikd))u 2n, j + α(1 + exp(ikd))u 2n, j 

2 2 + γ (1 − exp(ikd))vn, j − γ (1 − exp(ikd))vn, j ,

−ω2 m 2 u 2n, j = α(1 + exp(−ikd))u 1n, j − 2αu 2n, j , 

(6.21) (6.22)



−ω2 m 2 u 2n, j = α(1 + exp(−ikd))u 1n, j − 2αu 2n, j ,

(6.23)

2 1 2 −ω2 m 2 vn, j = γ (1 − exp(−ikd))u n, j − 2βvn, j ,

(6.24)





2 1 2 −ω2 m 2 vn, j = −γ (1 − exp(−ikd))u n, j − 2βvn, j .

(6.25) 



2 2 Using Eqs. (6.22)–(6.25), one can express the displacements u 2n, j ,u 2n, j , vn, j , and vn, j in terms of u 1n, j as

3

Note again that the displacement of mass m 1 in the y-direction need not be considered.

116

6 Metamaterials: Effective Property Realization 

u 2n, j = u 2n, j = 

2 2 vn, j = −vn, j =

α(1 + exp(−ikd)) 1 α(exp(ikd) + 1) 1 u n, j = u n+1, j , 2 2α − ω m 2 2α − ω2 m 2 γ (1 − exp(−ikd)) 1 γ (exp(ikd) − 1) 1 u n, j = u n+1, j . 2β − ω2 m 2 2β − ω2 m 2

(6.26) (6.27)

Finally, the substitution of Eqs. (6.26) and (6.27) into Eq. (6.21) yields the following dispersion relation for the longitudinal wave propagating in the x-direction: ω2 (m 1 +

  kd 4αm 2 2α 2 2γ 2 sin2 . ) = 4 − 2 2 2 2α − ω m 2 2α − ω m 2 2β − ω m 2 2

(6.28)

√ If the actuation frequency, ω, is much less than 2α/m 2 (this situation is conventional, because α is usually one order larger than β for slender members), one can write 2α − ω2 m 2 ≈ 2α.

(6.29)

Using (6.29), one can simplify Eq. (6.28) as  −ω2 (m 1 + 2m 2 ) = 2 α −

 2γ 2 (cos kd − 1). 2β − ω2 m 2

(6.30)

Equation (6.30) represents the dispersion relation for the metamaterial comprising Ck . To determine the characteristic impedance for the system, the force exerted on the (n + 1)-th mass by the n-th mass can be calculated as 



2 2 − f n,n+1 = −(2αu 1n+1, j − αu 2n, j − αu 2n, j + γ vn, j − γ vn, j ).

(6.31)

Thus, the substitution of Eqs. (6.26) and (6.27) into Eq. (6.31) yields − f n,n+1

2α 2 (1 + exp(−ikd)) 2γ 2 (1 − exp(−ikd)) 1 u n, j . = −2α exp(−ikd) + − 2α − ω2 m 2 2β − ω2 m 2 (6.32)

Upon approximating Eq. (6.29), Eq. (6.32) becomes − f n,n+1

2γ 2 (1 − exp(−ikd)) 1 u n, j = −2α exp(−ikd) + α(1 + exp(−ikd)) − 2β − ω2 m 2     2γ 2 1 − exp(−ikd) u 1n, j . = α− (6.33) 2β − ω2 m 2

6.4 Metamaterial with Frequency-Dependent Effective Mass and Stiffness

117

Therefore, the characteristic impedance Z A is evaluated as     2γ 2 1 − f n,n+1 1 − exp(−ikd) . α− = ZA = 1 2 iω 2β − ω m 2 ∂u n, j /∂t

(6.34)

Upon comparing Eqs. (6.30) and (6.34) with Eqs. (6.1) and (6.2), one can identify the effective properties for the metamaterial comprising C k . The results are as follows: for the metamaterial made of C k : effective mass : m eff (ω) = m 1 + 2m 2 effective stiffness : αeff = α −

2γ 2 2γ 2 /m 2 = α − , 2β − ω2 m 2 ω2y − ω2

(6.35)

(6.36)

where ω y corresponds to the resonance frequency of the y-resonating component of m 2 − β system:  ωy =

2β . m2

(6.37)

As observed in Eq. (6.36), the y-resonating frequency (ω y ) influences the effective stiffness in the x-direction without affecting the effective mass. The examination of Eqs. (6.35) and (6.36) shows that the effective stiffness (αeff ) becomes negative at frequencies slightly less than ω y . (One can find the frequency range for the negative effective stiffness using Eq. (6.36) by setting αeff (ω) < 0.)

6.4 Metamaterial with Frequency-Dependent Effective Mass and Stiffness In Sects. 6.2 and 6.3, the effective mass and stiffness for the metamaterials with the x- and y-resonating components are identified, respectively. As the unit cell Cmk can be regarded as a combination of Cm and Ck , the wave dispersion equation and the impedance equation for the metamaterial comprising Cmk can be rewritten as    2γ 2 /m 2 kd ωx2 m 3 =4 α− 2 , ω m 1 + 2m 2 + 2 sin2 ωx − ω2 ω y − ω2 2   1 2γ 2 /m 2  ZA = 1 − exp(−ikd) . α− 2 2 iω ωy − ω 2

(6.38)

(6.39)

Accordingly, the effective mass and stiffness for the metamaterial consisting of the unit cell Cmk can be derived as

118

6 Metamaterials: Effective Property Realization

Fig. 6.6 a Unit cell Cm , b Ck , and c Cmk (unit: mm, material: aluminum) (Oh et al. 2016)

m eff (ω) = m 1 + 2m 2 + αeff (ω) = α −

ωx2 m 3 , ωx2 − ω2

(6.40)

2γ 2 /m 2 . ω2y − ω2

(6.41)

The examinations of Eqs. (6.40) and (6.41) reveal that the effective mass and stiffness can be independently tuned, because ωx and ω y is independently tunable. Therefore, a wide range of values of the effective mass and stiffness can be realized. For instance, both the effective stiffness and mass become negative, or only one effective property (say, mass) is negative depending on the selected frequency. To present a metamaterial having independently tunable effective material properties, we utilize the unit cells of specific geometries as depicted in Fig. 6.6. The parameters, such as m 1 and α, may be analytically evaluated (see, e.g., Wang et al. (2011), Wang and Wang (2013), and Oh et al (2014)), or numerically estimated using the finite element analysis. The finite element model used for calculating the stiffness4 α, β, and γ of the unit cell C mk is depicted in Fig. 6.7b, and mass m 2 can be of the lumped block, as portrayed in Fig. 6.7a. approximated using the mass m block 2 Using the calculated results listed in Table 6.1, one can evaluate ωx and ω y as 1 ωx fx = = 2π 2π



ωy 1 2δ = = 23.44 kHz and f y = m3 2π 2π

 2β = 30.47 kHz. m2

To evaluate the spring coefficients α, β, and γ , only the part marked as the red box in Fig. 6.7a should be considered. Fixed boundary conditions are imposed on Boundary 1, and θz = 0 (no rotation with respect to z-axis) is imposed on Boundary 2, as illustrated in Fig. 6.7b. The structure shown in Fig. 6.7b can be discretized by the finite elements for the calculation of u and v (the xand y-directional displacements, respectively) at the center point of Boundary 2 with the independent application of F x and F y , which represent the x- and y-directional forces, respectively. The calculated displacements are employed to construct the compliance matrix B relating the forces to

T displacements as Bf = u, where f = Fx , Fy and u = {u, v}T . Ultimately, the spring coefficients α, β, and γ can be evaluated by using the inverse B−1 with α = (B−1 )11 , −γ = (B−1 )12 = (B−1 )21 , and β = (B−1 )22 .

4

6.4 Metamaterial with Frequency-Dependent Effective Mass and Stiffness

119

Fig. 6.7 a Unit cell Cmk and b finite element model to calculate the spring stiffness α, β, and γ . The boundary conditions used for the stiffness evaluation are illustrated. (Oh et al. 2016)

Table 6.1 The estimated mass and stiffness parameters using the three-dimensional finite element model in Fig. 6.7 for the unit cells shown in Fig. 6.6

α

Metamaterial (C m ) with x-resonating part

Metamaterial (C k ) with y-resonating part

Metamaterial (C mk ) with xand y-resonating parts

1.24e4 kN/m

1.02e4 kN/m

1.02e4 kN/m

β

1.15e3 kN/m

1.23e3 kN/m

1.23e3 kN/m

γ

1.10e-3 kN/m

2.44e3 kN/m

2.43e3 kN/m 1.04e3 kN/m

δ

1.04e3 kN/m

3.26e3 kN/m

m1

2.52e−4 kg

2.53e−4 kg

2.21e−4 kg

m2

1.85e−5 kg

6.71e−5 kg

6.71e−5 kg

m3

9.59e−5 kg

4.70e−5 kg

9.59e−5 kg

Upon substituting these results and the values of the other parameters listed in Table 6.1 into Eqs. (6.40) and (6.41), the frequency-dependent effective mass and stiffness described in Fig. 6.8 can be estimated using a retrieval method (Zhu et al. 2012; Lee et al. 2016). The dispersion curves for longitudinal waves propagating along the xaxis in the metamaterials consisting of the unit cells C m , C k , and C mk are compared in Fig. 6.9. Let us closely investigate the dispersion curves and frequency dependences of the effective mass m eff (ω) and stiffness αeff (ω) using the results in Fig. 6.9. In the case of the metamaterial composed of C m , only the effective mass can be negative at some frequencies near f x (= ωx /2π )—single negativity. On the other hand, the single negativity in the effective stiffness can be possible at some frequencies near f y (= ω y /2π ) for the metamaterial composed of C k . For the single-negative frequency zones, stopbands can be formed, which can be confirmed by the dispersion curves (Langlet et al. 1995); see the dispersion curves in Fig. 6.9. The figure also shows that besides these stopbands resulting from single-negative effective material property, additional stopband zones are formed in the frequency range for which either m eff (ω) is considerably large (positive) or αeff (ω) is extremely small (positive). These bandgaps, which are the Bragg-type bandgaps, are due to the periodicity of the system, not from the negative effective material properties. As evident from

120

6 Metamaterials: Effective Property Realization

Fig. 6.8 Effective density and stiffness for the metamaterials made of unit cells a C m , b C k , and c C mk . Solid lines: obtained by using the discrete mass–spring models using effective material properties. Solid circles: obtained by the retrieval method by Lee et al. (2016) using the detailed finite element model. The effective density is defined as the effective mass divided by d 2 h with d (lattice constant) = 20 mm and h (plate thickness) = 1 mm. (Oh et al. 2016)

Fig. 6.9c, a wave passing zone is formed where the effective mass and stiffness are simultaneously negative. In this zone, the signs of the group and phase velocities are different. The branch with negative group velocities and positive phase velocities appearing in Fig. 6.9c describes the waves propagating along the negative x-axis. Therefore, the branch corresponding to −π ≤ kd ≤ 0 should be used to describe the waves propagating along the positive x-axis; in this branch, the group velocities

6.4 Metamaterial with Frequency-Dependent Effective Mass and Stiffness

121

Fig. 6.9 a–c Dispersion curves, effective mass, and stiffness for longitudinal waves in metamaterials comprising unit cells Cm , Ck , and Cmk , respectively. d Mode shapes of unit cells. 0 = α (ω) = α.) (Oh et al. 2016) (m 0eff = m eff (ω) = m 1 + 2m 2 and αeff eff

are positive and the phase velocities are negative. Because f x and f y can be independently tuned in the developed metamaterial, it can be tailored to make one or both of its effective material properties (mass and stiffness) negative at the target range of frequencies. The deformation patterns of C m and C k around or near f x and f y are illustrated in Fig. 6.9d, which demonstrate the realization mechanisms of the negative density and stiffness. In case of C m , m 1 (fundamental mass elements at four corners of the unit cell) connects one unit cell to adjacent unit cells and m 3 is the mass component inducing f x , the resonant frequency. Note that when mass m 1 travels rightward, m 3 travels leftward. Thus, the total momentum of C m can be negative for a positive velocity of the unit cell because of the 180◦ out-of-phase motion of m 3 relative to the motion of the fundamental mass element m 1 . Therefore, the effective density can be negative. In contrast, the mode shape of the unit cell C k sketched at a frequency immediately under the resonance frequency f y exhibits a different behavior from that of the unit cell C m . Due to the large up- and down-ward y-directional motions of m 2 under a force (motion) at the left-hand side of C k , the right-hand side of C k moves leftward. This means that the right-hand side of C k moves in the opposite direction to the force (motion) at the left-hand side. Therefore, the effective stiffness can become negative. The appearance of negative effective stiffness cannot be possible without the presence of the x–y coupling stiffness γ . Finally, the metamaterials comprising C mk , which is the combination of C m and C k , exhibit the combined effects of the

122

6 Metamaterials: Effective Property Realization

Fig. 6.10 a Finite element simulation model of the metamaterial composed of C mk and measured displacements at the frequency of b 15 kHz (positive phase velocity), c 25 kHz (negative phase velocity), and d 35 kHz (positive phase velocity). (The S0 wave corresponds appropriately to the longitudinal wave.) (Oh et al. 2016)

two independent metamaterials composed of C m and C k , as clearly exhibited in Fig. 6.9d. To examine the propagation of waves in the metamaterial composed of Cmk at various frequencies, including the frequency (25 kHz) responsible for altered signs of the group and phase velocities, the numerical simulations performed with the finite element analysis are presented in Fig. 6.10. In Fig. 6.10a, the finite element model is used for numerical investigation of the phase velocities inside the metamaterial plate. The metamaterial layer comprising 4 Cmk unit cells is placed inside an aluminum plate. For the simulations, the S05 sinusoidal waves centered at 15, 25, and 35 kHz are incident from the left aluminum plate, and the x-directional displacements u j ( j = A, B, and C) were measured, where A, B, and C denote points located inside the metamaterial layer. In Fig. 6.9c, the metamaterial composed of Cmk should exhibit positive phase velocities at 15 and 35 kHz, but a negative phase velocity at 25 kHz (because the waves propagating along the positive x-axis are used for simulation). Although the dispersion curve in Fig. 6.9c is plotted for positive kd’s, its mirror image with respect to the ω-axis, i.e., the dispersion curve for negative kd’s, must be considered to explain the waves propagating toward the positive x-axis. This is because the group velocity in that branch (negative kd’s) is positive, whereas its phase velocity is negative. The displacements u A , u B , and u C at 15, 25, and 35 kHz are plotted in Fig. 6.10b–d, respectively, which are predicted by the finite element simulations. At 15 and 35 kHz, the crests move forward along the positive x-direction as the wave propagates. In 5

The S0 mode represents the lowest symmetric wave mode in a plate, which corresponds well to the longitudinal wave mode in a one-dimensional bar at low frequencies.

6.4 Metamaterial with Frequency-Dependent Effective Mass and Stiffness

123

contrast, the crests move backward along the negative x-direction at 25 kHz, as the wave propagates along the positive x-direction (the group velocity acts along the positive x-axis). This confirms that the phase velocity at 25 kHz is negative, and equivalently, the signs of the group and phase velocities are different. The phase velocities measured from the numerical simulations are 1739, –2500, and 3333 m/s, whereas the phase velocities evaluated from the dispersion curves are 1729, −2353, and 3294 m/s at 15, 25, and 35 kHz, respectively. The numerical simulations effectively demonstrate the formation of the negative phase velocity at 25 kHz. Experimental validation is presented below. The schematic illustration of the experimental setup performed on a thin aluminum plate is presented in Fig. 6.11. The modulated Gaussian pulse or Gabor pulse, in the form of exp(− f 2 t 2 /2G 2s ) cos(2π f t), was used as an actuation signal where f is the center frequency of the pulse (15, 25, and 35 kHz). The parameter G S adjusting the degree of pulse spread in time was set as 2.75. As a specific case, the excitation signals and those measured from the experiment performed with the metamaterial containing Cmk at the actuation frequency of 35 kHz are presented in Fig. 6.12a, b. The post-processed result of the first arrival pulse displayed in Fig. 6.12b by the STFT (short-time Fourier transform; e.g., Mallat (1999)) is presented in Fig. 6.12d, whereas the window function used for the STFT is plotted in Fig. 6.12c. The details of the STFT are not provided here, but in Fig. 6.12d, the horizontal axis corresponds to the arrival time ta , describing the traveling duration from the actuator to the sensor. Accordingly, it is found that the pulse centered at 35 kHz arrives at ta = 0.2 ms, as inferred from Fig. 6.12d. The arrival time can be theoretically calculated as ta =

meta d al 4dunit 0.1 0.08 + = al + meta , vgal vgmeta vg vg

(6.42)

where d al denotes the sum of the distances from the actuating and receiving transmeta represents the size of the unit cell of ducers to the metamaterial boundaries and dunit

Fig. 6.11 a Schematic illustration of the experiment set up in a 2.0 m × 1.2 m aluminum plate of 1 mm in thickness and b image of the piezoelectric patch transducer used as the actuator and sensor. (Oh et al. 2016)

124

6 Metamaterials: Effective Property Realization

Fig. 6.12 Experimental signals obtained with the metamaterial comprising C mk at the excitation frequency of f = 35 kHz. a Actuated signal, b measured signal, c Gaussian window function used for the short-time Fourier transform (STFT), and d STFT of the first arrival pulse in b. (Oh et al. 2016) meta the metamaterial in the x-direction. The distance of 4 dunit is used, because four metamaterial unit cells were used in the experiment. For evaluating ta through Eq. (6.42), we use the following values of the group velocities of the S0 wave modes:

  vgal  S0 ≈ 5200 m/s and vgmeta  S0 ≈ 2140 m/s (in the frequency ranges of experiment), where vgal and vgmeta denote the group velocities of the base plate and the metamaterial plate, respectively. The post-processed experimental results obtained from the STFT for the metamaterials composed of three distinct unit cells Cm , Ck , and Cmk at three center frequencies of 15, 25, and 35 kHz are presented in Fig. 6.13. As observed, the waves excited at the frequencies of 15 and 35 kHz belonging to the passbands were transmitted through all the metamaterials. At 25 kHz, wave transmission behaviors varied for the metamaterials of Cm , Ck , and Cmk . As portrayed in Fig. 6.13a, b, the Cm and Ck metamaterials do not allow wave transmission at 25 kHz because single negativity

6.4 Metamaterial with Frequency-Dependent Effective Mass and Stiffness

125

in the effective mass and stiffness, respectively; only the effective density is negative for Cm , and only the effective stiffness is negative for Ck at 25 kHz. (Refer to Figs. (6.8)) a, b or 6.9a, b.) In contrast, the experimental result in Fig. 6.13c shows that the wave excited at 25 kHz is transmitted through the Cmk metamaterial because the excitation frequency lies in the frequency band where both the effective mass and stiffness are negative. The analytically calculated arrival period ta is consistent with the experimental result. The second arrival pulse component of 25 kHz appears at 1.5 ms because of the internal reflections within the metamaterials, which cannot be predicted by the analytical analysis. The experimentally measured x-directional displacements u A , u B , and u C near 25 kHz inside the Cmk metamaterial are presented in Fig. 6.14b. For the measurement using a laser vibrometer, a thin highly reflective rectangular film is vertically installed at the measurement locations, as depicted in Fig. 6.14a. As observed from Fig. 6.14b, the wave peaks traveled backward, implying the negative phase velocity in the metamaterial composed of C mk . This result is favorably compared with the numerical result provided in Fig. 6.10c. Furthermore, the experimentally measured displacement fields agree well with those obtained with the numerical simulation presented in Fig. 6.10. The difference in magnitude between the displacements from the simulation and the experiment is caused by the difficulty of installing the thin film in the exact vertical orientation. Nevertheless, the experimental measurements revealed that the phase velocity in the metamaterial at 25 kHz associated with the negative-slope branch (having positive group velocity and negative phase velocity) was negative. The experimentally estimated dispersion curve in the passing band near 25 kHz was favorably compared in Fig. 6.14c with the numerically calculated dispersion curve. (The statistic errors indicated by the error bar in Fig. 6.14c were primarily caused by the fabrication imperfection in the metamaterial geometry.)

Fig. 6.13 Experimental results performed at 15, 25, and 35 kHz for metamaterials comprising unit cells: a C m , b C k , and c C mk as the function of the arrival time ta . The results are compared with analytically calculated dispersion curves plotted as the function of the dimensionless wave number (Bloch phase) kd. (The arrival time denotes the traveling duration from the actuator to the sensor in the setup shown in Fig. 6.11a.) (Oh et al. 2016)

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6 Metamaterials: Effective Property Realization

Fig. 6.14 a Image of the aluminum plate with a metamaterial embedded. Experiments were performed to measure the phase velocity inside C mk metamaterial at 26 kHz. b Experimentally measured x-displacements (uA , uB , and uC ) by a laser vibrometer. c Dispersion curves by analytical, numerical, and experimental predictions. (Oh et al. 2016)

References Achenbach JD (1976) Wave propagation in elastic solids. North-Holland Langlet P, Hladky-Hennion A-C, Decarpigny J-N (1995) Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method. J Acoust Soc Am 98:2792–2800 Lee HJ, Lee HS, Ma PS, Kim YY (2016) Effective material parameter retrieval of anisotropic elastic metamaterials with inherent nonlocality. J Appl Phys 120:104902 Mallat S (1999) A wavelet tour of signal processing. 2nd ed, Academic Miklowitz J (1978) The theory of elastic waves and waveguides. North-Holland Oh JH, Kwon YE, Lee HJ, Kim YY (2016) Elastic metamaterials for independent realization of negativity in density and stiffness. Sci Rep 6:23630 Oh JH, Seoung HM, Kim YY (2014) A truly hyperbolic elastic metamaterial lens. Appl Phys Lett 104:073503 Rose JL (2014) Ultrasonic guided waves in solid media. Cambridge University Press Wang YF, Wang YS (2013) Complete bandgap in three-dimensional holey phononic crystals with resonators. J Vib Acoust 135:041009 Wang YF, Wang YS, Su XX (2011) Large bandgaps of two-dimensional phononic crystals with cross-like holes. J Appl Phys 110:113520 Zhu R, Huang GL, Hu GK (2012) Effective dynamic properties and multi-resonant design of acoustic metamaterials. J Vib Acoust 134:031006

Chapter 7

Longitudinal Waves in 1D Continuum Bars

This chapter examines longitudinal waves in one-dimensional (1D) continuum bodies like an elastic bar. As depicted in Fig. 7.1, the longitudinal wave motion is described by the axial displacement u x (x, t) of an elastic bar (density ρ, cross-sectional area S, Young’s modulus E). To facilitate 1D analysis, u x (x, t) is assumed uniform over the bar cross-section.

7.1 Governing Wave Equation To derive the governing wave equation, we consider the momentum balance for an infinitesimal element of length x depicted in Fig. 7.2. The axial stress σx is assumed to act on the negative side of the element, and σx + σx is assumed to act on the positive side, as indicated in Fig. 7.2. For the infinitesimal element, the consideration of the equation of motion yields:

ρ Sx

∂ 2u x ≈ (σx + σx )S − σx S. ∂t 2

As x approaches d x, the above equation reduces to ∂ 2u x ∂σx =ρ 2 . ∂x ∂t

(7.1)

Using the constitutive relation between stress σx (x, t) and strain εx (x, t), σx = Eεx , where the strain can be defined as © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_7

127

128

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.1 Elastic bar carrying longitudinal waves described by axial displacement u x

Fig. 7.2 Forces acting on an infinitesimal element x

εx =

∂u x , ∂x

one can write the relationship between σx and u x as σx = Eεx = E

∂u x . ∂x

(7.2)

The combination of Eqs. (7.1) and (7.2) yields ∂ 2u x 1 ∂ 2u x = , 2 ∂x E/ρ ∂t 2 i.e., ∂ 2u x 1 ∂ 2u x = . ∂x2 c2 ∂t 2

(7.3)

In Eq. (7.3), the (1D) longitudinal wave speed c is defined as  c=

E . ρ

(7.4)

The values of the longitudinal wave speed for some materials are listed in Table 7.1. Equation (7.3) is the (1D) wave equation describing longitudinal waves propagating in an elastic bar. Instead of using the 2nd-order partial differential equation, Eq. (7.3), a system of two first-order differential equations can be used:

7.1 Governing Wave Equation

129

Table 7.1 Longitudinal wave speed (The material properties of epoxy and silicone rubber vary widely, so particular values are used) Material ρ

(kg/m3 )

Steel (low alloy)

Titanium

Silicon

Silicone rubber

Epoxy

7800

4500

2300

2000

3000

E (GPa)

200

110

112

0.5

5.0

c (m/s)

5063

4944

6978

500

1290

∂ ∂x



σx vx



 =

   0 ρ ∂ σx , 1/E 0 ∂t vx

(7.5)

where vx denotes the velocity defined as vx =

∂u x . ∂t

The first equation in Eq. (7.5) represents the momentum balance equation, whereas the second equation in Eq. (7.5) represents the constitutive relation expressed in terms of stress and velocity. It is interesting to see that the governing equation given by Eqs. (7.3) or (7.5) for longitudinal waves also describes other waves, such as transverse waves in a taut string, acoustic waves in a duct, and electromagnetic waves in a transmission line. Therefore, studying longitudinal waves in a bar helps understand wave phenomena in other similar systems. Below, we briefly introduce other wave systems that can be described by the governing equation of the same form as Eqs. (7.3) or (7.5). • Transverse Waves in Taut String For small transverse vibrations of a taut string of linear density ρ L 1 subjected to a tensile force T , the following equations can be derived (e.g., Elmore and Heald 1969) ∂ 2u y 1 ∂ 2u y = with c = ∂x2 c2 ∂t 2

 T ρL

(7.6)

or ∂ ∂x



Fy vy



   0 ρ L ∂ Fy = , 1/T 0 ∂t v y 

(7.7)

where u y (x, t) and v y (x, t) denote the transverse displacement and velocity along the y-direction, respectively, and Fy represents the force acting in the transverse (y) direction. • Acoustic Waves in Duct 1

It has the dimension of mass/length, not mass/volume.

130

7 Longitudinal Waves in 1D Continuum Bars

The acoustic waves propagating in a duct can also be described using a 1D wave equation. In this case, we use acoustic pressure p instead of stress σ with p = −σ and keep the velocity vx . Young’s modulus E is replaced by bulk modulus B. Consequently, the wave equation for the acoustic waves becomes (see, e.g., Kinsler et al. 2000)  ∂2 p B 1 ∂2 p , = 2 2 with c = ∂x2 c ∂t ρ       ∂ p 0 −ρ ∂ p = . −1/B 0 ∂ x vx ∂t vx

(7.8)

(7.9)

• Electromagnetic Waves in Transmission Line The electromagnetic wave propagating in a transmission line, which is a pair of electrical conductors, can be described by the same form of the longitudinal wave equation. If the inductance and capacitance per unit length are denoted by L  and C  , respectively, the wave equation is given as (see, e.g., Elmore and Heald 1969): ∂2V 1 ∂2V = with c = ∂x2 c2 ∂t 2



1 , L C 

(7.10)

or ∂ ∂x

  V I

   0 −L  ∂ V = . −C  0 ∂t I 

(7.11)

where V and I are the voltage and current on a transmission line.

7.2 General Solution of 1D Wave Equation The general solution to Eq. (7.3) may be obtained by assuming u x (x, t) as u x (x, t) = X (x)T (t).

(7.12)

Substituting Eq. (7.12) into Eq. (7.3) yields X  (x)T (t) = ··

1 X (x)T¨ (t), c2

(7.13)

where ( ) = ∂ 2 ( )/∂ x 2 and ( ) = ∂ 2 ( )/∂t 2 . Thus, Eq. (7.13) can be rearranged as

7.2 General Solution of 1D Wave Equation

c2

X  (x) T¨ (t) = ≡ −ω2 . X (x) T (t)

131

(7.14)

Because c2 X  (x)/ X (x), a function of x, should be equal to T¨ (t)/T (t), a function of t, they can be equated only when they become a scalar constant, say, −ω2 . If Eq. (7.14) is split into two equations, we obtain X  (x) + k 2 X (x) = 0,

(7.15)

T¨ (x) + ω2 T (x) = 0,

(7.16)

where k, called the wavenumber, is related to ω as ω2 = k 2 c2 ,

(7.17a)

ω = kc or ω = −kc.

(7.17b)

or

As ω = −kc can be obtained with k replaced by −k in the relation ω = kc, we primarily consider the ω = kc branch. The general solutions to Eqs. (7.15) and (7.16) can be written as X ∼ [sin kx, cos kx] or [eikx , e−ikx ],

(7.18a)

T ∼ [sin ωt, cos ωt] or [eiωt , e−iωt ].

(7.18b)

In Eq. (7.18b), the function T (t) varies harmonically with time at the angular frequency of ω. Therefore, for a given ω (ω = kc), u x (x, t) can be expressed as u x (x, t) = C1 cos kx cos ωt + C2 cos kx sin ωt + C3 sin kx cos ωt + C4 sin kx sin ωt,

(7.19)

where Ci (i = 1, 2, 3, 4) are real-valued constants. Alternatively, it can be expressed in complex form as u x (x, t) = Aei(ωt−kx) + Bei(ωt+kx) ,

(7.20)

where A and B are complex-valued constants. As explained in Chap. 3, ei(ωt−kx) and ei(ωt+kx) denote harmonic waves propagating along the +x- and −x-axis at the phase velocity of c and −c, respectively. Therefore, Eq. (7.20) is more convenient to analyze problems dealing with wave propagation. On the other hand, Eq. (7.19) is

132

7 Longitudinal Waves in 1D Continuum Bars

more convenient to analyze vibration problems as it uses standing vibration modes (such as cos kx and sin kx). If a wave signal contains multiple frequencies, Eq. (7.20) should be added over all frequencies of interest as u x (x, t) =

ωN

 A(ω)ei(ωt−kx) + B(ω)ei(ωt+kx) ,

(7.21)

ω=ω1

assuming that the signal involves a finite number of discrete frequencies, ω1 , ω2 , . . ., and ω N . Alternatively, Eq. (7.21) can be viewed as a Fourier series solution in ω. For an arbitrary signal, Eq. (7.21) should be interpreted as a Fourier integral (Sneddon 1995) as

∞ u x (x, t) =

 A(ω)ei(ωt−kx) + B(ω)ei(ωt+kx) dω.

(7.22)

−∞

Instead of expressing the general solution as a sum of harmonic functions expressed through Eqs. (7.21) or (7.22), the general solution to Eq. (7.3) can be expressed using another approach. To this end, we introduce two variables ξ and η such that ξ = x − ct, η = x + ct.

(7.23)

If ξ and η are used as the independent variables, the partial derivatives in x and t can be converted as ∂ξ ∂ ∂η ∂ ∂ ∂ ∂ = + = + , ∂ x ∂ x ∂ξ ∂ x ∂η ∂ξ ∂η    ∂u x ∂ 2u x ∂ 2u x ∂ ∂ 2u x ∂ ∂u x ∂ 2u x = = + 2 , + + + ∂x2 ∂ξ ∂η ∂ξ ∂η ∂ξ 2 ∂ξ ∂η ∂η2 and   1 ∂ξ ∂ ∂η ∂ ∂ ∂ 1 ∂ = + =− + , c ∂t c ∂t ∂ξ ∂t ∂η ∂ξ ∂η    1 ∂ 2u x ∂ ∂u x ∂u x ∂ 2u x ∂ 2u x ∂ ∂ 2u x + − + = + = − − 2 . c2 ∂t 2 ∂ξ ∂η ∂ξ ∂η ∂ξ 2 ∂ξ ∂η ∂η2 Using the second and fourth expressions derived above, Eq. (7.3) becomes ∂ 2 u x (ξ, η) = 0. ∂ξ ∂η

(7.24)

7.3 Characteristic Impedance

133

Fig. 7.3 Free body diagram of a 1D bar carrying a longitudinal harmonic wave. The bar is split into two components

Upon integrating Eq. (7.24) with respect to ξ and η, the general solution can be determined in the following form: u x = F(ξ ) + G(η) =F(x − ct) + G(x + ct),

(7.25)

where F and G can be any function. The solution given in Eq. (7.25) is called the D’Alembert solution. Note that the harmonic solution described in Eq. (7.20) is a special case of the general form given by Eq. (7.25). As we will primarily consider harmonic motions in this book, readers interested in the D’Alembert solution may find more related discussions elsewhere (e.g., Knobel 2000).

7.3 Characteristic Impedance Consider a bar carrying a harmonic wave propagating along the positive x-direction and examine a free body diagram of the bar split into two components, as depicted in Fig. 7.3. Referring to Fig. 7.3, the force2 f e exerted by the body of interest (the left component of the bar in this case) onto the adjacent body (the right component of the bar in this case) is expressed as f e = (−σx )S = −E

∂u x S. ∂x

(7.26)

We assume harmonic waves propagating along the x-axis as u x = Uei(ωt−kx) , vx = Vei(ωt−kx) , and f e = Fe ei(ωt−kx) ,

2

(7.27)

We use symbol f e to denote an external force applied to the system of interest. On the other hand, we use the symbol f to denote the internal force f = Sσ x .

134

7 Longitudinal Waves in 1D Continuum Bars

where U, V, and Fe are complex-valued amplitudes of the corresponding field variables u x , vx , f e . The amplitudes V and Fe are related to U as V = (iω)U,

(7.28a)

ω Fe = −E(−ik)SU = i E S U. c

(7.28b)

To obtain the last expression in Eqs. (7.28b), and (7.17b) was used. If the relation between Fe and V is expressed as Fe = ZV,

(7.29)

where Z is called the characteristic impedance. Using Eq. (7.28a, b), the characteristic impedance is given by Z=

sρc2 SE = = ρcS. c c

(7.30)

The relation E = ρc2 in Eq. (7.4) was used in Eq. (7.30). In the present wave problem, Z is frequency-independent. It is also real-valued because the wave system has no damping. Therefore, symbol Z (representing a real-valued quantity) is used instead of Z in the remainder of this chapter. The characteristic impedance physically represents the magnitude of force required to induce a unit velocity in a bar.3 Therefore, this parameter is an essential quantity for analyzing the wave phenomena. As discussed in the subsequent section, only some of the power (energy) of a traveling wave can be transmitted across the interface of the two bars when they have different characteristic impedances. Accordingly, the concept of impedance cannot be overemphasized in analyzing wave phenomena.

7.4 Reflection and Transmission We will consider two semi-infinite bars connected at x = 0, as shown in Fig. 7.4. They are made of different material properties (density ρ j and wave speed c j ) and 3

In the case of static problems, the applied force is related to the displacement through ‘stiffness’ for any given system. In the case of dynamic problems, including wave propagation, the applied force can also be related to displacement. However, it is more convenient to relate the applied force to the velocity; power (the product of force and velocity) is the primary quantity of interest needed in considering energy balance in dynamic problems. On the other hand, work (or energy itself) is the primary quantity of interest in considering energy balance in static problems. (In fact, power cannot be defined in static problems.) Refer to Sect. 7.5 to see the effects of impedance Z on the energy balance.

7.4 Reflection and Transmission

135

Fig. 7.4 One-dimensional system comprising two semi-infinite bars of different characteristic impedances

cross-sectional areas (S j ) ( j = 1, 2). Thereby, the characteristic impedance Z 1 of Bar 1 is generally different from the characteristic impedance Z 2 of Bar 2. Now, we consider the incidence of a time-harmonic wave from Bar 1 to Bar 2 where the displacement u i of the incident wave4 is assumed as incident wave: u i (x, t) = A1 ei(ωt−k1 x) .

(7.31)

In Eq. (7.31), A1 denotes the amplitude of the incident wave and k1 is the wavenumber in Bar 1, which is related to frequency ω as ω = k1 c1 (In Bar 2, ω = k2 c2 ). In case a longitudinal wave is incident from Bar 1 onto the interface of the two bars, two additional waves can be generated: a reflected wave u r (x, t) propagating along the negative x-axis in Bar 1 and a transmitted wave u t (x, t) along the positive x-axis in Bar 2: reflected wave: u r (x, t) = B1 ei(ωt+k1 x) ,

(7.32)

transmitted wave: u t (x, t) = A2 ei(ωt−k2 x) .

(7.33)

The force and displacement continuity conditions at the junction x = 0 of Bars 1 and 2 can be expressed as u i (0, t) + u r (0, t) = u t (0, t),

(7.34)

f (0, t)Bar 1 = f (0, t)Bar 2 .

(7.35)

The substitution of Eqs. (7.31)–(7.33) into Eq. (7.34) yields A1 + B1 = A2 .

(7.36)

We explicitly express the forces appearing in Eq. (7.35) as

4

The subscript x used to represent the longitudinal displacement u x will be omitted to simplify the notation. Instead, we will subscript i (r, or t) to indicate an incident (reflected or transmitted) wave.

136

7 Longitudinal Waves in 1D Continuum Bars

∂ [u i (x, t) + u r (x, t)]x=0 ∂x = S1 E 1 (−k1 A1 + k1 B1 )eiωt , (7.37a)

f (0, t)Bar 1 = S1 σx (0, t)|Bar 1 = S1 E 1

f (0, t)Bar 2 = S2 σx (0, t)|Bar 2 = S2 E 2

 ∂u t (x, t)  ∂ x x=0

= S2 E 2 (−k2 )A2 eiωt

(7.37b)

By substituting Eq. (7.37) into Eq. (7.35), we can obtain S1 E 1 (−k1 A1 + k1 B1 ) = −S2 E 2 k2 A2 .

(7.38)

If Eq. (7.17b) is used to replace k j with k j = ω/c j ( j = 1, 2) and Eq. (7.4) is used to replace E j with E j = ρ j c2j ( j = 1, 2), Eq. (7.38) becomes ρ1 c1 S1 (−A1 + B1 ) = −ρ2 c2 S2 A2 .

(7.39)

Using the definition of the characteristic impedance Z in Eq. (7.30) (Z j = ρ j c j S j , j = 1, 2), Eq. (7.39) can be expressed as Z 1 (−A1 + B1 ) = −Z 2 A2 .

(7.40)

Solving Eqs. (7.36) and (7.40) for B1 and A2 yields r

B1 Z1 − Z2 = , A1 Z1 + Z2

(7.41)

t

A2 2Z 1 = , A1 Z1 + Z2

(7.42)

where symbols5 r and t denote the reflection and transmission coefficients, respectively. If the amplitude A1 is given, A2 and B1 can be directly calculated from Eqs. (7.41) and (7.42). For general wave systems, r and t can be complex-valued and frequencydependent, but for the present longitudinal wave system, r and t are real-valued and frequency-independent. In addition, r = 0 and t = 1 (full transmission without any reflection) as long as the two bars have the same impedance, i.e., Z1 = Z2

5

(7.43)

Note that the same symbol, t, is used to represents time and the transmission coefficient, as there is no danger of confusion.

7.4 Reflection and Transmission

137

Equation (7.43) and the definition of the impedance Z in Eq. (7.39) suggest that if the impedances of two bars comprising different materials can be exactly matched, 100% transmission can be possible from one bar to another. As the limiting cases, we consider. (i) Z 2 = ∞ (simulating fixed end for Bar 1 at x = 0), r = −1 (B1 = −A1 ) and t = 0 (A2 = 0),

(7.44)

(ii) Z 2 = 0 (simulating free end for Bar 1 at x = 0), r = 1 (B1 = A1 ) and t = 2 (A2 = 2A1 ).

(7.45)

For the case of Z 2 = ∞, the incident wave is completely reflected with 180◦ outof-phase. In contrast, for the case of Z 2 = 0 simulating the traction-free boundary condition (i.e., σx = 0) at x = 0 for Bar 1, the incident wave is completely reflected without any phase variation. In two limiting cases considered above, the magnitude of the reflected wave (|B1 |) is equal to that of the incident wave (|A1 |). This result implies that the energy of the incident wave is completely transferred to the energy of the reflected wave. (Refer to the subsequent section for more detailed energy analysis.) Conversely, the result of A2 = 2A1 appears to be against the physical law as additional energy may be stored in Bar 2 by the transmitted wave of the amplitude A2 = 2A1 , although Bar 2 is truly nonexistent. However, the fact that A2 = 2A1 does not violate the physical law of energy balance, because the transmitted wave possesses no energy because Z 2 = 0 (refer to Problem 7–2). This can be simply viewed as an artifact by solving a single-bar problem based on a two-bar system. As the above analysis is concerned with a single harmonic wave, the Fourier integral or series should be used to deal with nonharmonic waves. Alternatively, the D’Alembert solution in Eq. (7.25) is convenient to solve the reflection of a pulsetype wave. For instance, let us consider an incident wave pulse in a semi-infinite bar (−∞ < x ≤ 0) from x = −∞ toward its fixed end (x = 0). The displacement distribution at certain instant t1 is shown on the left-hand side of Fig. 7.5a. It is denoted by F(x − ct1 ) and cannot be expressed by a single harmonic wave. Guided by the result in Eq. (7.44), we consider an incident wave from x = +∞ toward x = 0, which is the mirror image of “−F(x − ct1 )” with respect to x = 0, as indicated by the dotted line in the right-hand side of Fig. 7.5a. This method is called the method of image. At a later instant t = t2 where the nonzero front end of F(x − ct2 ) impacts the end point, x = 0, the mirror image of the “−F(x − ct2 )” impacts the end point as well, ensuring that the sum of the two solution is zero, i.e., u(x = 0, t2 ) = 0; refer to Fig. 7.5b. At a later instant t = t3 displayed in Fig. 7.5c, the solution field (for −∞ < x ≤ 0) can be graphically sketched using the superposition of the two solutions. At t = t4 and t = t5 (t5 > t4 > t3 ), the displacement distributions are sketched in Fig. 7.5d and e, respectively. The use of the D’Alembert solution may help understand how the reflected wave is generated.

138

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.5 Reflection of a propagating wave rightwards in a semi-infinite bar at its fixed end (x = 0)

7.5 Energy Perspective Assume that a wave propagates through a bar between x = x1 and x = x2 . To derive the energy balance equation, we use the equation of motion (or the momentum balance equation), given by the first equation of Eq. (7.5): ∂(ρvx ) ∂σx = . ∂x ∂t

(7.46)

Multiplying both sides of Eq. (7.46) with vx S and integrating the resulting equation over x between x = x1 and x = x2 yields

x2 x1

∂σx vx Sd x = ∂x

x2 vx x1

∂(ρvx ) Sd x. ∂t

If we write vx (∂σx /∂ x) and vx ∂(ρvx )/∂t appearing in Eq. (7.47) as vx

  ∂ ∂σx ∂ ∂vx ∂ ∂u x = σ σx σ = σ − − (vx x ) (vx x ) x ∂x ∂x ∂x ∂x ∂ x ∂t ∂ ∂εx ∂εx ∂ = = (vx σx ) − σx (vx σx ) − Eεx ∂x ∂t ∂x ∂t

(7.47)

7.5 Energy Perspective

=

139

∂ ∂ (vx σx ) − ∂x ∂t



1 E(εx )2 2



and vx

∂ ∂(ρvx ) = ∂t ∂t



 1 ρ(vx )2 , 2

Equation (7.47) can be simplified as

x2 0=− x1

∂ (vx σx )Sd x + ∂x

x2 x1

∂ ∂t



 

x2  1 ∂ 1 2 2 ρ(vx ) Sd x + E(εx ) Sd x. 2 ∂t 2 x1

(7.48) To interpret the Eq. (7.48), we introduce the instantaneous power P defined as P = fe · v = (−tS) · v = (−tx Sex ) · (vx ex ) = −tx Svx .

(7.49)

In Eq. (7.49), fe denotes the force exerted by the system of interest to its adjacent element, and t denotes the traction vector (unit of force/area) acting on the system of interest (Body B in Fig. 7.6b), and v denotes the particle velocity. The symbol ex denotes the unit vector along the x-axis. Referring to Fig. 7.6b, tx = σx . Therefore, P represents the power transmitted to the adjacent element, i.e., Body R through x = x2 is Px=x2 = (−σx Svx )x=x2 .

Fig. 7.6 a Model used to consider energy balance for a bar segment between x = x1 and x = x2 . b Illustration of force F B→R = [(−σx )Sex ] acted by the bar segment (Body B) onto Body R ( x ≥ x2 )

140

7 Longitudinal Waves in 1D Continuum Bars

In contrast, the power transmitted to Body L at x = x1 is Px=x1 = [(−tx S)vx ]x=x1 = (σx Svx )x=x1 , because tx = −σx . Using the instantaneous power P, the first integral in Eq. (7.48) can be written as

x2 − x1

∂ (vx σx )Sd x = (−σx vx S)at x=x2 + (σx vx S)at x=x1 ∂x =Px=x2 + Px=x1  Ptotal .

(7.50)

Note the integral on the left side of Eq. (7.50) represents the power flowing out from Body B shown in Fig. 7.6b. Body B occupying the bar lying between x = x1 and x = x2 is the system of interest. In contrast, the sum of the second and third integrals can be interpreted as the time variation of the total energy UT stored in the system of interest:

x2 x1

∂ ∂t



 ∂UT 1 1 2 2 ρ(vx ) + E(εx ) Sd x = , 2 2 ∂t

(7.51)

where UT is defined as

x2 UT =

x2 (u k + u s )Sd x =

x1

u t Sd x.

(7.52)

x1

In Eq. (7.52), u k and u s denote the kinetic and strain energy density,6 respectively, and u t denotes the total energy density, where u k and u s can be defined as uk =

1 1 ρ(vx )2 ; u s = E(εx )2 . 2 2

Inserting Eqs. (7.50) and (7.51) into Eq. (7.48) yields, 0 = Ptotal +

∂UT . ∂t

(7.53a)

or Ptotal = −

6

∂UT . ∂t

Energy density is defined as energy stored in unit volume.

(7.53b)

7.5 Energy Perspective

141

Fig. 7.7 a Analysis model to investigate power balance at the junction of two dissimilar bars. A1 and B1 : the amplitudes of incident and reflected harmonic waves in Bar 1, respectively; A2 : amplitude of transmitted harmonic wave in Bar 2. b Illustration of an infinitesimal region of size

at location x

Equation (7.53b) implies that the power (Ptotal ) flowing out of the system is equal to the reduction rate of the total energy stored in the system (−∂UT /∂t). Note that all the above quantities, such as velocity, force, and energy, are real-valued. Now, we utilize Eq. (7.53a) to derive the power balance equation at the junction of two bars composed of dissimilar media illustrated in Fig. 7.7a, where a harmonic wave of the angular frequency ω is assumed to be incident from Bar 1 onto the interface of Bar 1 and Bar 2. Therefore, both the reflected wave in Bar 1 and the transmitted wave in Bar 2 exist. Referring to Fig. 7.7b, we select an infinitesimal region between x − and x + surrounding the interface located at x as the system of interest and apply Eq. (7.53a) to obtain 

 ∂UT + Pout = 0, ∂t

(7.54)

where the symbol · denotes the time average over a period T defined as 1 · = T

T o

  2π for a harmoinc wave . (·)dt T = ω

In evaluating the time average of the product of two field quantities G and F, the following relation is used (refer to Eq. 3.36). (ReG(t))(ReF(t)) T =

 1  Re G(t)F(t)∗ , 2

(7.55)

where G and F are assumed to vary harmonically in time and (·)∗ denotes the complex conjugate. Considering the limit of → 0 in Fig. 7.7b, Eq. (7.54) reduces to Pout = 0, i.e., Px=x + + Px=x − = 0.

(7.56)

To calculate Px=x + , we consider the transmitted wave (u t ) expressed in Eq. (7.33). The corresponding velocity vt and stress (σx )t at x = x + are expressed as

142

7 Longitudinal Waves in 1D Continuum Bars

vt = (iω)A2 ei(ωt−k2 t) , (σx )t = E 2

∂u t = E 2 A2 (−ik2 )ei(ωt−k2 t) . ∂x

(7.57) (7.58)

Therefore, 1  Px=x + = Px=x + A2 =−Re(vt )Re((σx )t )S2 = − Re vt (σx )∗t S2 2  1  1 ∗ = − Re (iωA2 )[E 2 A2 (−ik2 )∗ ] S2 = ωk2 E 2 |A2 |2 S2 2 2 1 2 1 2 2 2 = ω ρ2 c2 S2 |A2 | = ω Z 2 |A2 | , (7.59) 2 2 where the relations Z j = ρ j c j S j , k j = ω/c j , and E j = ρ j c2j were used. The power term Px=x − can be expressed as Px=x − = Px=x − A1 + Px=x − B1 , where A1 corresponds to the incident wave in Bar 1 propagating along the + x-axis and B1 corresponds to the reflected wave in Bar 1 propagating along the –x-axis. To calculate Px=x − A1 , we use the incident wave u i (x, t) in Eq. (7.31) to express vi and (σx )i in terms of A1 : vi = (iω)A1 ei(ωt−k1 t) , (σx )i = E 1

∂u i = E 1 A1 (−ik1 )ei(ωt−k1 t) . ∂x

(7.60)

Using Eq. (7.60), we can express Px=x − A1 as 1  Re vi (σx )i∗ S1 2  1  1 = Re (iωA1 )[E 1 A∗1 (−ik1 )∗ ] S1 = − ωk1 E 1 |A1 |2 S1 2 2 1 1 = − ω2 ρ1 c1 S1 |A1 |2 = − ω2 Z 1 |A1 |2 . 2 2

Px=x − A1 = Re(vi )Re((σx )i )S1 =

(7.61)

To calculate Px=x − B1 , we use the reflected wave (u r ) in Eq. (7.32). In this case, the corresponding velocity vr and stress (σx )r at x = x − are vr = (iω)B1 ei(ωt+k1 t) , (σx )r = E 1 Therefore,

∂u r = E 1 B1 (ik1 )ei(ωt+k1 t) . ∂x

(7.62)

7.6 Impedance Matching

143

1  Re vt · (σx )∗t S1 2  1  1 ∗ = Re (iωB1 )[E 1 B1 (ik1 )∗ ] S1 = ωk1 E 1 |B1 |2 S1 2 2 1 1 = ω2 ρ1 c1 S1 |B1 |2 = ω2 Z 1 |B1 |2 . 2 2

Px=x − B1 = Re(vt )Re((σx )t )S1 =

(7.63)

The substitution of Px=x + in Eq. (7.59), Px=x − A1 in Eq. (7.61), and Px=x − B1 in Eq. (7.63) into Eq. (7.56) yields the power balance equation at the junction of two dissimilar bars as: 1 2 1 1 ω Z 1 |A1 |2 = ω2 Z 2 |A2 |2 + ω2 Z 1 |B1 |2 . 2 2 2

(7.64)

The left-side term in Eq. (7.64) represents the incoming power flow toward the junction, and the two right-side terms represent the outgoing power flow farther from the junction. Note that three field equations are applicable at the junctions of two dissimilar bars: Eqs. (7.36), (7.40), and (7.64). However, only two equations are independent. Refer to Problem 7–5 to demonstrate that Eq. (7.64) can be derived using Eqs. (7.36) and (7.40).

7.6 Impedance Matching In the previous section, we learned that no full-wave transmission is possible between bars (say, Bar 1 and Bar 3) composed of different characteristic impedances Z 1 and Z 3 (Z 1 = Z 3 ). To achieve full transmission between dissimilar bars, we consider inserting a finite-sized bar of length d and characteristic impedance Z 2 , as depicted in Fig. 7.8. Our objective is to find d and Z 2 that enable complete transmission from Bar 1 to Bar 3. To derive the impedance-matching condition, we assume that a wave is incident from Bar 1 with amplitude A1 , as shown in Fig. 7.8. In this case, wave fields in all bars can be expressed as Bar 1: u 1 (x, t) = A1 ei(ωt−k1 x) + B1 ei(ωt+k1 x)

(7.65a)

Bar 2: u 2 (x, t) = A2 ei(ωt−k2 x) + B2 ei(ωt+k2 x)

(7.65b)

Bar 3: u 3 (x, t) = A3 ei(ωt−k3 x)

(7.65c)

144

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.8 Impedance-matching element (Bar 2) inserted between two dissimilar bars (Bar 1 and Bar 3) for possible full transmission. Bar j is assumed to have the material properties ρ j , E j , c j and cross-sectional area S j ( j = 1, 2, 3). (A1 and B1 : amplitudes of the incident and reflected waves in Bar 1; A2 and B2 : amplitudes of waves propagating rightward and leftward in Bar 2, respectively; A3 : amplitude of the transmitted wave in Bar 3)

For subsequent analysis, we introduce the reflection and transmission coefficients7 defined as r=

B1 A3 and t = . A1 A1

(7.66a,b)

To facilitate the analysis, we set A1 = 1. Then, we can replace B1 by r and A3 by t. In addition, we will write A2 and B2 simply as A and B. Using these symbols, Eq. (7.65a, b and c) is written as u 1 (x, t) = 1ei(ωt−k1 x) + rei(ωt+k1 x) ,

(7.67a)

u 2 (x, t) = Aei(ωt−k2 x) + Bei(ωt+k2 x) ,

(7.67b)

u 3 (x, t) = tei(ωt−k3 x) .

(7.67c)

To satisfy the continuity conditions in displacement and force at the interfaces x = 0 and x = d, the following conditions must hold: at x = 0, u 1 (0, t) = u 2 (0, t) → 1 + r = A + B, F1 (0, t) = F2 (0, t) → k1 E 1 S1 (−1 + r) = k2 E 2 S2 (−A + B),

(7.68) (7.69)

at x = d, u 2 (d, t) = u 3 (d, t) → Ae−ik2 d + Beik2 d = te−ik3 d , 7

(7.70)

Unlike in Sec. 7.4 where the reflection and transmission coefficients are assumed to be real-valued, they will be treated to be complex-valued to consider both magnitude and phase change. Therefore, we now use r and t of r and t.

7.6 Impedance Matching

145

F2 (d, t) = F3 (d, t) → k2 E 2 S2 (−Ae−ik2 d + Beik2 d ) = k3 E 3 S3 (−te−ik3 d ), (7.71)  where force F j ( j = 1, 2, 3) can be calculated as F j = E j ∂u j /∂ x S j . Using the relationship that k j E j S j = (ω/c j )(ρ j c2j )S j = ωZ j , Eqs. (7.69) and (7.71) can be rewritten as F1 (0, t) = F2 (0, t) → Z 1 (−1 + r) = Z 2 (−A + B), F2 (d, t) = F3 (d, t) → Z 2 (−Ae−ik2 d + Beik2 d ) = Z 3 (−te−ik3 d ),

(7.72) (7.73)

Using Eqs. (7.72) and (7.73), the continuity conditions are rewritten as at x = 0: displacement continuity: 1 + r = A + B

(7.74)

force continuity: Z 1 (−1 + r) = Z 2 (−A + B)

(7.75)

displacement continuity: Ae−ik2 d + Beik2 d = te−ik3 d

(7.76)

force continuity: Z 2 (−Ae−ik2 d + Beik2 d ) = Z 3 (−te−ik3 d )

(7.77)

at x = d:

To find the values of d and Z 2 for full transmission from Bar 1 to Bar 3, we set r = 0 in Eqs. (7.74) and (7.75) to obtain 1 = A + B,

(7.78)

−Z 1 = Z 2 (−A + B).

(7.79)

Solving Eqs. (7.78) and (7.79) for A and B yields A=

    1 Z1 Z1 1 ,B= . 1+ 1− 2 Z2 2 Z2

(7.80)

Using Euler’s formula, Eqs. (7.76) and (7.77) can be expressed as (A + B) cos k2 d + (B − A)i sin k2 d = te−ik3 d , Z 2 [(−A + B) cos k2 d + (A + B)i sin k2 d] = Z 3 (−te−ik3 d ).

(7.81) (7.82)

We will now examine Eq. (7.81) ×(Z 3 /Z 2 ) + Eq. (7.82) ×(1/Z 2 ) to eliminate t. The resulting equation becomes

146

7 Longitudinal Waves in 1D Continuum Bars

 −A + B +

   Z3 Z3 (A + B) cos k2 d + i sin k2 d (A + B) + (B − A) = 0. Z2 Z2 (7.83)

Considering the real and imaginary parts of Eq. (7.83) yields   Z3 (A + B) cos k2 d = 0, −A + B + Z2   Z3 (A + B) + (B − A) sin k2 d = 0. Z2

(7.84a) (7.84b)

There are two possible cases for which Eqs. (7.84a, b) can be satisfied. Case 1: sin k2 d = 0 (cos k2 d = 0) In this case, Eq. (7.84a) can be satisfied if −A + B +

Z3 (A + B) = 0. Z2

(7.85)

The substitution of Eq. (7.80) into Eq. (7.85) yields Z1 = Z3.

(7.86)

This result violates the current assumption that Z 1 = Z 3 , and thus, no solution exists for Case 1. Let us consider the other case. Case 2:cos k2 d = 0 (sin k2 d = 0) In this case, Eq. (7.84b) can be satisfied if (A + B) +

Z3 (B − A) = 0. Z2

(7.87)

Substituting Eq. (7.80) into Eq. (7.87) yields a nontrivial solution as follows: Z 22 = Z 3 Z 1 or Z 2 =



Z3 Z1.

(7.88)

Equation (7.88) states that the characteristic impedance of Bar 2 should be equal to the geometric mean of the characteristic impedances of Bar 1 and Bar 3. The condition stated in Eq. (7.88) is known as the impedance-matching condition. If cos k2 d = 0 is satisfied, the wavenumber k2 for Bar 2 (the matching element) should be k2 d =

2n − 1 π (n : integer). 2

(7.89)

7.6 Impedance Matching

147

This condition is called the phase-matching condition. If the wavelengthwavenumber relationship, which is λ2 k2 = 2π , is used, Eq. (7.89) can be expressed as an equation relating d and the wavelength λ2 in Bar 2 as d=

2n − 1 λ2 (n = 1, 2, 3, · · ·). 4

(7.90)

Equation (7.90) expresses the phase-matching condition (7.89) more conveniently as the relation between the length d of Bar 2 and the wavelength λ2 in Bar 2. The length d can be directly calculated from Eq. (7.90) if ω and c2 are given because λ2 is related to ω as ω = k2 c2 = 2π c2 /λ2 . Based on the above analysis, complete transmission across dissimilar media requires the insertion of a matching element that simultaneously satisfies. (i) the phase-matching condition: d = (2n − √ 1)λ2 /4 (Eq. 7.90), (ii) the impedance-matching condition: Z 2 = Z 3 Z 1 (Eq. 7.88). As we are most interested in the shortest length of d denoted as dmin , n = 1 is inserted in Eq. (7.90) to obtain dmin =

λ2 . 4

(7.91)

The resulting impedance-matching element is called the quarter-wave impedancematching element. If the two conditions stated in Eqs. (7.88) and (7.90) are satisfied, the amplitudes A and B become       Z1 Z1 1 1 ,B= . (7.92) A= 1+ 1− 2 Z3 2 Z3 The substitution of Eq. (7.89) and Eq. (7.92) into Eq. (7.67b) yields,   1 Z 1  i(ωt−k2 x)  1  i(ωt−k2 x) i(ωt+k2 x) + e +e − ei(ωt+k2 x) u 2 (x, t) = e 2 2 Z3    Z1 sin k2 x eiωt = cosk2 x − i Z3    Z1 πx πx −i = cos sin (7.93) eiωt for n = 1 (d = dmin ). 2d Z3 2d The real and imaginary components of the spatial field of u 2 (x, t) in Eq. (7.93) (i.e., with exclusion of eiωt ) are plotted in Fig. 7.9. Using Eqs. (7.76) and (7.92), the transmission coefficient t for dmin = λ2 /4 (k2 d = π/2) can be determined as

148

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.9 Spatial distribution of u 2 in the impedance-matching element (Bar 2) with dmin = λ2 /4. Real component: cos(π x/2d); imaginary component: √ − Z 1 /Z 3 sin(π x/2d)

    Z 1 t = eik3 d Ae−ik2 d + Beik2 d =eik3 d cosk2 d − i sin k2 d Z3   Z1 Z1 π ik3 d ik3 d = − ie sin = −ie , Z3 2 Z3

(7.94)

where the condition of k2 d = π/2 is used to obtain the last expression in Eq. (7.94). The full-power transmission from Bar 1 to Bar 3 can be verified √ by calculating the power flow in Bars 1 and 3 with A1 = 1 and A3 = t = −ieik3 d Z 1 /Z 3 : 1 1 1 2 ω Z 1 |A1 |2 = ω2 Z 1 (1)2 = ω2 Z 1 , 2 2 2 1 1 Power flow in Bar 3 = ω2 Z 3 |A3 |2 = ω2 Z 3 |t|2 , 2 2  2    Z1 1 Z 1  1 2 1 2  ik3 d = ω2 Z 1 . = ω Z 3 −ie  = ω Z3  2 Z3  2 Z3 2 (7.95) Power flow in Bar 1 =

Equation (7.95) shows that the power flow from Bar 1 is perfectly transmitted to Bar 3, confirming full transmission.

7.7 Fabry–Pérot Resonance Figure 7.10 presents a situation where Bar 2 is inserted in the base bar of medium 1. The medium (medium 2) of Bar 2 is assumed to be different from that of the base bar. When the characteristic impedance of Bar 2 is different from that of the base bar, an incident wave from the base bar (medium 1) to Bar 2 (medium 2) cannot be completely transmitted. However, full transmission can be possible through Bar 2 if the length d of Bar 2 is appropriately selected depending on the frequency ω of an

7.7 Fabry–Pérot Resonance

149

incident harmonic wave. This phenomenon is known as the “Fabry–Pérot resonance” (Pérot and Fabry 1899), and the frequencies at which the resonance occurs are called the Fabry–Pérot resonance frequencies. To investigate the Fabry–Pérot resonance phenomenon, we employ the general solution stated in Eqs. (7.65a–c) with k3 = k1 . To determine the condition for complete transmission (no reflected wave in Bar 1), we can use the analysis given in Sect. 7.6, as given by Case 1 and Case 2. The present case corresponds to Case 1 because Z 1 = Z 3 . sin k2 d = 0 → k2 d = nπ.

(7.96)

In terms of the wavelength λ2 , d=

n λ2 (n = 1, 2, 3, . . .). 2

(7.97)

Equation (7.97) states that the Fabry–Pérot resonance can occur if the length d of the inserted bar is a multiple of a half wavelength corresponding to the excitation frequency ω. Using Eqs. (7.65b), (7.80), and (7.96), the displacement of Bar 2 can be written as  1 Z 1  i(ωt−k2 x)  1 e − ei(ωt+k2 x) u 2 (x, t) = ei(ωt−k2 x) + ei(ωt+k2 x) + 2 2 Z2   Z1 nπ x nπ x iωt −i e . = cos sin (7.98) d Z2 d The real and imaginary components of the spatial field of u 2 (x, t) (i.e., with the exclusion of eiωt ) are plotted for n = 1 in Fig. 7.11. In Fig. 7.12, the square of the transmission coefficient |t|2 = |A3 /A1 |2 is plotted as a function of frequency ω for a given length d, where A1 and A3 denote the amplitudes of the waves defined in Eq. (7.65a–c) with k3 = k1 and Z 3 = Z 1 . In the plot, the frequencies where peaks appear are called the Fabry–Pérot resonance frequencies. They are denoted by ωFBR or f FBR = ωFBR /2π . They can be explicitly derived from Eq. (7.96) as Fig. 7.10 Bar made of medium 2 inserted in a bar of the base medium, medium 1

150

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.11 Spatial distribution of u 2 in the inserted bar (Bar 2) of length d = λ2 /2 when the Fabry–Pérot resonance occurs. Real component: cos(π x/d); imaginary component: −(Z 1 /Z 2 ) sin(π x/d)

Fig. 7.12 |t|2 = |A3 /A1 |2 as the function of ω

c  2 , (n = 1, 2, 3, . . .) d n  c2  (n = 1, 2, 3, . . .). = 2 d

ωFBR = nπ

(7.99a)

f FBR

(7.99b)

7.8 Standing Waves A standing wave is a wave that oscillates in time but does not propagate through space, and consequently, cannot transport energy from one point to another. Therefore, the locations of the maximum and minimum amplitudes in space remain unchanged. In principle, the nodes and antinodes refer to the locations corresponding to zero and maximal amplitude, respectively. We will show that a standing wave can be constructed by adding two propagating waves to the right and left with the same magnitude and specially adjusted phases. As a specific example, we consider a finite bar of length l, as depicted in Fig. 7.13. Here, we primarily consider the bar with both ends fixed, but the analysis used for this case equally applies to the case with other boundary conditions such as a traction-free end. The fixed boundary conditions for the bar shown in Fig. 7.13 are u x (x = 0, t) = 0, u x (x = l, t) = 0.

(7.100a,b)

7.8 Standing Waves

151

Fig. 7.13 Finite bar of length (l) carrying the longitudinal wave

According to the general solution stated in Eq. (7.20), the conditions (7.100a,b) can be expressed as 

1 1 e−ikl eikl



A B

 =

  0 . 0

(7.101)

To find a nontrivial solution, the determinant of the 2×2 matrix in Eq. (7.101) should be set as zero to yield eikl − e−ikl = 2i sin kl = 0 → kl = nπ  kn l(n = 1, 2, 3, . . .).

(7.102)

Using the relation ω = kc, one can find the frequencies satisfying Eq. (7.102) as ω=

nc nπ c  ωn , or f =  f n (n = 1, 2, 3, . . .). l 2l

(7.103)

In Eq. (7.103), the symbol ωn (rad/s) or f n (Hz) represents the n-th eigenfrequency and the corresponding wavelength λn is λn = 2l/n. From Eq. (7.103), one can see that the boundary condition (7.100a, b) is satisfied only at discrete frequencies. This also means that the bar length l becomes a multiple of half the wavelength (l = n(λn /2)) at an eigenfrequency. Thus, placing the result in Eq. (7.102) into Eq. (7.20) with B = −A (as obtained from the first equation of Eq. (7.101)) yields the following solution satisfying the boundary condition (7.100a, b): u n (x, t) = X n (x)eiωn t = An sin kn xeiωn t  An Un (x)eiωn t ,

(7.104)

where kn = nπ/l, ωn = kn c, Un (x) = sin(kn x). The function Un (x) is called the n-th normal mode or eigenmode and it is plotted in Fig. 7.14. Let us examine the standing wave from the perspective of power flow. As the solution in Eq. (7.104) can be regarded as the sum of two waves, Aei(ωt−kx) which denotes a wave propagating rightward and Bei(ωt+kx) which denotes a wave propagating leftward in the bar, one can observe that their powers can be negated inside the bar, because B = −A. A more direct approach involves the computation of the time-averaged power flow in the bar using Eq. (7.49) with tx = σx and Eq. (7.55):

152

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.14 Lowest three eigenmodes (Un , n = 1, 2, and 3) for a fixed–fixed bar. A node or nodal point of zero displacement always remains unchanged during oscillatory motion

 1  < P >=< −Re(vx )Re(σx )S >= − Re vx (σx )∗ S. 2

(7.105)

The velocity and stress can be calculated using the displacement u n in Eq. (7.104): ∂u n = An Ekn cos kn xeiωn t vn = (iω)An sin kn xeiωn t ; σn = E ∂ x  nπ  kn = l where vx and σx are replaced by vn and σn to indicate its relation to the nth mode. The substitution of these results into Eq. (7.105) yields  1  < P > = − Re [(iω)An sin kn x][An Ekn cos kn x]∗ 2  1  = − Re (iω)Ekn |An |2 sin kn x cos kn x = 0. 2

(7.106)

This signifies that the standing wave does not transport any power in space. In the case of standing waves (< P >= 0), the total energy UT is independent of time at all instances because of Eq. (7.54). This can be confirmed if the total energy is explicitly calculated. Because the real component of the displacement u n (x, t) in Eq. (7.104) should be used for energy calculation, we explicitly express the real part of u n (x, t) as     Re[u n (x, t)] = Re An Un (x)eiωn t = Re Arn + i Ain Un (x)eiωn t  = Arn cos ωn t − Ain sin ωn t Un (x) = |An | cos(ωn t + φ) sin kn x, where Arn and Ain denote the real and imaginary parts of A and φ = − tan−1 (Ain /Arn ). The kinetic energy Uk and strain energy Us of the system can be written as

7.9 Dispersive Longitudinal Waves-Metamaterial Interpretation

l Uk = 0

153

 

l ∂Re(u n ) 2 1 1 ρ ρ[|An |(−ωn ) sin(ωn t + φ) sin kn x]2 Sd x Sd x = 2 ∂t 2 0

1 = ρ S|An |2 (ωn )2 sin2 (ωn t + φ) 2

l sin2 kn xd x 0

ρl S |An |2 (ωn )2 sin2 (ωn t + φ), = 4

l Us =

1 2 Eε Sd x = 2 x

0

(7.107)

l 0

  ∂Re(u n ) 2 1 E Sd x 2 ∂x

1 = E S|An |2 (kn )2 cos2 (ωn t + φ) 2 

l cos2 kn xd x 0

  ωn l 1 cos2 (ωn t + φ) = E S|An |2 √ 2 2 E/ρ ρl S |An |2 (ωn )2 cos2 (ωn t + φ). = (7.108) 4 √ In Eq. (7.108), we used kn = ωn /c = ωn / E/ρ. Upon employing Eqs. (7.107) and (7.108), the total energy can be derived as U T = Uk + Us = =

2

  ρl S |An |2 (ωn )2 sin2 (ωn t + φ) + cos2 (ωn t + φ) 4

ρl S |An |2 (ωn )2 , 4

(7.109)

where ρl S denotes the total mass of the bar. Equation (7.109) shows that the total energy of the system is independent of time t.

7.9 Dispersive Longitudinal Waves-Metamaterial Interpretation The longitudinal wave considered thus far exhibits a nondispersive wave phenomenon, i.e., the phase velocity (v p = ω/k as defined in Eq. 3.28) is frequency-independent (see Eq. 7.4): v p = c or v p = −c.

154

7 Longitudinal Waves in 1D Continuum Bars

In this section, we consider a longitudinal wave system with observable dispersive phenomena. Specifically, we will consider longitudinal waves in a bar with distributed longitudinal resonators along the x-axis, as shown in Fig. 7.15. As indicated in Fig. 7.15, the displacement of the bar is denoted by u x and the displacement of the distributed mass m d connected to the bar by the distributed spring K d is denoted by Ux . The free body diagram is considered for an infinitesimal element x of the bar in Fig. 7.16a to derive the equation of motion.    ∂σx ∂ 2u x x − Sσx x − K d (u x − Ux )x + O (x)2 ρ Sx 2 = S σx + ∂t ∂x  ∂σx x − K d (u x − Ux )x + O (x)2 =S ∂x  ∂ 2u x =S E (7.110) x − K d (u x − Ux )x + O (x)2 , 2 ∂x where σx = E∂u x /∂ x is used. If x approaches its limit d x in Eq. (7.110) and the resulting equation is divided by Sd x, we can obtain ρ

Kd

∂ 2u x Kd ∂ 2u x (u x − Ux ). =E − 2 ∂t ∂x2 S Kd

Kd

(7.111)

Ux

md

ux

x

Fig. 7.15 Bar with distributed longitudinal resonators. K d : distributed longitudinal stiffness (force/length2 ) and m d : distributed mass (mass/length)

Fig. 7.16 Free body diagram for a an infinitesimal bar element and b infinitesimal distributed mass element

7.9 Dispersive Longitudinal Waves-Metamaterial Interpretation

155

Upon considering the free body diagram for an infinitesimal element m d x of the distributed mass element displayed in Fig. 7.16b, we can obtain m d x

 ∂ 2 Ux = −K d (Ux − u x )x + O (x)2 , 2 ∂t

(7.112)

which yields md

∂ 2 Ux = −K d (Ux − u x ). ∂t 2

(7.113)

To solve Eqs. (7.111) and (7.113), u x and Ux are assumed to represent waves propagating along the + x-axis, u x (x, t) = Au ei(ωt−kx) ,

(7.114)

Ux (x, t) = AU ei(ωt−kx) .

(7.115)

As the waves propagating along the −x-axis can be observed by replacing k by −k, we will consider the wave form in Eqs. (7.114) and (7.115) only. The substitution of Eqs. (7.114) and (7.115) into Eqs. (7.111) and (7.113) yields ρ(iω)2 Au = EAu (−ik)2 −

Kd (Au − AU ), S

(7.116)

m d (iω)2 AU + K d AU = K d Au .

(7.117)

From Eq. (7.117), the relationship between AU and Au can be found as AU =

Kd Au . K d − m d ω2

(7.118)

The substitution of Eq. (7.118) into Eq. (7.116) yields  ρω = Ek + 1 − 2

2

Kd K d − m d ω2



Kd = Ek 2 − S





ω2 Kd md

−

ω2

Kd . S

(7.119)

Introducing the resonance frequency ω0 and the distributed spring stiffness K˜ d per unit mass of the bar, which are defined as ω02 = and using the definition of c =

√

Kd Kd , and K˜ d = md ρS

E/ρ, Eq. (7.119) can be written as

(7.120a,b)

156

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.17 Dispersion curve for a bar with distributed mass–spring resonators, a when m d = ∞, and b m d is finite (k: wavenumber, ω: angular frequency)

ω2 = c2 k 2 −

K˜ d ω2 . ω02 − ω2

(7.121)

Equation (7.121) shows that ω − k relation is no longer linear. Compared with Eqs. (7.17a), (7.121) contains an additional term K˜ d ω2 /(ω02 − ω2 ), which becomes the source of wave dispersion. The dispersion relations expressed by Eq. (7.121) are plotted in Fig. 7.17. In the case of m d → ∞ (i.e., ω0 → 0), the system in Fig. 7.15 represents a bar supported by an elastic foundation with the foundation stiffness K d , because the limiting case (m d → ∞) simulates the fixed condition. Accordingly, the dispersion relation, Eq. (7.121), becomes ω2 = c2 k 2 + K˜ d (for m d → ∞).

(7.122)

From Eq. (7.122) and the dispersion curve for m d → ∞ in Fig. 7.17a, one can see that no wave can propagate until the frequency ω reach the cutoff frequency ωcutoff 8 defined as  ωcutoff = K˜ d (for m d → ∞). The edge frequencies of a stopband can be found by finding ω satisfying dω/dk = 0. Using Eq. (7.121), dω/dk is found as (ω2 − ω2 )2 k(ω) dω = c2 · · 2 0 . dk ω (ω0 − ω2 )2 + K˜ d ω02

8

The cutoff frequency ωcutoff can be obtained from Eq. (7.122) with k = 0.

(7.123)

7.9 Dispersive Longitudinal Waves-Metamaterial Interpretation

157

Two frequencies satisfying dω/dk = 0 are ω0 and nonzero ωs satisfying k(ωs ) = 0. From k(ωs ) = 0, ω = ωs is found as ωs2 = ω02 + K˜ d > ω02 .

(7.124)

Therefore, the stopband formed in the following frequency range: stopband: ω0 ≤ ω ≤ ωs . 

The stopband is marked in Fig. 7.17b. When m d → ∞, ω0 → 0 and ωs → K˜ d . In this case, the stopband starts from ω = 0 as shown in Fig. 7.17a. Using the definition of the phase velocity (v p = ω/k) and Eq. (7.121), the phase velocity is obtained as  ω02 − ω2 ω . vp = = c k ω02 + K˜ d − ω2 At ω = 0 (for finite m d ),  v p |ω=0 = c

ω02 ω02

< c.

+ K˜ d

Because of the presence of the distributed resonators, the phase velocity at ω = 0 is also affected. We will now analyze the dispersion relation Eq. (7.121) or (7.119) using the notion of effective density. This analysis would aid in understanding the origin of the stopband. To this end, Eq. (7.119) is rewritten as follows:  K˜ d ω2 =Ek 2 . ρ 1+ 2 ω0 − ω2 

(7.125)

If the following effective density ρe f f (ω) is defined as 

ρeff

K˜ d ρ 1+ 2 ω0 − ω2



 ωs2 − ω2 , =ρ ω02 − ω2 

(7.126)

Equation (7.125) can be written in a compact form that resembles the dispersion relation of the nondispersive case (for which ρeff (ω) = ρ): ρeff (ω)ω2 = Ek 2 .

(7.127)

158

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.18 a Effective density, and b dispersion relation as a function of frequency ω for a bar with distributed mass–spring resonators

In Fig. 7.18, the effective density and dispersion curve are plotted as a function of ω. The figure shows that the stop band is formed between ω = ω0 and ω = ωs due to the presence of the negative effective density. One may compare Eq. (7.127), the dispersion relation for a bar with distributed resonators, and the dispersion relation for the mass-in-mass discrete lattice system considered in Sect. 5.1. To this end, the dispersion relation for the mass-in-mass lattice system will be written here (see Eq. 5.19): 2 2 m eff 1 (ω)ω = 4s1 sin

  β , 2

(7.128)

where β is the dimensionless wavenumber. The effective mass m eff 1 (ω) in Eq. (7.128) is defined as  2  ω12 − ω2 eff , (7.129) m 1 (ω) = m 1 ω22 − ω2 with ω22 =

s2 s2 , ω2 = ω22 + . m 2 12 m1

Comparing Eqs. (7.127) and (7.128), their left-hand sides have the same form in ω if we set ω2 = ω0 and ω12 = ωs while their right-hand sides have different forms in the wavenumber (k or β). For small β’s, however, Eq. (7.128) becomes,

7.10 Transfer Matrix Approach

159

Fig. 7.19 Comparison of the dispersion curve for the continuous bar with distributed resonators and that for the mass-in-mass lattice system. a Effective density or mass, and b dispersion curves as the function of frequency ω

2 2 m eff 1 (ω)ω ≈ s1 β .

(7.130)

Therefore, Eqs. (7.127) and (7.128) become identical for small β’s, showing the resemblance between the two systems (the continuous bar with distributed resonators and the mass-in-mass lattice system). Keeping this resemblance in mind, we examine Fig. 7.19. For the two systems, a stop band, the resonance-induced stop band, is formed between ω = ω0 (= ω2 ) and ω = ωs (= ω12 ) because they have negative effective mass or density. However, additional stopbands (known as the Bragg gaps) are formed outside the resonance-induced stop band in the mass-in-mass system unlike the continuous bar with distributed resonators because the mass-in-mass system is a periodic system. From this analysis, one can see that the introduction of a local resonance in a mechanical system, whether it is a continuous or discrete system, a stop band associated with the local resonance can be formed.

7.10 Transfer Matrix Approach As illustrated in Fig. 7.19, the Bragg gap appearing in the lattice is caused by periodicity. Based on this observation, the occurrence of the Bragg-type gap for a periodic system of continuum bodies will be investigated (see Hussein et al. 2006, 2007). For example, we can consider a continuum bar of periodically arranged material properties with a period d. Specifically, we consider a bar of repeated unit cells, each of

160

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.20 Continuum bar with periodically varying material properties

Fig. 7.21 Model used to the explain transfer matrix approach

which has two dissimilar layers—layers 1 and 2; see Fig. 7.20. It is assumed that layer j of length d j is composed of medium j ( j = 1, 2). To analyze the wave phenomena of this periodic continuum system, we employ the transfer matrix approach and the Bloch–Floquet theorem. We look into the transfer matrix approach in this section as a preliminary study. (The Bloch–Floquet theorem is presented in the following section.) To explain the transfer matrix approach, let us consider a bar segment between x + and x + d − , depicted in Fig. 7.21. Note that x + and x − are equal to x, but x + (x − ) refers to the location pertaining to the region in the right (left) side. The transfer matrix T is defined as     V(x + d − ) V(x + ) =T , (7.131) F(x + d − ) F(x + ) where V and F represent the (possibly complex-valued) spatial components of the velocity and the force appearing in vx (x, t) = V(x)eiωt ,

(7.132)

f (x, t) = Sσx (x, t) = F(x)eiωt .

(7.133)

As defined in Eq. (7.131), the transfer matrix T relates the field quantities at two distinct locations. To derive the components of T, we begin with the displacement field u x (x, t) in Eq. (7.20) and obtain the expressions for vx (x, t) and f (x, t) as follows: vx (x, t) =

  ∂u x (x, t) = iω Aei(ωt−kx) + Bei(ωt+kx) , ∂t

(7.134)

7.10 Transfer Matrix Approach

161

  ∂u x (x, t) = Sρc2 A(−ik)ei(ωt−kx) + B(ik)ei(ωt+kx) f (x, t) = S E ∂x   =iωZ −Aei(ωt−kx) + Bei(ωt+kx) . (7.135) To obtain the last expression in Eq. (7.135), we used k = ω/c and Z = ρcS. Upon comparing Eqs. (7.132) and (7.134) along with Eqs. (7.133) and (7.135), we can derive     −ikx  V(x) 1 1 Ae , (7.136) = iω Beikx F(x) −Z Z and 

V(x + d) F(x + d)



 = iω

1 1 −Z Z



e−ikd 0 0 eikd



 Ae−ikx . Beikx

(7.137)

Solving Eq. (7.136) for {Ae−ikx , Beikx }T and substituting the result into Eq. (7.137) yields 

V(x + d) F(x + d)

 −1    −ikd 0 1 1 V(x) 1 1 e eikd −Z Z F(x) −Z Z 0       −ikd 1 0 Z −1 V(x) 1 1 e = eikd Z 1 F(x) −Z Z 0 2Z    1 Z (e−ikd + eikd ) −e−ikd + eikd V(x) = 2Z Z 2 (−e−ikd + eikd ) Z (e−ikd + eikd ) F(x)    i sin kd V(x) cos kd Z . (7.138) = i Z sin kd cos kd F(x)





=

Therefore, the transfer matrix T can be written as 

T11 T12 T(d; k, Z ) = T21 T22





 i sin kd cos kd Z = . i Z sin kd cos kd

(7.139)

To facilitate subsequent discussions, Eq. (7.138) can be written as y(x + d) = T(d)y(x),

(7.140)

where y(x) is defined as  y(x) =

 V(x) . F(x)

(7.141)

162

7 Longitudinal Waves in 1D Continuum Bars

Also, we will write T(d; k, Z ) simply as T(d). The examination of T in Eq. (7.139) yields that  det T = cos kd − 2

 i sin kd (i Z sin kd) Z

= cos2 kd + sin2 kd = 1.

(7.142)

The result in Eq. (7.142) can be also confirmed using the following observation. First, we solve for y(x) using Eq. (7.140) to obtain y(x) = T−1 (d)y(x + d), and place x − d in x to obtain the following relationship: y(x − d) = T−1 (d)y(x).

(7.143)

Next, we replace d by −d in Eq. (7.140) to obtain y(x − d) = T(−d)y(x).

(7.144)

Comparing Eqs. (7.143) and (7.144) yields T−1 (d) = T(−d).

(7.145)

If we find the expression for T−1 (d),   1 T11 (d) −T12 (d) T (d) = det T −T21 (d) T22 (d) −1

and use the fact that T11 (d) = T11 (−d), −T12 (d) = T12 (−d), −T21 (d) = T21 (−d), and T22 (d) = T22 (−d), T−1 (d) is related to T(−d) as   1 1 T11 (−d) T12 (−d) T(−d). T (d) = = det T T21 (−d) T22 (−d) det T −1

(7.146)

Comparing Eqs. (7.145) and (7.146) confirms that det T = 1 must hold.

7.11 Bloch–Floquet Theorem The system illustrated in Fig. 7.20 exhibits a periodic material distribution with a period d. In this case, the general solution contains a certain periodicity and it can be expressed as (by the Bloch–Floquet theorem)

7.11 Bloch–Floquet Theorem

163

  u x (x, t) = eiωt e−ikx g(x) ,

(7.147)

where g(x) is a periodic function that satisfies g(x + d) = g(x).

(7.148)

The Bloch–Floquet Theorem (Floquet 1883; Bloch 1929) is essential for the analysis of periodic systems (see, e.g., Kittel 2004). Here, we will present a simplified proof of the theorem. Bloch–Floquet Theorem Consider a 1st-order ordinary differential equation given by d y(x) = A(x)y(x), dx

(7.149)

y(x) = {y1 (x), y2 (x), y3 (x), . . . yn (x)}T ,

(7.150)

with

where the coefficient matrix A(x) exhibits periodicity with a period d, A(x + d) = A(x).

(7.151)

Then, the solution y(x) has a special form such that y(x + d) = λy(x),

(7.152)

where λ corresponds to an eigenvalue independent of x. The solution can be alternatively written as y(x) = eμx g(x) or y(x) = eikx g(x),

(7.153a,b)

with g(x + d) = g(x).

(7.154)

Note that Eq. (7.149) with Eq. (7.150) is equivalent to the nth-order differential equation for a single variable (e.g., y1 ). To prove the theorem,9 let (x) represents the fundamental solutions to Eq. (7.149) such that d (x) = A(x)(x), dx 9

The proof is not fully rigorous and intended to demonstrate the validity of the theorem.

(7.155)

164

7 Longitudinal Waves in 1D Continuum Bars

where (x) = {1 (x), 2 (x), 3 (x), · · ·, n (x)}T . The placement of x + d for x in Eq. (7.155) yields d (x + d) = A(x + d)(x + d). d(x + d)

(7.156)

Using (7.151) and noting d(·)/d(x + d) = [d x/d(x + d)]d(·)/d x = d(·)/d x, Eq. (7.156) becomes d (x + d) = A(x)(x + d). dx

(7.157)

As shown by Eq. (7.157), (x+d) satisfies Eq. (7.149). Therefore, (x+d) can be another set of the fundamental solutions. As Eq. (7.149) provides only n independent solutions, i (x + d) must be expressed in terms of  j (x) ( j = 1, 2, . . . , n) as i (x + d) =

n

C ji  j (x) (i = 1, 2, . . . , n),

(7.158)

j=1

where Ci j is the expansion coefficient. As (x) denotes the fundamental solutions, a general solution to Eq. (7.149) can be written as y(x) =

n

di i (x).

(7.159)

i=1

Substituting x + d for x in Eq. (7.159) yields y(x + d) =

n

i=1

di i (x + d) =

n n



di C ji  j (x) =

i=1 j=1

 n n



j=1

 C ji di  j (x).

i=1

(7.160) T nIf d = {d1 , d2 , . . . , dn } is an eigenvector of a square matrix C = [C ji ], then i=1 C ji di appearing in Eq. (7.160) can be simplified as n

C ji di = λd j ,

(7.161)

i=1

where λ denotes the eigenvalue of the matrix C. With the application of Eqs. (7.161) and (7.160) can be reduced to

7.12 Dispersion Analysis of Periodic Continuum Body

y(x + d) = λ

n

d j  j (x) = λy(x),

165

(7.162)

j=1

which is the same as Eq. (7.152). If λ is set as λ = e−dμ or λ = e−ikd ,

(7.163)

and the following function g(x) is introduced, g(x) = eμx y(x) or g(x) = eikx y(x),

(7.164)

we can show that g(x) is a periodic function with period d: g(x + d) = eμ(x+d) y(x + d) = eμ(x+d) λy(x) = eμ(x+d) e−μd y(x) = eμx y(x) = g(x), or g(x + d) = eik(x+d) y(x + d) = eik(x+d) λy(x) = eik(x+d) e−ikd y(x) = eikx y(x) = g(x).

(7.165)

Equations (7.164) and (7.165) are the same as Eqs. (7.153) and (7.154). In case of multidimensional problems with x = {x1 , x2 , x3 }T , y(x) can be placed into y(x) = e−ik·x g(x).

7.12 Dispersion Analysis of Periodic Continuum Body The underlying techniques required for the dispersion analysis of a bar with periodic material properties are presented in Sects. 7.10 and 7.11. (A periodic structure carrying mechanical vibratory motions is called a phononic crystal.) Using these techniques, we will investigate the wave phenomenon in the periodic bar illustrated in Fig. 7.20. For the field variable y(x) defined in Eq. (7.141), the Bloch–Floquet theorem can be applied as y(x + d) = λy(x) = e−ikd y(x),

(7.166)

166

7 Longitudinal Waves in 1D Continuum Bars

where λ or kd is an unknown to be determined. As such, the field variable y(x + d) can be related to y(x) through the transfer matrix. Noting that d = d1 +d2 , y(x +d) = y(x + (d1 + d2 )) can be related to y(x + d1+ ) as y(x + d) = T(d2 ; k2 , Z 2 )y(x + d1+ )  T(2) y(x + d1+ ).

(7.167)

Similarly, y(x + d1− ) can be related to y(x) as y(x + d1− ) = T(d1 ; k1 , Z 1 )y(x)  T(1) y(x).

(7.168)

Owing to the continuity at x = d, the following relation holds y(x + d1− ) = y(x + d1+ ).

(7.169)

Combining Eqs. (7.167) and (7.169) yields y(x + d) = T(2) T(1) y(x)  T(21) y(x),

(7.170)

where  T(21) = T(2) T(1) =  =

i cos β2 sin β2 Z2 i Z 2 sin β2 cos β2

cos β1 cos β2 −

Z1 Z2

sin β1 sin β2



 i cos β1 sin β 1 Z1 i Z 1 sin β1 cos β1   β1 i cos βZ2 1sin β1 + sin β2Zcos 2

i(Z 1 cos β2 sin β1 + Z 2 sin β2 cos β1 ) cos β1 cos β2 −  (21) (21)  T11 T12  (21) (21) , T21 T22

Z2 Z1



sin β1 sin β2 (7.171)

and β j is defined as β j (ω) = k j (ω)d =

ωd ( j = 1, 2). cj

(7.172)

As β j is the explicit function of ω, T(21) should be regarded as a function of ω. Therefore, we express T(21) as T(21) (ω) to emphasize that T(21) is known for specified ω. Now, we equate Eqs. (7.166) and (7.170) to set up an eigenvalue problem involving an eigenvalue λ = e−iβ : T(21) (ω)y(x) = λIy(x),

(7.173a)

T(21) (ω)y(x) = e−ikd Iy(x) = e−iβ Iy(x),

(7.173b)

and equivalently,

7.12 Dispersion Analysis of Periodic Continuum Body

167

where I represents a 2 × 2 identity matrix and β is the Bloch phase that is a dimensionless quantity related to the wavenumber k: β = kd.

(7.174)

In the subsequent analysis, β is regarded as the eigenvalue (dimensionless wavenumber) to be determined, whereas symbol β j is regarded as a known function of ω, as stated by Eq. (7.172). To obtain a nontrivial solution for Eqs. (7.173a, b), the following equation must hold, det[T(21) (ω) − λI] = 0,

(7.175)

  (21)   T (ω) − λ T (21) (ω) 12  = 0.  11 (21)  T (21) (ω) T22 (ω) − λ  21

(7.176)

i.e.,

The expansion of Eq. (7.176) yields, ! (21) (21) λ2 − T11 (ω) + T22 (ω) λ + det T(21) (ω) = 0.

(7.177)

(21) (21) (21) (21) Because det T(21) = T11 T22 − T12 T21 = det T(2) det T(1) = 1 (refer to Eqs. (7.142) and (7.177)) can be written ascan be written as

λ2 − 2a(ω)λ + 1 = 0,

(7.178)

where a(ω) =

! 1 (21) (21) T11 (ω) + T22 (ω) . 2

(7.179)

The solution to Eq. (7.178) can be expressed as follows: for a 2 (ω) ≤ 1:  λ = e−iβ = cos β − i sin β = a(ω) ± i 1 − a 2 (ω);

(7.180a)

for a 2 (ω) > 1: λ = e−iβ = cos β − i sin β = a(ω) ±

 a 2 (ω) − 1.

(7.180b)

If a 2 (ω) > 1, β = kd should be imaginary or complex. Therefore, no wave can propagate. For a 2 (ω) ≤ 1, β = kd is real-valued for which waves can propagate. In this case, we obtain

168

7 Longitudinal Waves in 1D Continuum Bars



cos β = a(ω),  sin β = ± 1 − a 2 (ω).

(7.181)

Equation (7.181) represents the dispersion relation (ω − β relation) for propagating waves.10 For actual calculations to determine the dispersion relation, the expression for a(ω) can be further simplified as     Z1 Z2 1 sin β1 sin β2 2 cos β1 cos β2 − + 2 Z2 Z1 1 = {[cos(β1 + β2 ) + cos(β1 − β2 )] 2    1 Z1 Z2 + + [cos(β1 + β2 ) − cos(β1 − β2 )] 2 Z2 Z1       1 Z1 1 Z1 1 1 Z2 Z2 = cos(β1 + β2 ) 1 + − cos(β1 − β2 ) −1 + . + + 2 2 Z2 Z1 2 2 Z2 Z1 (7.182)

a(ω) =

If a symbol ζ is introduced such that      Z 1 − Z 2   1 − Z 2 /Z 1  =  < 1, ζ =  Z 1 + Z 2   1 + Z 2 /Z 1 

(7.183)

certain terms in Eq. (7.182) can be simplified as   1 Z1 (Z 1 + Z 2 )2 Z2 = + 2 Z2 Z1 2Z 1 Z 2 2 (Z 1 + Z 2 ) 2 2 = = 1 , 2 =  2 − (Z − Z )2 1 − ζ2 (Z + Z ) Z −Z 1 2 1 2 1 2 2 1 − Z 1 +Z 2   Z2 1 Z1 (Z 1 − Z 2 )2 (Z 1 − Z 2 )2  = + = 1 −1+ 2 Z2 Z1 2Z 1 Z 2 + Z 2 )2 − (Z 1 − Z 2 )2 2 (Z 1  2 2 2 ZZ 11 −Z +Z 2 2ζ 2 = . 2 =  1 − ζ2 2 1 − ZZ 11 −Z +Z 2

1+

Therefore, a(ω) can be stated as an explicit function of ω: The dispersion relation β = cos−1 a(ω) can be used to find any β’s, either real-valued or complexvalued, if it is solved numerically (say, using Matlab). For a given ω, real-valued and complex-valued β’s will be automatically obtained for propagating and evanescent waves, respectively, without a need to separately solve when a(ω) < 1 or not. Solve Problem 7–8.

10

7.12 Dispersion Analysis of Periodic Continuum Body

a(ω) =

cos[β1 (ω) + β2 (ω)] − ζ 2 cos[β1 (ω) − β2 (ω)] , 1 − ζ2

169

(7.184)

where β j (ω) is expressed in Eq. (7.172). Using Eq. (7.181) (for a 2 (ω) ≤ 1), the dispersion curve for a typical bar plotted in Fig. 7.20 is further detailed in Fig. 7.22. It demonstrates the periodic nature of the dispersion curve with a period of 2π in β. This periodicity can be directly checked if Eq. (7.181) with β replaced by β +2π yields the original equation. The zone between β = −π and β = π is called the 1st Brillouin zone and that between β = −2π and β = −π and between β = π and β = 2π , the 2nd Brillouin zone. (The 3rd and 4th zones can be similarly defined.) The periodic nature of the dispersion curve demonstrated in Fig. 7.22 is similar to that found in the dispersion curve for a monatomic lattice system depicted in Fig. 3.6. The fundamental difference is that |β| > π cannot be measured for the lattice system, because the particles are located only at discrete points distanced by d. Therefore, the dispersion curve beyond |β| = π is meaningless for the lattice system. However, the dispersion curve for |β| > π in the present continuum system is meaningful, because |β| can be measured even if |β| > π . Moreover, infinitely many higher branches are possible, because a continuum system contains an infinite number of degrees. Certain stopbands observed in Fig. 7.22 will be investigated in the following section.

Fig. 7.22 Dispersion curve for a bar with periodic material properties illustrated in Fig. 7.20 where ω denotes the angular frequency and β = kd represents the dimensionless wavenumber

170

7 Longitudinal Waves in 1D Continuum Bars

7.13 Analysis of Stop Band in Periodic Continuum Body To analyze the stopbands indicated in. Figure 7.22 for the periodic continuum bar in Fig. 7.20, Eq. (7.180b) is considered for a 2 (ω) > 1. To determine the cutoff frequencies corresponding to the formation of stop bands, a 2 (ω) = 1 is solved. As observed in Fig. 7.22, there are the upper and lower cutoff frequencies denoted by U L and ωcutoff for each stopband. ωcutoff 2 If a (ω) = 1, i.e., a(ω) = ±1, Eq. (7.180b) becomes cos β = cos kd = ±1.

(7.185)

The wavenumbers satisfying Eq. (7.185) can be stated as  β = kd =

0, ±2π, ±4π, . . . , for a(ω) = 1, ±π, ±3π, ±5π, . . . , for a(ω) = −1.

(7.186)

Owing to the periodicity in β = kd, we will primarily consider the 1st Brillouin U zone by selecting β = kd = π (or −π because of symmetry) to determine ωcuotff L and ωcutoff . Thus, a(ω) = −1 is put into Eq. (7.184): −1 + ζ 2 = cos[β1 (ω) + β2 (ω)] − ζ 2 cos[β1 (ω) − β2 (ω)], which reduces to 1 + cos[β1 (ω) + β2 (ω)] = ζ 2 {1 + cos[β1 (ω) − β2 (ω)]}.

(7.187)

Using 1 + cos α = 2 cos2 (α/2), Eq. (7.187) can be further simplified to  cos2

   β1 (ω) + β2 (ω) β1 (ω) − β2 (ω) = ζ 2 cos2 , 2 2

(7.188)

where βi (ω) is defined in Eq. (7.172). The solutions ω satisfying Eq. (7.188) can be numerically evaluated. U L and ωcutoff , a special case As a demonstrative example for determining ωcutoff β1 (ω) = β2 (ω) is considered. Based on Eq. (7.172), this case requires that ωd1 ωd2 d1 d2 = → = . c1 c2 c1 c2

(7.189)

The condition (7.189) is alternatively interpreted as t1 = t2 , where ti denotes the time to travel the distance of di at the speed of ci in layer i (composed of medium i). Substituting the condition (7.189) into Eq. (7.188) yields

7.13 Analysis of Stop Band in Periodic Continuum Body

171

Fig. 7.23 Graphical approach to determine solutions of Eq. (7.191)

       1 d1 d1 d2 d2 2 2 ω = ζ or cos π f = ζ 2. cos + + 2 c1 c2 c1 c2 2

(7.190)

Because ζ 2 < 1, Eq. (7.190) provides solutions in all cases. To facilitate solving Eq. (7.190), the following modified frequency f is introduced,  f =

 d1 d2 π f, + c1 c2

to rewrite Eq. (7.190) as cos f = ±|ζ |.

(7.191)

The solutions to Eq. (7.191) can be graphically found using Fig. 7.23; they can be determined as the intersection points of two curves, y = F1 ( f ) and y = ±F2 ( f ), where F1 ( f ) = cos f and F2 ( f ) = |ζ |. The least values of the lower and upper cutoff frequencies (in terms of f ) are L

f cutoff = cos−1 |ζ |,

U

f cutoff = − cos−1 |ζ |.

If the following identity is used, cos−1 x = Equation (7.192) can be simplified to

π − sin−1 x 2

(7.192)

172

7 Longitudinal Waves in 1D Continuum Bars L

f cutoff =

π − sin−1 |ζ |, 2

U

f cutoff =

π + sin−1 |ζ | 2

(7.193)

π + 2 sin−1 |ζ | .  2π dc11 + dc22

(7.194)

or L f cutoff =

π − 2 sin−1 |ζ | ,  2π dc11 + dc22

U = f cutoff

U U L L and f cutoff in Eq. (7.194) correspond to ωcutoff and ωcutoff , respectively, Note that f cutoff which are marked in Fig. 7.22. C of the stop band Using Eq. (7.194), we can calculate the center frequency f stop U L formed between f = f cutoff and f = f cutoff as C f stop =

1 1 L U f cutoff + f cutoff =  2 2 dc11 +

d2 c2

=

1 . 2(t1 + t2 )

(7.195)

C Equation (7.195) indicates that f stop depends on the lengths and wave speeds of the selected layer stacks. The lager d1 and d2 are, the lower the center frequency becomes. The bandwidth  f stop of the stopband becomes

U L  f stop = f cutoff − f cutoff =

2 sin−1 |ζ | 4 C   = f stop sin−1 |ζ |. π d1 + d2 π c1 c2

(7.196)

Instead of  f stop , it is more convenient to use the following relative bandwidth to characterize the stop band,      f stop 4 4 −1 −1  1 − Z 2 /Z 1  . = sin |ζ | = sin  relative bandwidth: c f stop π π 1 + Z 2 /Z 1 

(7.197)

As depicted in Fig. 7.24, the relative bandwidth increases with the impedance contrast, i.e., as Z 2 /Z 1 approaches 0 or ∞. Thus, the bandwidth reduces to zero in case of no impedance contrast (Z 2 = Z 1 ).

7.14 Periodic Bar Structure Containing Quarter-Wave Stacks With the condition of d1 /c1 = d2 /c2 (i.e., β1 = β2 ) stated in Eq. (7.189), we can advance one step further to impose β1 = β2 =

π , 2

(7.198a)

7.14 Periodic Bar Structure Containing Quarter-Wave Stacks

173

Fig. 7.24 Relative bandwidth as the function of impedance contrast (Z 2 /Z 1 )

resulting in β = β1 + β2 = π,

(7.198b)

where β denotes the phase change through the unit cell containing layers 1 and 2. If β1 = β2 = π/2 is imposed, the following layer thickness (d j ) should be selected: dj =

λj βj π λj = ( j = 1, 2). = kj 2 2π 4

(7.199)

Accordingly, the resulting unit cell consists of two quarter-wave stacks. The corresponding wavelength becomes λ = 2d,

(7.200)

because β is set as β = kd = π . In the case of the present periodic system of quarterwave stacks, the phase change β = π through the unit cell can be decomposed into π/2 in layer 1 and π/2 in layer 2. While the corresponding center frequency of the stopband is expressed in Eq. (7.195), its formation of the stopband can be explained by the Bragg condition, which was discovered in a crystalline solid. It is depicted in Fig. 7.25. In Fig. 7.25, two beams with identical wavelength and phase are incident onto a crystalline solid and scattered from the lattice planes. They are separated by distance d, which is equivalent to the length of a unit cell in a periodic structure. The scattered wave fields form constructive inference if the following condition is satisfied:

174

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.25 Illustration of the Bragg diffraction phenomenon

AO  B = mλ, (m = 1, 2, . . .).

(7.201)

In terms of geometric parameters, Eq. (7.201) can be written as 2d sin θ = mλ (m = 1, 2, . . .).

(7.202)

Equation (7.202) is known as the Bragg condition that dictates the so-called Bragg phenomenon (Bragg and Bragg 1913). Specifically, at the normal incidence (θ = π/2) with m = 1, Eq. (7.202) becomes d=

λ , 2

(7.203)

which is exactly the same as Eq. (7.200). Accordingly, the stopband formed at f = C f stop is found to be due to the Bragg phenomenon, and thus, the stopband formed U L between f = f cutoff and f = f cutoff is called a Bragg stopband. An advantage of using quarter-wave stacks in forming a periodic structure is that it is relatively convenient to determine a periodic structure that exhibits the desired value C of the first stopband at a specific target frequency, e.g., of the center frequency f stop T f stop . Specifically, the following procedure can be implemented. T , determine the layer thickness d j as d j = λ j /4 = Step 1: For a given f stop T c j /4 f stop ( j = 1, 2). c Step 2: Determine Z 2 /Z 1 for which the desired relative bandwidth  f stop / f stop can be satisfied. Thereafter, select media 1 and 2 to yield the desired Z 2 /Z 1 value. If no natural material satisfies the required Z 2 /Z 1 value, artificial metamaterials can be considered.

A medium with voids is a typical example of an artificial metamaterial that can exhibit the desired effective material properties for a given target frequency. To evaluate the effective homogenized material property of an artificial material, the so-called S-parameter method can be used, as explained in the following section.

7.15 Material Characterization Using S Parameters

175

7.15 Material Characterization Using S Parameters As described in Sect. 7.14, a material exhibiting the desired properties, such as Z , is required. As a design method for such a material, an inhomogeneous medium can be considered as depicted in Fig. 7.26a. This may contain inclusions composed of dissimilar media (including voids) in a base medium. To determine the effective material properties ρeff and E eff (or ceff and Z eff for the homogenized medium shown in Fig. 7.26b), we can employ the method using the S-parameters (scattering parameters). Note that the effective material properties can be generally frequency-dependent. The definition of the S-parameters is provided below. To characterize the material properties (c2 and Z 2 ) of an artificial inhomogeneous medium of length d, we insert it inside a homogeneous infinitely long medium with known material properties (c1 and Z 1 ), as indicated in Fig. 7.27. With the incidence of a wave (u i (x1 , t) = A1 ei(ωt−k1 x1 ) ) onto the unknown medium, we can use numerical or experimental methods to obtain the reflected (u r (x1 , t) = B1 ei(ωt+k1 x1 ) ) and transmitted (u t (x3 , t) = A3 ei(ωt−k1 x3 ) ) waves in the homogeneous medium. The wavenumber in the homogeneous medium is denoted by k1 . By evaluating r = B1 /A1 and t = A3 /A1 , we can estimate the effective material properties c2 and Z 2 . Instead of u i (x1 , t) = A1 ei(ωt−k1 x1 ) , one can alternatively use u i (x3 , t) = B3 ei(ωt+k1 x3 ) . The material characterization process explained above can be more conveniently conducted once the scattering matrix S is obtained. It is defined in the following equation: 

A3 B3



     A1 S11 S12 A1 =S = , B1 S21 S22 B1

(7.204)

where Si j are called scattering parameters. To derive Si j , we use the field relation expressed in the transfer matrix as given by Eq. (7.138):

Fig. 7.26 a Artificial inhomogeneous medium. b Equivalent homogenized medium with effective material properties at a specific frequency

176

7 Longitudinal Waves in 1D Continuum Bars

Fig. 7.27 Model used to explain the S-parameter method. Coefficients A1 , etc., are related to waves in the homogeneous medium as in u(x1 , t) = A1 ei(ωt−k1 x1 ) + B1 ei(ωt+k1 x1 ) and u(x3 , t) = A3 ei(ωt−k1 x3 ) + B3 ei(ωt+k1 x3 )



V(x1 ) F(x1 )



 = x1 =d −

i cos k2 d sin k2 d Z2 i Z 2 sin k2 d cos k2 d



V(x1 ) F(x1 )



  T2 x1 =0+

V(x1 ) F(x1 )

 , x1 =0+

(7.205) where k2 denotes the (unknown) wavenumber of the homogenized version of the artificial inhomogeneous medium (effective medium 2) of length d. Here, 0+ and d − are used to indicate that the quantities in consideration are associated to the artificial inhomogeneous medium. Now, we impose the field continuity relations such that 



V(x1 ) F(x1 )

 = x1 =0−

V(x1 ) F(x1 )



 and x1 =0+

V(x1 ) F(x1 )



 = x1 =d −

V(x3 ) F(x3 )

 . x3 =0+

(7.206a, b) Using Eq. (7.136), {V(x1 ), F(x1 )}Tx1 =0− and {V(x3 ), F(x3 )}Tx3 =0+ can be explicitly related to A1 , B1 , A3 , and B3 as 



V(x1 ) F(x1 ) V(x3 ) F(x3 )





x1 =0−

1 1 = iω −Z 1 Z 1 

 = iω x3 =0+

1 1 −Z 1 Z 1



A1 e−ik1 x1 B1 eik1 x1



A3 e−ik1 x3 B3 eik1 x3



 A1 , B1 (7.207a)    1 1 A3 = iω . B3 −Z 1 Z 1 (7.207b) 

x1 =0−

 x3 =0+

1 1 = iω −Z 1 Z 1



Using Eqs. (7.205), (7.206) and (7.207a, b), the relation between {A3 , B3 }T and {A1 , B1 }T can be found as 

A3 B3





1 1 = −Z 1 Z 1

−1

 T2

1 1 −Z 1 Z 1



A1 B1



  A S 1 , B1

(7.208)

7.15 Material Characterization Using S Parameters

177

where the scattering matrix S is defined as  S=

1 1 −Z 1 Z 1

−1

 T2

 1 1 . −Z 1 Z 1

(7.209)

The scattering matrix S identified in Eq. (7.209) can be explicitly written as  S=

S11 S12 S21 S22



 =

cos k2 d −

i 2

∗ S12 = −S12



Z1 Z2

+

Z2 Z1



i 2

sin k2 d



∗ S11

Z1 Z2

−

Z2 Z1



 sin k2 d

, (7.210)

where the starred quantities represent the complex conjugates. The evaluation of the determinant of S from Eq. (7.208) shows that −1   1 1 1 1 T2 det S = det −Z 1 Z 1 −Z 1 Z 1  −1   1 1 1 1 = det = det T2 · det T2 · det −Z 1 Z 1 −Z 1 Z 1 

Because the determinant of the transfer matrix is unity (see Eq. 7.142), i.e., det T2 = 1, det S = S11 S22 − S12 S21 = 1.

(7.211)

We will now explain the material characterization process. Considering that a wave is incident from the left-hand side, we can select A1 = 1 and B3 = 0.

(7.212)

Then, B1 and A3 can be replaced by the reflection coefficient r and the transmission coefficient t 11 as B1 = r and A3 = t.

(7.213)

Substituting Eqs. (7.212) and (7.213) into Eq. (7.208) yields

11

t = S11 · 1 + S12 · r,

(7.214a)

0 = S21 · 1 + S22 · r.

(7.214b)

Instead of S ij , the reflection and transmission coefficients r and t are often called the scattering parameters. Also note that the use of r and t may be preferred over S ij because the physical significance can be extracted more directly from r and t.

178

7 Longitudinal Waves in 1D Continuum Bars

Solving Eqs. (7.214a, b) for r and t yields S21 , S22

r =− t = S11 + S12 r =

(7.215a)

S11 S22 − S12 S21 1 = , S22 S22

(7.215b)

where det S = S11 S22 − S12 S21 = 1 (see Eq. (7.211)) is used to obtain the expression 1 / S 22 in Eq. (7.215b). Substituting Si j derived in Eq. (7.210) into Eqs. (7.215a, b) yields i 2

r=

Z1 Z2

cos k2 d +

= t=



− 

i 2

Z2 Z1 Z1 Z2

 sin k2 d  + ZZ 21 sin k2 d

Z 12 − Z 22 , Z 12 + Z 22 − 2i Z 1 Z 2 cot k2 d

cos k2 d +

i 2



1 +

Z1 Z2

Z2 Z1



(7.216) ,

(7.217)

sin k2 d

To explicitly express S11 , S12 , S21 , and S22 in terms of r and t that can be computed or experimentally measured, we use Eq. (7.215b) to obtain S22 =

1 . t

(7.218)

Using Eqs. (7.218) and (7.215a), we can express S21 as r S21 = − . t

(7.219)

∗ If the relation (S21 = S12 ) given by Eq. (7.210) is used, S12 is determined as,

S12 = −

r∗ . t∗

(7.220)

∗ ) given in Eq. (7.210) to express S11 as Finally, we use the relation (S22 = S11

S11 =

1 . t∗

Equations (7.218)–(7.221) can be put into a compact form as

(7.221)

7.15 Material Characterization Using S Parameters

 S=

S11 S12 S21 S22



179

 =

1 t∗

− rt

∗

− rt ∗ 1 t

 .

(7.222)

We will now explain the procedure to estimate the phase velocity c2 of the effective medium using r and t. To this end, we use Eqs. (7.216) and (7.217) to obtain the following relations:  2 cos2 k2 d + sin2 k2 d + 41 ZZ 21 − ZZ 21 sin2 k2 d t2 − r2 = ,   !2 cos k2 d + 2i ZZ 21 + ZZ 21 sin k2 d  2 cos2 k2 d + 41 ZZ 21 + ZZ 21 sin2 k2 d =   !2 cos k2 d + 2i ZZ 21 + ZZ 21 sin k2 d    ! cos k2 d + 2i ZZ 21 + ZZ 21 sin k2 d cos k2 d − 2i ZZ 21 + =   !2 cos k2 d + 2i ZZ 21 + ZZ 21 sin k2 d   cos k2 d − 2i ZZ 21 + ZZ 21 sin k2 d   = cos k2 d + 2i ZZ 21 + ZZ 21 sin k2 d   ! − cos k2 d − 2i ZZ 21 + ZZ 21 sin k2 d + 2 cos k2 d   = cos k2 d + 2i ZZ 21 + ZZ 21 sin k2 d 

= −1 + 2t cos k2 d.

Z2 Z1



! sin k2 d

(7.223)

To obtain the last expression in Eq. (7.223), Eq. (7.217) is used. Using Eq. (7.223), one can express k2 d in terms of r and as  1 − r2 + t2 , cos k2 d = 2t

(7.224a)

ort k2 d = 2mπ ± cos−1



 1 − r2 + t2 , (m = 0, 1, 2, . . .). 2t

Because the following relation holds k2 d =

ωd , c2

(7.224b)

180

7 Longitudinal Waves in 1D Continuum Bars

one can calculate c2 once k2 is determined from Eqs. (7.224a, b). In Eq. (7.224b), m = 0 is generally used, but a proper value12 of m can be chosen. The parameter remaining to be evaluated is the effective characteristic impedance Z 2 or ξ = Z 2 /Z 1 . For its evaluation, the following relation is used:   1 i 1 = cos k2 d + ξ+ sin k2 d, t 2 ξ   i 1 r = − ξ sin k2 d. t 2 ξ

(7.225) (7.226)

Because cos k2 d is evaluated as Eq. (7.224a), we can consider the following relation: 1 t

i 2

− cos k2 d



= 

r t

i 2

1 ξ

+ξ

1 ξ

−ξ

 .

(7.227)

The simplification of the left- and right-hand sides of Eq. (7.227) yields LHS =

  1 1 + r2 − t2 1 1 1−t · 1 − r2 + t2 = , (1 − t cos k2 d) = r r 2t 2r RHS =

1 + ξ2 . 1 − ξ2

Solving for ξ 2 using LHS = RHS yields the equation needed to calculate ξ or Z 2 :  ξ2 =

Z2 Z1

2 =

(1 − r )2 − t 2 . (1 + r )2 − t 2

(7.228)

Equations (7.224a, b) and (7.228) are the equations necessary to estimate c2 and Z 2 , which are the effective material properties of a unknown inhomogeneous medium whose unit cell is sized by d. Once c2 and Z 2 are determined, the effective density and stiffness can be determined. For more general cases of the characterization of effective material properties of elastic metamaterials, refer to Lee et al. (2016).

12

Because the evaluated c2 is frequency-dependent in general, integer m should be selected to ensure the continuous variation of c2 as a function of frequency.

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

181

Fig. 7.28 Bar (plate) with a pair or resonators attached. a 1D bar model and b its realization with a plate and a pair of U-channel plates functioning as point resonators in the bar model. (t 0 = 2 mm, 2L = 36.6 mm, 2L T = 30.0 mm. The other dimensions to be determined after analysis for a given target frequency) (Kim et al. 2018)

7.16 Lowered Effective Impedance by Resonators (Advanced Topic) In this section, we consider a new approach to manipulate the effective material properties of a bar segment using a pair of resonators. In particular, Kim et al. (2018) showed that if the effective characteristic impedance of a bar segment is manipulated to be near zero, the far-field output by an actuator or the sensitivity of a sensor installed right on the bar segment can be substantially increased. Figure 7.28a illustrates a bar of thickness t0 with a pair of resonators attached at x = ±W . In the subsequent discussion, we assume that the bar is excited by two point harmonic forces Finp (t) at x = L and x = −L, with L < W . The two point forces simulate the forces excited by a piezoelectric patch installed at the center (x = 0) of the bar13 and the actual realization of the bar model is presented in Fig. 7.28b. The main reasons for installing a pair of resonators for possible magnification of the far-field output strain (stress) under a given magnitude of Finp (t) are as follows. (1) As the output velocity in the bar segment is inversely proportional to the effective impedance of a system to which Finp (t) is applied, reducing the effective impedance will increase the output. 13

Alternatively, we can consider sensing through a piezoelectric patch installed on the bar. Because signal sensing by a PZT patch is reciprocal to actuation by a piezoelectric patch, we will mainly consider the actuation described here.

182

7 Longitudinal Waves in 1D Continuum Bars

(2) Therefore, the output velocity can be increased for the same magnitude of Finp (t) if the effective characteristic impedance of the bar region bounding the actuation points for the actuation frequency is reduced in comparison with the nominal impedance of the bar. (3) However, the increased output cannot be fully transmitted outside of the bounded bar region because of the impedance mismatch. This problem can be overcome and full transmission can be possible if the actuation frequency is tuned at the Fabry–Pérot resonance frequency. (4) At the same time, the selected frequency should also be the frequency maximizing the output of the piezoelectric actuating element. This condition has nothing to do with the installation of the resonators but is directly related to the actuation mechanism. Before analyzing the involved physics of the resonator-pair-based actuator output magnification, we examine the actual realization of the bar model illustrated in Fig. 7.28b. Two U-channel box beams attached to a thin plate (simulating the bar in Fig. 7.28a) serve as point resonators. The eigenfrequencies of the resonators should be elaborately determined for magnifying the far-field output. Suppose the plate size in the y-direction is considerably greater than the plate thickness t0 , and the wave fields generated by a piezoceramic patch (denoted as PZT) with a size of 2L T are almost uniform along the y-axis. In that case, the actual wave propagating in the plate is the lowest symmetric Lamb wave (S 0 ) in the frequency range of interest. Refer to Achenbach (1976), Miklowitz (1978), or Rose (2014) for more details on the Lamb waves propagating in a plate. As demonstrated in earlier works (Oh et al. 2016; Lee et al. 2016), the one-dimensional longitudinal wave corresponds to the S 0 wave. Therefore, the wave motion in the plate can simulate one-dimensional longitudinal waves in a bar portrayed in Fig. 7.28. At each of the installation locations x = ±W in Fig. 7.28a, two U-channel box beams, one on the top and the other on the bottom, are symmetrically arranged to couple with the longitudinal wave mode of the bar. Because the bar is exited by two point forces Finp (t) at x = ±L (with L < W ), the generated longitudinal waves (of the same magnitude) propagate both along the +x and −x axes in the bar. Note that the point force model (known as the pin-force model) employed in Fig. 7.28a describes the actual actuation mechanism of the PZT with fair accuracy (see, e. g., Chaudhry and Rogers 1994). To more accurately characterize the actual actuation mechanism with the pin-force model, the locations of the pin-forces were adjusted to (−L, L) from the installation locations (−L T , L T ) of the actual PZT in such a way that the frequency response of the PZT-plate system obtained with the analytic pin-force model matches that of the finite element model. Accordingly, we found 2L = 36.6 mm for 2L T = 30 mm. To reveal why the generated longitudinal wave at a specific frequency can be increased if a pair of resonators surrounding the excitation points are installed, we employ an equivalent homogenized model utilizing effective material properties; see Fig. 7.29. In this model, the bar segment (−W ≤ x ≤ W ) surrounded by the

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

183

Fig. 7.29 Equivalent systems. a Bar with pair of resonators attached, and b equivalent continuum bar comprising two-bar segments having the nominal (impedance: z 0 ) and effective (impedance:zunknown to be determined) material property (Kim et al. 2018)

resonators is replaced by an equivalent metamaterial14 bar segment (−W  ≤ x ≤ W  , W  : to be determined) without resonators installed. The effective impedance z of the metamaterial bar, different from the impedance z 0 of the original bar, is expected to be frequency-dependent because of the installed resonators. A somewhat complicated analysis is needed to determine W  , z, and other physical quantities of interest (such as the amplified far-field output). First, we analyze the longitudinal wave motion in the original one-dimensional model with two resonators. To this end, we use the field variables discussed in Fig. 7.28a, such as u R exp(iωt), ω denotes angular frequency. It denotes the longitudinal displacement of the mass m of the resonator. Points Q and Q  denote the bar locations at which the resonators of mass m and stiffness s are attached onto the bar. We will use u(x, t) to denote the displacement field in the bar and u Q exp(iωt) to denote u(x, t) at point Q. Upon considering the field symmetry, the displacement in the bar can be expressed as ⎧ − U eikx for x ≤ −W ⎪ ⎪ ⎪ ⎪ ⎪ ikx −ikx ⎪ for − W ≤ x ≤ −L ⎪ ⎨ − u2e − u3e iωt −ikx ikx u(x, t) = e , (7.229) u1e − u 1 e for − L ≤ x ≤ L ⎪ ⎪ ⎪ ⎪ u 2 e−ikx + u 3 eikx for L ≤ x ≤ W ⎪ ⎪ ⎪ −ikx ⎩ Ue for x ≥ W

14

Typically, a metamaterial has unit cells, the size of which is within a subwavelength scale. In this respect, we cannot use the terminology ‘metamaterial’ here because the original system in consideration has two point resonators, and it is not possible to define even unit cells. However, the far-field output strain amplification can be explained by the effective material properties of an equivalent system with the effects of resonators reflected. Thus, the resulting equivalent system will be called an equivalent metamaterial.

184

7 Longitudinal Waves in 1D Continuum Bars

where k represents the wavenumber. The symbol U denotes the amplitude of waves for x ≥ W . The symbols u 2 and u 3 denote the amplitude of the wave (e−ikx ) propagating rightward and that of the wave (eikx ) propagating leftward, respectively, for L ≤ x ≤ W . In the bar segment between x = −L and x = L, we use the same symbol u 1 to denote the amplitudes of both the waves propagating rightward and leftward. The displacement field for x < 0 is also stated in Eq. (7.229), which can be readily found considering the symmetry with respect to x = 0. Accordingly, we will consider the wave field only for x ≥ 0. In consideration of the displacement continuity and equilibrium conditions at x = L, the following equations can be obtained displacement continuity: u 1 e−ik L − u 1 eik L = u 2 e−ik L + u 3 eik L ,

(7.230)

  equilibrium: − i z 0 ω u 1 e−ik L + u 1 eik L = Finp − i z 0 ω u 2 e−ik L − u 3 eik L . (7.231) Note that the left- and right-hand sides of Eq. (7.231) represent the internal force F = E A0 ∂u/∂ x, where E is Young’s modulus of elasticity, and A0 = b0 t0 (b0 : width) denotes the cross-sectional area of the bar. If the volume density is denoted by ρ, the characteristic impedance z 0 can be expressed as  z 0 = A0 ρ E.

(7.232)

The continuity and equilibrium conditions at x = W (point Q) yields displacement continuity: u Q = U e−ikW with u Q = u 2 e−ikW + u 3 eikW , (7.233)  equilibrium: − i z 0 ω u 2 e−ikW − u 3 eikW = s(u R − u Q ) − i z 0 ωU e−ikW . (7.234) In contrast, the equation of motion for resonator mass m can be expressed as −mω2 u R + s(u R − u Q ) = 0.

(7.235)

To explicitly write U , we rearrange Eq. (7.235) using u Q = U e−ikW : uR − uQ =

mω2 mω2 uQ = U e−ikW . 2 s − mω s − mω2

If Eq. (7.236) is used, the right-hand side of Eq. (7.234) becomes s(u R − u Q ) − i z 0 ωU e

−ikW

 =

 smω2 − i z 0 ω U e−ikW s − mω2

(7.236)

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)



185

 smω2 · = − i z0 ω + 1 U e−ikW z 0 ω s − mω2 i

 −i z 0 ω(iα + 1)U e−ikW ,

(7.237)

where α=

1 smω sω 1 = . 2 2 z 0 s − mω z 0 ω R − ω2

(7.238)

In Eq. (7.238), the resonance frequency ω R of the resonator is defined as ω R = 2π f R =

s . m

(7.239)

The substitution of Eq. (7.237) into Eq. (7.234) yields u 2 e−ikW − u 3 eikW = (iα + 1)U e−ikW .

(7.240)

The combination of Eqs. (7.233) and (7.240) yields u2 =

1 1 (iα + 2)U, u 3 = − iαU e−2ikW . 2 2

(7.241)

Eliminating u 1 from Eqs. (7.230) and (7.231) yields Finp e−ik L + eik L  −ik L + u 2 e−ik L − u 3 eik L , u2e + u 3 eik L = − −ik L ik L e −e i z0 ω which simplifies to Finp 2 2 . u 2 + −ik L u3 = − e−ik L − eik L e − eik L i z0 ω

(7.242)

Substituting u 2 and u 3 in Eq. (7.241) into Eq. (7.242) yields e−ik L

   Finp 1 . iα 1 − e−2ikW + 2 U = − ik L −e i z0 ω

(7.243)

Noting that e−ik L − eik L = −2i sin k L, the expression for U in Eq. (7.243) can be obtained as U=

2Finp sin k L . i z 0 ω α(1 − e−2ikW ) − 2i

(7.244)

When no resonator is installed (i.e., α = 0), the corresponding displacement U0 can be obtained from Eq. (7.244):

186

7 Longitudinal Waves in 1D Continuum Bars

U0 =

Finp sin k L . z0 ω

(7.245)

If U0 expressed in Eq. (7.245) is used, U in Eq. (7.244) can be written in a more concise form: U = U0

−2i . α(1 − e−2ikW ) − 2i

(7.246)

Because the piezoelectric element (PZT) induces strain as an actuator in the bar (plate), we will check if the strain εx is magnified at a far field (x >> W ) due to the installation of resonators. The strain εx is calculated from the displacement u as εx =

∂u = −ikU ei(ωt−kx)  Sei(ωt−kx) , ∂x

(7.247)

where the complex-valued strain amplitude S can be explicitly written as (using Eq. (7.244)) 2Finp sin k L z 0 ω α(1 − e−2ikW ) − 2i 2Finp sin k L . =− E A0 α(1 − e−2ikW ) − 2i

S = −k

(7.248)

The last expression in Eq. (7.248) is obtained using z 0 given in Eq. (7.232) and √ the relation ω = kc = k E/ρ. Evidently, S depends on ω, L, W , and the resonator parameters. The amplitude S0 of the nominal strain in a bar without installed resonators is obtained by placing α = 0 in Eq. (7.248): S0 = −ikU0 = −i

Finp k Finp sin k L = −i sin k L . z0 ω E A0

(7.249)

Using S0 in Eq. (7.249), one may write S as S = S0

−2i . α(1 − e−2ikW ) − 2i

(7.250)

Note the magnitude |S0 | becomes maximized at15 k L = π/2, i.e. f =

c  fT . 4L

where the frequency that maximizes |S0 | is defined as f T :

15

The frequency maximizing |S0 | has nothing to do with resonator installation. It depends on the actuation mechanism by a piezoelectric patch element.

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

fT =

c (the corresponding wavelength: 2L). 4L

187

(7.251)

At f = f T , |S0 |max becomes |S0 |max = |S0 | f = f T =

   Finp  E A0

   f   S0 T .

(7.252)

This analysis indicates that the maximum output is obtained upon selecting the excitation frequency in such a way that the PZT actuator length is half the wavelength of the excitation harmonic wave. Therefore, the excitation frequency maximizing the strain output by a PZT patch of a specific size (L) can be uniquely determined. We will now analyze the wave behavior observed in the original model in Fig. 7.29a using an equivalent model displayed on the right side of Fig. 7.29b. For the equivalent model, we must estimate the effective impedance z of the metamaterial bar region between x = −W  and x = W  . Because the effective material properties of the metamaterial bar segment are different from those of the nominal bar, the magnitude F˜inp of the applied force should be adjusted to ensure that the power input to the system by Finp is equal to the power input to the system by F˜inp . We also need to determine the size W  of the effective metamaterial bar segment because W  cannot be directly determined when discrete resonators are attached as in the present case and, therefore, will present the analysis to determine F˜inp , W  , and z. To analyze the wave motion using the equivalent model in Fig. 7.29b, we use the symbol (&) and write the displacement field as ⎧ − U˜ eikx for x ≤ −W  ⎪ ⎪ ⎪ ⎪ ⎪ ikx −ikx ⎪ for − W  ≤ x ≤ −L ⎪ ⎨ − u˜ 2 e − u˜ 3 e . u˜ = eiωt u˜ 1 e−ikx − u˜ 1 eikx for − L ≤ x ≤ L ⎪ ⎪ ⎪ u˜ e−ikx + u˜ eikx for L ≤ x ≤ W  ⎪ 2 3 ⎪ ⎪ ⎪ ⎩ ˜ −ikx for x ≥ W  Ue

(7.253)

As the size L of the PZT patch remains the same in the equivalent and the original systems, the wavenumber k for the region of −L ≤ z ≤ L should be the same in the equivalent and original systems. Using the field symmetry with respect to x = 0, the subsequent analysis will be primarily conducted for x ≥ 0. The continuity conditions at x = L can be written as displacement continuity: u˜ 1 e−ik L − u˜ 1 eik L = u˜ 2 e−ik L + u˜ 3 eik L ,

(7.254)

  equilibrium: − i zω u˜ 1 e−ik L + u˜ 1 eik L = F˜inp − i zω u˜ 2 e−ik L − u˜ 3 eik L . (7.255) The continuity conditions at x = W  yield

188

7 Longitudinal Waves in 1D Continuum Bars

displacement continuity: u˜ 2 e−ikW + u˜ 3 eikW = U˜ e−ikW ,

(7.256)

     equilibrium: − i zω u˜ 2 e−ikW − u˜ 3 eikW = −i z 0 ωU˜ e−ikW .

(7.257)







Our approach for evaluating F˜inp , W  , and z is to ensure that the wave field in the equivalent system is equal to that in the original system with two point resonators installed. Accordingly, we require that the following conditions should be fulfilled: U˜ = U

 for x ≥ W  ,

(7.258)

u˜ 1 u˜ 2 u˜ 3 = = = g(ω) = |g(ω)|eiθ , u1 u2 u3

(7.259)

F˜inp = h(ω) = |h(ω)|eiγ , Finp

(7.260)

where g(ω) and h(ω) are unknown functions of ω to be determined. The substitution of Eqs. (7.259) and (7.260) into Eq. (7.254) recovers Eq. (7.230), which represents the displacement continuity in the original system. The substitution of Eqs. (7.259) and (7.260) into Eq. (7.255) yields −i zω

g(ω)  −ik L g(ω)  −ik L u1e u2e + u 1 eik L = Finp − i zω − u 3 eik L . h(ω) h(ω)

(7.261)

For Eq. (7.261) to represent Eq. (7.231), the following condition must hold g(ω) z0 = . h(ω) z

(7.262)

Now, we require that the input power Pin |orignal to the original system by the external force Finp should be equal to the input power Pin |equiv of the equivalent system by the external force F˜inp : Pin |orignal = Pin |equiv ,

(7.263)

where (  ∗ ( 1 ' 1 ' ∗ Re Foriginal voriginal = Re Finp iω(u 1 e−ik L − u 1 eik L ) x=L 2 2  1  −iη (7.264) = Finp ω sin k L|u 1 | cos η, = Re Finp 2ω sin k L|u 1 |e 2

Pin |original =

and

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

( 1 ' ∗ Re Fequiv vequiv x=L 2  ∗ ( 1 ' = Re Finp h(ω) iωg(ω)(u 1 e−ik L − u 1 eik L ) 2 = Finp ω sin k L|u 1 | cos(η + θ − γ )|h(ω)||g(ω)|.

189

Pin |equiv =

(7.265)

In Eqs. (7.264) and (7.265), v denotes the particle velocity, Foriginal = Finp eiωt , Fequiv = F˜inp eiωt , and u 1 is expressed as u 1 = |u 1 |eiη . The symbol * denotes the complex conjugate. For Eq. (7.263) to be valid, the following relations should hold satisfied: |h(ω)||g(ω)| = 1,

(7.266)

θ = γ.

(7.267)

By using Eqs. (7.262), (7.266), and (7.267), and noting that g(ω) = |g(ω)|eiθ and h(ω) = |h(ω)|eiγ , we can determine g(ω) and h(ω) as g(ω) = h(ω) =

z 0 iθ e , z

(7.268)

z iθ e , z0

(7.269)

where z and θ are yet to be determined. To determine z, we eliminate U˜ from Eqs. (7.256) and (7.257) to obtain 

z − z0 u˜ 3 eikW .  = −ikW u˜ 2 e z + z0

(7.270)

If Eq. (7.259) is used, Eq. (7.270) becomes 

u 3 eikW z − z0 = . u 2 e−ikW  z + z0

(7.271)

The ratio between u 3 and u 2 appearing in Eq. (7.271) can be explicitly calculated using the solution in the original system. To this end, we use Eqs. (7.233)–(7.235). Based on Eq. (7.235), we derive the following relation: s(u R − u Q ) =

msω2 sω2 uQ = 2 uQ. 2 s − mω ω R − ω2

Upon combining Eqs. (7.233), (7.234), and (), we obtain

(7.272)

190

7 Longitudinal Waves in 1D Continuum Bars



 sω2 − i z0 ω u Q ω2R − ω2    sω2 = − i z 0 ω u 2 e−ikW + u 3 eikW . (7.273) 2 2 ωR − ω

 −i z 0 ω u 2 e−ikW − u 3 eikW =

The rearrangement of Eq. (7.273) yields7.272 u 3 2ikW u 3 eikW = ·e  r, −ikW u2e u2

(7.274)

where r is explicitly expressed as r = |r |eiβ = |r |(cos β + i sin β) =

sω . 2i z 0 (ω2R − ω2 ) − sω

(7.275)

Eventually, considering Eqs. (7.271) and (7.274), the following relation can be established:   z 1 + r e2ik(W −W ) 1 + |r |ei [β+2k(W −W )] = = . z0 1 − r e2ik(W  −W ) 1 − |r |ei[β+2k(W  −W )]

(7.276)

To demonstrate that z is positive and real-valued, we use the fact that the power conservation relation should be valid through x = W for the original system. We also examine the power conservation relation through x = W  for the equivalent system. For the original system, we obtain (e.g., Eq. 7.59) 1 2 1 1 ω z 0 |u 2 |2 = ω2 z 0 |u 3 |2 + ω2 z 0 |U |2 for the original system, 2 2 2

(7.277)

and for the equivalent system, we obtain  2 1 2 1 1   ω Re(z)|u˜ 2 |2 = ω2 Re(z)|u˜ 3 |2 + ω2 z 0 U˜  for the equivalent system. 2 2 2 (7.278) Substituting Eqs. (7.258), (7.259) and (7.268) in Eq. (7.278) yields 1 2 1 1 ω Re(z)|g(ω)u 2 |2 = ω2 Re(z)|g(ω)u 3 |2 + ω2 z 0 |U |2 , 2 2 2 which becomes 1 2 Re(z) 1 Re(z) 1 ω z 0 |u 2 |2 = ω2 z 0 |u 3 |2 + ω2 z 0 |U |2 . |z| |z| 2 2 2

(7.279)

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

191

Because Eq. (7.279) should be equal to Eq. (7.277), the following relation must hold: Re(z) = 1. |z|

(7.280)

From this relation, one can see that z is positive and real-valued. To impose this condition on z in Eq. (7.276), the following conditions should be satisfied: 



β + 2k(W − W ) =

2nπ (n : integer). (2n + 1)π

(7.281)

If β + 2k(W  − W ) = (2n + 1)π , i.e., if W  can be expressed as W = W +

1 [(2n + 1)π − β], 2k

(7.282)

Equation (7.276) yields z 1 − |r | = < 1. z0 1 + |r |

(7.283)

As mentioned at the beginning of this section, the effective impedance z must be smaller than z 0 to increase the sensitivity. Therefore, the case expressed by Eqs. (7.282) and (7.283) should be chosen. If β + 2k(W  − W ) = 2nπ , on the other hand, Eq. (7.276) yields z/z 0 = (1 + |r |)/(1 − |r |) > 1 because z > z 0 . Therefore, this case is discarded. To evaluate g(ω) (i.e., |g(ω)| and θ ) in Eq. (7.259), we calculate U˜ /u˜ 2 = U/gu 2 , using Eqs. (7.270) and (7.257) as follows: 2z/z 0 U 2z 2(1 − |r |)/(1 + |r |) U˜ = = 1 − |r |. (7.284) = = = u˜ 2 gu 2 z0 + z 1 + z/z 0 1 + (1 − |r |)/(1 + |r |) From Eq. (7.284), g(ω) is determined as g=

U 1 · . u 2 1 − |r |

(7.285)

The term U/u 2 in Eq. (7.285) can be obtained by dividing Eq. (7.233) by u 2 exp(−ikW ) and using the definition of r in Eq. (7.274), as follows: U u3 = 1 + e2ikW = 1 + r. u2 u2 Substituting Eq. (7.286) into Eq. (7.285) yields

(7.286)

192

7 Longitudinal Waves in 1D Continuum Bars

g(ω) =

1+r 1 + |r |(cos β + i sin β) =  |g(ω)|eiθ . 1 − |r | 1 − |r |

(7.287)

To explicitly express |g(ω)| and θ , the following relationship should be used: 1 = |r |2 + |1 + r |2 .

(7.288)

This relation can be obtained by substituting |u 3 /u 2 | = |r | from Eq. (7.274) and |U/u 2 | = |1 + r | in Eq. (7.286) into the energy conversation relation, Eq. (7.277). Considering the absolute value of g(ω) in Eq. (7.287) yields |1 + r | |g(ω)| = = 1 − |r |



1 − |r |2 = 1 − |r |



1 + |r | = 1 − |r |



z0 , z

(7.289)

where the third expression is obtained by using Eq. (7.288) and the last expression results from Eq. (7.283). To obtain the explicit expression for θ , we first express β appearing in Eq. (7.287) as cos β = −|r |.

(7.290)

Equation (7.290) can be obtained by substituting r = |r | cos β + i|r | sin β into Eq. (7.288). According to the definition of r in Eq. (7.275), the range of β varies depending on ω as 

π < β < 3π/2 if ω < ω R . π/2 < β < π if ω > ω R

(7.291)

By using Eqs. (7.287) and (7.290), θ can be expressed as tan θ =

|r | sin β − cos β sin β 1 = =− . 2 1 + |r | cos β 1 − cos β tan β

(7.292)

Using Eq. (7.292) and the range of β, θ can be explicitly expressed as  θ=

β − π/2, for ω < ω R . β + π/2, for ω > ω R

(7.293)

This result can be more concisely represented as θ = β + pπ/2, p = sign(ω − ω R ).

(7.294)

Then, g(ω) and h(ω) are explicitly written as g(ω) =

z 0 i(β+ pπ/2) , e z

(7.295)

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

h(ω) =

z i(β+ pπ/2) e , z0

193

(7.296)

where z is given in Eq. (7.283) (with r in Eq. (7.275)) and β is expressed in Eqs. (7.290) and (7.291). To explicitly derive U = U˜ (at a far field for x  W  ), a similar procedure employed to derive Eq. (7.246) can be used. Based on Eqs. (7.256) and (7.257), u˜ 2 and u˜ 3 can be expressed in terms of U˜ as u˜ 2 =

    z0 ˜ z 0 ˜ −2ikW  1 1 1+ 1− . U , u˜ 3 = Ue 2 z 2 z

(7.297)

Thus, expressing u˜ 1 using Eq. (7.254) as u˜ 1 = (u˜ 2 e−ik L + u˜ 3 e+ik L )/(e−ik L − e+ik L ) and substituting it into Eq. (7.255) yields F˜inp e−ik L + eik L  −ik L + u˜ 2 e−ik L − u˜ 3 eik L , u˜ 2 e + u˜ 3 eik L = − −ik L ik L e −e i zω which can be simplified as F˜inp 2 . + u ˜ u ˜ = − ( ) 2 3 e−ik L − eik L i zω

(7.298)

Substituting Eq. (7.297) into Eq. (7.298) yields −

1 2i sin k L

     F˜inp z0 z 0 −2ikW  ˜ 1+ + 1− e U =− . z z −i zω

(7.299)

From Eq. (7.299), U˜ can be expressed as U˜ (= U ) =

2 F˜inp sin k L  zω 1+

z0 z



1  + 1−

z0 z

e−2ikW 

 F˜inp sin k L z 0 eikW z0 ω i z 0 sin kW  + z cos kW  √  zz 0 ei(kW +β+ pπ/2) = U0 , i z 0 sin kW  + z cos kW 

=

(7.300)

where p = sign(ω − ω R ) and U0 denotes the nominal displacement defined in Eq. (7.245). Using U in Eq. (7.300) and the definition of strain, the strain magnitude S can be expressed as

194

7 Longitudinal Waves in 1D Continuum Bars

S = S˜ = S0

√



zz 0 ei(kW +β+ pπ/2) , i z 0 sin kW  + z cos kW 

p = sign(ω − ω R ),

(7.301)

which yields √ |S| = |S0 |

z 02

sin

2

kW 

zz 0 . + z 2 cos2 kW 

(7.302)

Equation  (7.302) shows that |S| can be maximized if both |S0 | and √ zz 0 / z 02 sin2 kW  + z 2 cos2 kW  are simultaneously maximized. In the first place, we should opt for f = f T = c/4L to maximize|S0 |based on the results stated in  f  Eq. (7.252). This maximum value is denoted by S0 T . (Note that |S0 | is the output without installing a resonator.) To observe  the behavior S as a function of ω (or f ),  fT  the normalized strain magnitude S/S0  can be expressed as      S (ω)   S(ω)   0     f T  = K (ω, m, s, W ) f T ,  S   S  0

(7.303)

0

where √ K (ω, m, s, W )  

zz 0

z 02 sin2 kW  + z 2 cos2 kW 

.

(7.304)

The examination of the term K (ω, m, s, W ) in Eq. (7.304) suggests that if the following condition is satisfied: sin kW  = 0,

(7.305)

K (ω, m, s, W ) in Eq. (7.304) becomes K (ω, m, s, W )  Therefore, |S| can be amplified by a factor of amplitude |S0 |, |S| =

z0 . z

 ) z 0 z > 1 compared to the nominal

z0 |S0 | > |S0 |. z

(7.306)

The condition in Eq. (7.305) represents the Fabry-Pérot resonance condition derived earlier as Eq. (7.96). Because W  depends on several parameters (ω, m, s, and W ), both f = f T and sin kW  = 0 can be simultaneously satisfied (where ω is set as

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

195

 2π f T and other parameters are adjusted to  sin kW = 0). Therefore, the strain  satisfy  fT  output magnitude |S| can be larger than S0 . Let us investigate the extent of increase in |S| achieved by installing a pair of frequency-tuned resonators on a bar carrying longitudinal waves. As a specific example, we consider utilizing the U-channel box aluminum beams as the resonators installed on a base aluminum plate, as depicted in Fig. 7.28b. The geometries and material properties of the plate and box beam are as follows:

t(base plate thickness) = 2 mm, z 0 = 28350 kg/s, c = 5250 m/s, t R = 3 mm, w R = 6 mm, b R = 1.5 mm, h R = 4.5 mm, 2W = 78.8 mm, L = 18.3 mm, E = 69 GPa, ν = 0.3, ρ = 2700 kg/m3 .

(7.307)

Using the data in (7.307), f T can be found as f T = c/4L = 71.7 kHz. The mass (m) and stiffness (s) of the resonator and its eigenfrequency ( f R ) corresponding to the longitudinal wave mode in the plate can be estimated from a finite element analysis (Kim et al. 2018) as: m = 103.0 g, s = 35.3 GN/m, f R = 93.1 kHz.

(7.308)

   f  Based on the listed data, S/S0 T  and z/z 0 are plotted as the functions of the excitation frequency f in Fig. 7.30a and b, respectively. As the phase velocity (c) remains constant in the equivalent system, its effective material properties ρ and E behave identically with the function of frequency, as displayed in Fig. 7.30b. Evidently, z/z 0 approaches zero as f approaches f R . As observed in Fig. 7.30a, the signal magnification factor reaches the value of 4.24 at f = f T :    f  |S| = 4.24 S0 T  at f = f T = 71.7 kHz.

(7.309)

As the magnification aims to satisfy sin kW  = 0 (i.e., the Fabry–Pérot resonance condition is satisfied), additional magnifying frequencies are presented in Fig. 7.30a. These Fabry–Pérot resonance frequencies (satisfying sin kW  = 0) can be more prominently identified by examining the transmission coefficient |T | = |C/A| depicted in Fig. 7.30c. In the figure, A, B, and C denote the amplitudes of the incident, reflected, and transmitted waves, respectively, in the case when a slab of width 2W  and impedance z is assumed to be inserted into a background homogeneous medium of impedance z 0 . (Refer to the analysis model described in the figure inset.)  f  The effects of f R on S/S0 T  near f = f T = 71.7 kHz are presented in Fig. 7.30d. As f R approaches f T , |S| increases rapidly because zr = z/z 0 diminishes and the quality

196

a

7 Longitudinal Waves in 1D Continuum Bars S

5

d

fT = 71.7 kHz

4

C A

93.1 kHz

Nominal (No resonator)

2

0

C A

Effective properties (Arb. Units)

C A

0

0

e

fR = 111.8 kHz (

C A

= 0.18)

fT = 71.7 kHz

68

70

72 Frequency

fR = 93.1 kHz (fixed) W = 38.4 mm

74

76 kHz

W = 37.5 mm

W = 39.4 mm W = 40.5 mm

fT = 71.7 kHz

C A

1

4

2

0.5 2W

0

6

200 kHz

Frequency

93.1 kHz = 0.056)

5

E / E0

0.5

C A

Without resonators (S = S0)

z / z0 ρ / ρ0

fR = 93.1 kHz

0

f (

fR = 85.7 kHz C ( A = 0.023)

200 kHz

Frequency

1

0

c

fT 0

fR = 82.0 kHz C ( A = 0.012)

3

1

b

( S = z/z0 at f = fT)

10

A B

fR = 93.1 kHz 0

Frequency

z0

C

200 kHz

0

Without resonators

68

70

fT = 71.7 kHz

72 Frequency

74

76 kHz

Fig. 7.30 Variations in strain fields and effective properties as the function of frequency using the model depicted in Fig. 7.28b. a Radiated strain field |S|, b effective impedance z (density: ρ and stiffness: E), c transmission coefficient in the bar system shown in the inset, d effects of resonance frequency f R on |S|, and e effects of distance W between two resonators on |S|. (Kim et al. 2018)

factor Q (Eq. 2.48) enlarges. Therefore, by tuning the value of z using an appropriate value for  f R , a tradeoff can be achieved between the amplitude and bandwidth in   fT  S/S0  at the target frequency of f T . The influence of the distance (2W ) between the two resonators on |S| is depicted in Fig. 7.30e, where f R is assumed to be fixed. In the case of varying only W, the effective impedance z does not vary. However, the Fabry– Pérot resonance frequencies in the medium of impedance z in the region confined between 2W  are varied, because W  varies with W , as expressed in Eq. (7.282). Therefore, the peak frequency of the locally maximized |S| is significantly affected by W. Note that the phenomenon in Fig. 7.30e cannot be observed if only a single resonator is installed, because it only functions as a dynamic absorber (see, e.g., Tse et al. 1963). The wave interference occurring between the paired resonators is unique because it can reduce the effective impedance z of the region surrounded by the resonators, even rendering it near zero. Because the bar region surrounded by the resonators has unique frequency-dependent material properties due to the resonators, the region is regarded to be made of a metamaterial. The phenomenon of the actuator output magnification using a pair of resonators is evidenced by experiments, as presented in Fig. 7.31. This experiment was designed to estimate the effective impedance z and demonstrate the magnification of |S| in the radiated wave field. The magnitude |S| was measured for frequencies between 68 and 76 kHz covering the target frequency f T = 71.7 kHz and normalized with

7.16 Lowered Effective Impedance by Resonators (Advanced Topic)

197

Fig. 7.31 a Experimental setup for longitudinal wave experiments in a thin plate. Three PZT patches were used to ensure the generation of a plane longitudinal wave mode in the plate to simulate a longitudinal wave in a bar. b Comparison of experimental and theoretical/numerical results for the radiated far-field strain |S|. (The geometric and material data are expressed in Eqs. (7.307) and (7.308)) (Kim et al. 2018)

   f  respect to S0 T . For the detailed experimental procedure, refer to Kim et al. (2018). As portrayed   in Fig. 7.31b, |S| at f = f T = 71.7 kHz is increased by 307% compared  f  with S0 T . The finite element result obtained without considering any damping effect is denoted as “FEM” in Fig. 7.31b, which is adequately consistent with the theoretical result calculated using Eq. (7.309). COMSOL Multiphysics was used for the simulation. To account for the damping effect occurring in the experiment, the loss factor of 0.065 was estimated. The results denoted by “FEM + Damping” match well with the experimental results. Furthermore, the impedance value of z|exp from the experimental results at f = f T can be extracted using the following formula:  2  2  S0f T  z = sin k L   ,  S  z0

(7.310)

   f  which is valid if sin kW  = 0. If the value of S/S0 T  estimated from Fig. 7.31b is substituted into Eq. (7.310), is estimated as z/z 0 |exp ≈ 0.053. The influence of damping is considered for the estimation. This value agreed well with the t z/z 0 |exp heoretical value z/z 0 |Theory = 0.056 at f = f T .

198

7 Longitudinal Waves in 1D Continuum Bars

7.17 Problem Set Problem 7.1. Prove that the characteristic impedance Z for transverse waves propagating in a taut string is Z = ρc, where Z is defined as the ratio of the complex amplitude of force Fy to the complex amplitude of vertical velocity v y in the string. Problem 7.2. Consider the two-bar system discussed in Sect. 7.4. If Z 2 is assumed as zero for simulating a traction-free end at x = 0 for Bar 1, we obtain that A2 = 2A1 . Prove that the power flow in the Bar 2 of Z 2 = 0 is identically zero, even with the nonzero-value of A2 . Problem 7.3. Consider a uniform bar (ρ, E, S) in which a harmonic wave in the form of u x = Aei(ωt−kx) propagates. Prove that ∂UT /∂t = 0 for the bar segment bounded by x = 0 and x = L. (Hint: Pout = Pout x=0 + Pout x=L = 0 and use Eq. (7.54)). This problem shows that unless an energy source or a dissipation mechanism exists inside the bar segment, the power flow at every cross section remains the same. Problem 7.4. The result that det T = 1 is proven in Sect. 7.10. Alternatively, one can prove det T = 1 examining the power flows at locations x and x + d because the power flow at every cross section remains constant unless an energy source or a dissipation mechanism exists in a bar. To this end, calculate the time-averaged rightward power flows < P >x and < P >x+d , and use < P >x = < P >x+d . Problem 7.5. Derive the power balance equation, Eq. (7.64) at the junction of two dissimilar bars with distinct characteristic impedances. Use Eqs. (7.36) and (7.40). Problem 7.6. The Fabry–Pérot resonance phenomenon can be understood from the interference perspective involving an infinite number of reflections and transmissions, as illustrated in the figure below. To this end, we consider the reflection and transmission at the interfaces x = 0 and x  = 0. Assume that the impedances of the base medium 1 and the inserted medium 2 are Z 1 and Z 2 , respectively. We will use symbol ξ = Z 2 /Z 1 to denote the impedance ratio.

7.17 Problem Set

199

Fabry–Pérot resonance phenomenon analyzed from interference perspective. a Illustration of infinite reflections and transmissions at interfaces at x = 0 and x  = 0. b and c reflection and transmissions at x = 0 and x  = 0, respectively.

(a) Using the wave fields considered in the figure above, show that the reflection (r0 ) and transmission (t0 ) coefficients at x = 0 for the incident wave from the base medium 1 to medium 2 and the reflection (r  ) and transmission (t  ) coefficients at x  = 0 for the incident wave from medium 2 to medium 1 are expressed as r0 =

1−ξ 2 ξ −1  2ξ , t0 = ; r = ,t = . 1+ξ 1+ξ 1+ξ 1+ξ

(b) Prove that the transmission coefficient t1 of the initial incident longitudinal wave from medium 1 on the left to the same medium on the right through medium 2 can be given by t1 = t0 (e−ik2 x )x=d t  = t0 t  e−ik2 d . (c) Prove that the transmission coefficient t2 for a wave experiencing a reflection at x  = 0 and another reflection at x = 0 is 

t2 = t0 (e−ik2 x )x=d r  (e+ik2 x )x  =−d r  (e−ik2 x )x=d t 

200

7 Longitudinal Waves in 1D Continuum Bars

   = (r  )2 t0 t  e−3ik2 d = t0 t  e−ik2 d (r  )2 e−2ik2 d   = t1 (r  )2 e−2ik2 d  t1rˆ , where rˆ = (r  )2 e−2ik2 d . (d) Referring to Figure (a) above, establish that t j = t j−1rˆ ( j = 2, 3, . . .). (e) Then show that the total transmission T becomes t=

∞

j=1

tj =

to t  e−ik2 d . 1 − (r  )2 e−2ik2 d

(f) Prove that the imposition of full transmission (|T | = 1) yields e−2ik2 d = 1 → k2 d = nπ , or d =

n λ2 (n = 1, 2, 3, . . .). 2

Problem 7.7. Determine the eigenfrequencies and eigenmodes for a bar of length l with one end fixed at x = 0 and the other free at x = l. The motion of the bar is assumed to be longitudinal. Problem 7.8. There is an infinitely long periodic structure carrying longitudinal waves, as shown in the figure below. Each unit cell consists of two dissimilar layers (layer a and layer b).

Infinitely long periodic structure carrying longitudinal waves

7.17 Problem Set

201

(a) Plot the dispersion curve for ω ∈ [0, 3 × 105 ] rad/s. Use ω as the x-axis, Reβ as the positive y-axis, and Imβ (β = kd) as the negative y-axis. For the plot, use the following data: ρa = 2713 kg/m3 , ρb = 1050 kg/m3 , A = π(0.05)2 m2 , E a = 70GPa, E b = 2.3GPa, da = 0.04m, db = 0.06m. (b) Assume that the periodic structure is finite. Calculate the transmittance (T = |t|2 ) through a single unit cell (N = 1) where the unit cell (consisting of layers a and b) is inserted in a base medium (medium 1, ρ1 = 2200 kg/m3 , E 1 = 50 GPa). Use Eq. (7.215b) to calculate t and S in Eq. (7.209) with T2 given by T2 = T(b) · T(a) . The matrix T( j) is the transfer matrix for layer j ( j = a, b). Plot T for ω ∈ [0, 0.75 × 105 ] rad/s. (c) Plot T in the range of ω ∈ [0, 0.75×105 ] rad/s when N = 3 and N = 10. Make some comments on the obtained results by comparing them with the dispersion curve obtained in part (a). Problem 7.9. Consider an infinitely long one-dimensional elastic rod (density: ρ, Young’s modulus: E, cross-sectional area: A). Assume that a harmonic wave (exp[i(ωt − kx)]) is incident from −∞. Determine the reflection (r ) and transmission (t) coefficients when a point mass or (and) spring is (are) inserted in the rod as described below. (a) A point mass M is inserted in the rod at x = 0. Determine r and t. Use the symbols Z = ρc A and q = Mω/2Z in your results. Examine what happens as M → 0 or M → ∞. What are the physical significances of these limits?

Rod with a point mass M inserted

(b) A point spring S is inserted in the rod at x = 0. Determine r and t. Use the symbols Z = ρc A and p = 2S/ωZ in your results. Examine what happens as S → 0 or S → ∞. What are the physical significances of these limits?

202

7 Longitudinal Waves in 1D Continuum Bars

Rod with a point spring S inserted

(c) A point mass M is inserted at x = 0 and a point spring is inserted at x = d in the rod, as shown below. Referring to the definitions of a and b in the figure below, derive four equations from which the coefficients r, t, a, and b can be determined. Use the symbols M = Mω2 /jk E A and S = S/jk E A in your result.

Rod with a point mass M and a point spring S inserted

References Achenbach JD (1976) Wave propagation in elastic solids. North-Holland Bloch F (1929) Uber die Quantenmechanik der Electronen in Kristallgittern. Z Phys 52:555–600 Bragg WH, Bragg WL (1913) The reflection of X-rays by crystals. In: Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, vol 88, pp 428–38 Chaudhry Z, Rogers CA (1994) The pin-force model revisited. J Intell Mater Syst Struct 5:347–354 COMSOL Multiphysics®, www.comsol.com. COMSOL AB, Stockholm, Sweden Elmore WHM, Heald M (1969) Physics of waves. Dover Floquet G (1883) On linear differential equations with periodic coefficients. Annales Scientifiques De L’école Normale Supérieure 12:47–88 Hussein MI, Hulbert GM, Scott RA (2006) Dispersive elastodynamics of 1D banded materials and structures: analysis. J Sound Vib 289:779–806 Hussein MI, Hulbert GM, Scott RA (2007) Dispersive elastodynamics of 1D banded materials and structures: design. J Sound Vib 307:865–893 Kim K, Park CI, Lee H, Kim YY (2018) Near-zero effective impedance with finite phase velocity for sensing and actuation enhancement by resonator pairing. Nat Commun 9:1–10 Kinsler LE, Frey AR, Coppens AB, Sanders JV (2000) Fundamentals of acoustics. John Wiley & Sons Kittel C (2004) Introduction to solid state physics, 8th edn. Wiley

References

203

Knobel R (2000) An introduction to the mathematical theory of waves. American Mathematical Society Lee H, Oh JH, Seung HM, Cho SH, Kim YY (2016) Extreme stiffness hyperbolic elastic metamaterial for total transmission subwavelength imaging. Sci Rep 6:24026 Miklowitz J (1978) The theory of elastic waves and waveguides. North-Holland Oh JH, Kwon YE, Lee HJ, Kim YY (2016) Elastic metamaterials for independent realization of negativity in density and stiffness. Sci Rep 6:23630 Pérot A, Fabry C (1899) On the application of interference phenomena to the solution of various problems of spectroscopy and metrology. Astrophys J 9:87–115 Rose JL (2014) Ultrasonic guided waves in solid media. Cambridge University Press Sneddon IN (1995) Fourier transforms. Courier Corporation Tse FS, Morse IE, Hinkle RT (1963) Mechanical vibrations. Allyn and Bacon

Chapter 8

Flexural Waves in a Beam

A beam is a long, slender member with bending as its predominant mode of deformation. Beams and strings both experience transverse displacements, but beams exhibit a structural resistance known as flexural (or bending) rigidity, whereas strings do not. Figure 8.1 depicts a beam whose height (h) and width (b) are significantly smaller than its length (L). For the purposes of its efficient mechanical analysis, it can be considered a one-dimensional member. When a beam is subjected to a timedependent vertical force (Fy ) or bending moment (Mz ), waves can propagate through it. These waves are known as flexural waves.1 Depending on the degree of approximation, either the Euler–Bernoulli beam theory or the Timoshenko beam theory may be employed. (For a comparative analysis of these theories, see, e.g., Labuschangne et al. (2009)) After explaining the underlying wave phenomena of flexural waves in a beam using these beam theories, the manipulation of flexure waves using the metamaterial concept will be discussed.

8.1 Wave Analysis by Euler–Bernoulli Beam Theory Figure 8.1 depicts the underlying kinematics adopted in a beam theory. The transverse displacement of the neutral axis2 (or middle plane) in the y-direction is denoted by v(x, t), and the rotation of the surface normal to the neutral axis is denoted by φ(x, t), as depicted in Fig. 8.2. In the rectangular beam shown in Fig. 8.1, the neutral axis is the x-axis. As h > b, h) exhibiting dominant flexural motion in the y-direction



u x (x, y, t) = −yφ(x, t), u y (x, y, t) = v(x, t).

(8.1)

Using Eq. (8.1), the strain εx (x, y, t) can be expressed as ∂φ(x, t) ∂u x (x, y, t) = −y ∂x ∂x  −yκ(x, t),

εx (x, y, t) =

(8.2)

where κ represents the curvature of the neutral axis as follows: κ(x, t) =

∂φ(x, t) . ∂x

(8.3)

Accordingly, the normal stress σx at a generic point can be expressed as σx (x, y, t) = −E yκ(x, t),

(8.4)

where E indicates Young’s modulus. The shear strain γ (x, t) along the neutral axis can be defined as

Fig. 8.2 Deformed shape of a beam (If the Euler–Bernoulli beam theory is used, the rotation φ of the surface normal is set equal to the rotation ∂v/∂ x of the neutral axis.)

8.1 Wave Analysis by Euler–Bernoulli Beam Theory

γ (x, t) =

207

∂v(x, t) − φ(x, t), ∂x

(8.5)

where ∂v(x, t)/∂ x denotes the rotation (angle) of the neutral axis. Generally, γ (x, t) cannot be zero if a beam is subjected to a vertical force, but in the Euler–Bernoulli beam theory (especially for very thin beams), the shear strain is assumed to be zero,4 resulting in φ(x, t) =

∂v(x, t) . ∂x

(8.6)

Therefore, the surface normal to the neutral axis is assumed to remain normal to the neutral axis for all instances in the Euler–Bernoulli beam theory and only v(x, t) is treated as the independent variable. Note that in more advanced beam theories5 than the Euler–Bernoulli beam theory, such as the Timoshenko beam theory (to be discussed in Sect. 8.3), the two variables (v(x, t) and φ(x, t)) are treated as independent variables so that the assumption (8.6) is not valid. If Eq. (8.6) is inserted in Eq. (8.3), the curvature for the Euler–Bernoulli beam theory can be expressed only in terms of v(x, t) as follows: κ(x, t) =

∂ 2 v(x, t) . ∂x2

(8.7)

To derive the governing (wave) equation, the equations of motion involving the vertical shearing force V (v, t) and the bending moment M(x, t) acting along the neutral axis of a beam should be considered. As indicated in Fig. 8.3a, one can obtain the following equations considering the equation of motion for an infinitesimal beam element (x): • force equilibrium:

(V + V ) − V + qx = ρ Ax

∂ 2v ∂t 2

(8.8)

• moment equilibrium:

(M + M) − M + (V + V ) 4

x ∂ 2φ x +V = ρI 2 , 2 2 ∂t

(8.9)

The shear stress is not necessarily zero, even if the shear strain is assumed to be zero. If the shear stress is given by Eq. (8.43), the shear stress can still be nonzero for zero shear strain if the shear modulus G is being treated as ∞. 5 There are advanced beam theories using many independent variables, such as a higher-order beam theory developed by Kim et al. (2023).

208

8 Flexural Waves in a Beam

Fig. 8.3 a Free-body diagram of an infinitesimal beam element; b stress acting on an infinitesimal area A producing bending moment M(x, t)

where ρ, I , and A denote the density, rotary inertia, and cross-sectional area of the  beam in consideration. The rotary inertia I is defined as I = A y 2 dA. (We will explain below how this expression is obtained.) In the Euler–Bernoulli beam theory, the term ρ I ∂ 2 φ/∂t 2 will be ignored in Eq. (8.9) because the influence of the rotary inertia is neglected. The equations of motion should be considered in the deformed configuration, but they can be considered in the undeformed configuration in a linear beam theory in which small deformations are assumed. Considering the limit of x → d x, Eqs. (8.8) and (8.9) become ∂ 2v ∂V = ρ A 2 − q, ∂x ∂t

(8.10)

∂M + V = 0. ∂x

(8.11)

and

Referring to Fig. 8.3b, the bending moment caused by σx acting on the cross section of a beam is evaluated as   M = (−σx dA) · y = −(−E yκ)ydA A

⎛

= E⎝



⎞

A

y 2 dA⎠κ = E I κ,

A

where I, the moment of inertia about the z-axis, is defined as

(8.12)

8.1 Wave Analysis by Euler–Bernoulli Beam Theory

209

 I =

y 2 dA.

(8.13)

A

For instance, I becomes • for rectangular cross section (width: b, thickness: h):

h/2

 y dA =

I =

y 2 bdy =

2

−h/2

A

bh 3 , 12

• for circular cross section (radius: a)6 :

 I =

2π a y 2 dA =

(r sin θ )2 (r dr dθ ) 0

A

2π

0

a (sin θ ) dθ ·

=

r 3 dr =

2

0

πa 4 . 4

0

The quantity E I appearing in Eq. (8.12) is called the bending rigidity that serves as the primary stiffness resisting bending motion. Using Eqs. (8.10), (8.11), (8.12), and (8.3), the flexural or bending wave equation in a beam can be expressed in terms of v(x, t) as

∂ 2 v(x, t) ∂ 2 v(x, t) ∂2 = q(x, t). ρA + 2 EI ∂t 2 ∂x ∂x2

(8.14)

To derive the dispersion relation, we assume that EI is not a function of x and also q = 0. Assuming v(x, t) as v(x, t) = X (x)T (t),

(8.15)

and substituting Eq. (8.15) into Eq. (8.14), the following relation is obtained: −

T¨ (t) E I X  (x) = , ρ A X (x) T (t)

(8.16)

To calculate I for a circular cross section, we used the polar coordinates (r, θ) with y = rsinθ and dA = rdrdθ.

6

210

8 Flexural Waves in a Beam

where () = ∂ 4 ()/∂ x 4 and (·· ) = ∂ 2 ()/∂t 2 . Because the left side of Eq. (8.16) is a function of x only and the right side of Eq. (8.16) is a function of t only, Eq. (8.16) can hold only if they become a scalar constant, say, ω2 : −

T¨ (t) E I X  (x) =  −ω2 . ρ A X (x) T (t)

(8.17)

Equation (8.17) can be split into two equations as X  (x) − k 4 X (x) = 0,

(8.18)

T¨ (x) + ω2 T (x) = 0,

(8.19)

where k is related to ω as k4 =

ρA 2 ω . EI

(8.20)

The constant k actually represents the wavenumber. From Eq. (8.20), k is determined as: k = ±β(ω), and k = ±iβ(ω),

(8.21)

where β is defined as β(ω) 

ρA EI

1/4

√

ω.

(8.22)

Equation (8.20) (or Eq. (8.21)) represents the dispersion relation of the flexural wave described by the Euler–Bernoulli beam theory. The dispersion relation (ω − k) relation for bending waves based on the Euler–Bernoulli theory is plotted in Fig. 8.4. The general solutions to Eqs. (8.18) and (8.19) can be written as Fig. 8.4 Dispersion relation (ω − k relation) for bending waves based on the Euler–Bernoulli beam theory for Rek > 0 and Imk > 0

8.1 Wave Analysis by Euler–Bernoulli Beam Theory

211

X ∼ [eiβx , e−iβx , eβx , e−βx ], T ∼ [eiωt , e−iωt ]. Thus, for a given ω, v(x, t) can be expressed as  v(x, t) = Ae−iβx + Be−βx + Ce+iβx + De+βx eiωt ,

(8.23)

where the coefficients A, B, C, and D are complex-valued amplitudes. As the complex-embedded form is used in Eq. (8.23), the actual solution is obtained by taking the real part of v(x, t) in Eq. (8.23). Each of the terms appearing in Eq. (8.23) represents four different waves: Aei(ωt−βx) , Cei(ωt+βx) : propagating waves along the +x- and −x- directions, respectively; Be−βx+iωt , De+βx+iωt : decaying waves along the +x- and −x- directions, respectively. The decaying waves are nonpropagating waves. Using the propagating branch (with real-valued k’s), the phase (c p ) and group velocities (cg ) can be calculated as



E I 1/4 √ EI ω k= cp = = ω, k ρA ρA



E I 1/4 √ EI dω cg = =2 k=2 ω = 2c p . dk ρA ρA

(8.24)

(8.25)

The two velocities are presented as a function of ω in Fig. 8.5a. Although these velocities compare favorably with those calculated by the theory of elasticity (see, e.g., Miklowitz (1978) and Achenbach (1976)) in a low-frequency range, they are not accurate at relatively high frequencies. In particular, the Euler–Bernoulli beam theory predicts cg growing indefinitely as ω approaches ∞. Because energy can be transmitted only at a finite group (energy) velocity, cg |ω→∞ = ∞ violates the law of physics. A remedy to this problem is to use the Timoshenko beam theory, which is elaborated in Sect. 8.3. Fig. 8.5 a Group and phase velocities as the function of frequency (ω) and b frequency-arrival time (ω−t a ). The results are based on the Euler–Bernoulli beam theory

212

8 Flexural Waves in a Beam

Fig. 8.6 Comparison of wave propagation phenomena in dispersive and nondispersive wave systems for a harmonic pulses and b arbitrary-shaped pulses

As evident from Eqs. (8.24) and (8.25), the bending wave is dispersive, i.e., c p and cg are frequency-dependent. In Fig. 8.5b, the arrival time ta = d/cg of a wave with the harmonic component ω is presented, where d indicates the distance between the excitation and sensing points. (Recall that the longitudinal wave in a onedimensional continuum bar studied in the previous chapter is nondispersive because c p is frequency-independent.) To investigate the effects of wave dispersion on measured signals at a far field, the results are depicted in Fig. 8.6. The propagation of two pulses with the center frequencies at ω1 and ω2 (ω2 > ω1 ) is illustrated in Fig. 8.6a. Unlike the nondispersive wave system, the pulse centered at ω2 propagates faster than the pulse centered at ω1 in the dispersive wave system. For the illustrations in Fig. 8.6, the dispersion is based on the result of the flexural wave displayed in Fig. 8.5, where a wave of a higher-frequency component travels faster. Suppose the traveling distance is cg (ω)t ∗ , where t ∗ is elapsed time. In this case, the traveling distance of the pulse centered at ω2 (ω2 > ω1 ) is longer than that of the pulse centered at ω1 because of the dispersion relation in Fig. 8.5. Therefore, an arbitrarily shaped pulse experiences “dispersion,” i.e., it is distorted at later time t ∗ in a dispersive wave system, as depicted in Fig. 8.6b. On the other hand, the pulse shape is preserved at any instance if the wave system is nondispersive. (The longitudinal wave in a continuum bar studied in the previous chapter is nondispersive.) The dispersive nature of bending waves is presented in Fig. 8.7. The shape of the signal measured 10 cm away from the impact site is different from that measured 20 cm away due to dispersion. The dispersion effects7 are prominently observed in the propagating waves toward the end and the reflected waves from the end. 7

The dispersive characteristics of the flexural waves similar to those given in Fig. 8.7b, c can be effectively analyzed using the continuous wavelet transform (Kim and Kim 2001).

8.2 Reflection and Transmission

213

Fig. 8.7 Flexural wave experiments in a simply-supported beam subjected to a ball-drop impact. a Experimental setup. b Strain measured at 10 cm and c strain measured at 20 cm from the excitation point

8.2 Reflection and Transmission Due to the dispersive nature of bending waves, the phenomena of reflection and transmission in a beam are different from those observed in a bar carrying nondispersive longitudinal waves. To study the wave reflection phenomena, we consider four boundary conditions depicted in Fig. 8.8. They are explicitly stated as follows. (a) Simply supported (or hinged) end:

Fig. 8.8 Various boundary conditions at x = 0. a Simply-supported, b free, c clamped, d elastically supported for the vertical motion, and e elastically supported for the rotational motion

214

8 Flexural Waves in a Beam

v(0, t) = 0 and M(0, t) = 0

(8.26a)

V (0, t) = 0 and M(0, t) = 0

(8.26b)

v(0, t) = 0 and φ(0, t) = 0

(8.26c)

(b) Free end:

(c) Clamped end:

(d) Elastically supported end for the transverse motion:

V (0, t) + K t v(0, t) = 0 and M(0, t) = 0

(8.26d)

(e) Elastically supported end for the rotational motion:

V (0, t) = 0 and M(0, t) + αr φ(0, t) = 0,

(8.26e)

where K t indicates the translation stiffness and αr denotes the rotational stiffness. The field variables appearing in Eq. (8.26) can be explicitly written in terms of the vertical displacement v(x, t) as ∂v(0, t) = 0, ∂x ∂ 2 v(0, t) M(0, t) = E I = 0, ∂x2 ∂ 3 v(0, t) V (0, t) = −E I = 0. ∂x3

φ(0, t) =

(8.27)

Among the various boundary conditions illustrated in Fig. 8.8, we consider the reflection from a free end as a specific example. Using Eqs. (8.26b) and (8.27), we can write the free-end condition explicitly as ∂ 3 v(0, t) ∂ 2 v(0, t) = 0, = 0. ∂x2 ∂x3

(8.28)

8.2 Reflection and Transmission

215

Before applying the boundary conditions (8.28), we use Eq. (8.23) to obtain the following expressions: ⎧ ⎪ ⎪ ⎨

⎫ ⎡ 1 v ⎪ ⎪ ⎬ ⎢ −ik ∂v/∂ x =⎢ ⎣ −k 2 ⎪ ∂ 2 v/∂ x 2 ⎪ ⎪ ⎪ ⎩ 3 ⎭ ∂ v/∂ x 3 ik 3

⎫ ⎤⎧ Ci e−ikx ⎪ 1 1 1 ⎪ ⎪ ⎪ ⎨ −kx ⎬ −k ik k ⎥ ⎥ Di e , k 2 −k 2 k 2 ⎦⎪ Cr eikx ⎪ ⎪ ⎪ ⎩ ⎭ −k 3 −ik 3 k 3 Dr ekx

(8.29)

where A, B, C, and D in Eq. (8.23) are replaced by Ci , Di , Cr , and Dr in Eq. (8.29). The subscripts i and r are associated with the incident (propagating or decaying along the +x-axis) and reflected (propagating or decaying along the −x-axis) waves. Upon applying the two conditions in Eq. (8.28) with Eq. (8.29), we obtain 

⎧ ⎫ ⎪ ⎪ ⎪ Ci ⎪ ⎬   ⎨ −k k −k k 0 Di = , ⎪ 0 C r ik 3 −k 3 −ik 3 k 3 ⎪ ⎪ ⎪ ⎭ ⎩ Dr 2

2

2

2

(8.30)

where Ci and Di , related to the incident wave field, are assumed to be prescribed. To determine Cr and Dr , Eq. (8.30) is written as 

−1 1 −i 1



Cr Dr



 =

1 −1 −i 1



 Ci , Di

from which the following result can be found: 

Cr Dr



 =

−i 1 + i 1−i i



 Ci . Di

(8.31)

If a wave is incident from a far field without an evanescent wave (i.e., Di = 0), Eq. (8.31) reduces to Cr = −iCi ,

(8.32a)

Dr = (1 − i)Ci .

(8.32b)

Based on the results in Eq. (8.32), the following observations can be made: ◦

1. The reflected propagating wave is 270 out-of-phase with respect to the incident propagating wave (due to the term “−i” in Eq. (8.32a)). 2. Even if only a propagating wave is incident, an evanescent wave associated with Dr is generated to satisfy the boundary condition. 3. The reflected coefficients Cr and Dr are independent of the excitation frequency.

216

8 Flexural Waves in a Beam

4. The magnitude of Cr is equal to the magnitude of Ci , i.e., |Cr | = |Ci |, which satisfies the power balance law (no power is transmitted by the evanescent wave associated with Dr .) The results in Eqs. (8.31) and (8.32) may be compared with those for the simplysupported boundary condition, which are Cr = −Ci , Dr = −Di

(8.33)

Cr = −Ci . Dr = 0

(8.34)

If Ci = 0 and Di = 0,

Equation (8.34) shows that no evanescent wave is generated in the case of the simplysupported boundary condition if only a propagating wave is incident. The wave phenomena occurring at the interface of two dissimilar beams involve both transmission and reflection upon the incidence of a wave onto their interface. As depicted in Fig. 8.9, the incident wave from Beam 1 is expressed as v1i (x, t) = C1i ei(ωt−k1 x) .

(8.35)

If the reflected wave in Beam 1 and the transmitted wave in Beam 2 are denoted by v1r (x, t) and v2t (x, t), respectively, they can be expressed as  v1r (x, t) = C1r eik1 x + D1r ek1 x eiωt ,

(8.36)

 v2t (x, t) = C2t e−ik2 x + D2t e−k2 x eiωt ,

(8.37)

where C1r , D1r , C2r , and C2t are unknown amplitudes to be determined. In Eqs. (8.35)–(8.37), the wavenumber ki is related to ω as Fig. 8.9 Reflection and transmission at the interface of two dissimilar beams (v1i : incident wave, v1r : reflected wave, v2t : transmitted wave)

8.2 Reflection and Transmission

217

ki =

ρA EI

1/4

√

ω (i = 1, 2).

i

To determine C1r , D1r , C2r , and C2t , the continuity conditions at the interface x = 0 are imposed: displacement: v1i + v1r = v2t , rotation: φ1r + φ2r = φ2t , shea force: V1i + V1r = V2t , moment: M1i + M1r = M2t .

(8.38)

Equation (8.38) yields four independent equations for C1r , D1r , C2t , and D2t , which can be written in a matrix form as ⎡

1 ⎢iR ⎢ ⎣ −1 −i

⎫ ⎧ ⎫ ⎤⎧ −1 ⎪ C1r ⎪ 1 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎬ R iQ Q ⎥ ⎥ D1r = i R C1i , 1 ⎪ C ⎪ ⎪ 1 R 2 Q 2 −R 2 Q 2 ⎦⎪ ⎪ ⎪ ⎩ ⎩ 2t ⎪ ⎭ ⎪ ⎭ −i 1 −i R Q 3 R Q 3 D2t

(8.39)

where R=

E 2 I2 E 1 I1

1/4

and Q =

ρ2 A2 ρ1 A1

1/4 .

(8.40)

Solving Eq. (8.39) for the unknowns yields ⎧ ⎫ ⎧ C1r ⎪ 2R Q(R 2 −  Q 2 ) − i(1 − R 2 Q 2 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨  2 2 R 2 Q 2 (1 − i), D1r 1 − R Q 1 + = C1i  ⎪ ⎪ C ⎪ 2 1 + R 2 Q 2 (R + Q)/Q, ⎪ ⎪ ⎩ 2t ⎪ ⎭ ⎩ D2t −2 1 − R 2 Q 2 (R + i Q)/Q.

(8.41)

  2  = 2R Q R 2 + Q 2 + 1 + R 2 Q 2 (real-valued).

(8.42)

where

Based on the results expressed in Eq. (8.41), the following observations can be made. First, the transmitted propagating wave (related to C2t ) is in phase with the incident wave. The reflected propagating wave (related to C1r ) is not in phase with the incident wave. As the transmitted wave is frequency-independent, the stepped beam cannot be used as a frequency filter. The fixed and free boundary conditions for Beam 1 can be obtained by considering the limits of (R → ∞, Q → ∞) and (R → 0, Q → 0), respectively.

218

8 Flexural Waves in a Beam

8.3 Wave Analysis by Timoshenko Beam Theory As discussed in Sect. 8.1, the wave analysis based on the Euler–Bernoulli beam theory can accurately describe the flexural waves for relatively low frequencies. Therefore, an improved beam theory, which can be used for up to relatively high frequencies, is needed. To this end, this section introduces the Timoshenko beam theory (e.g., Labuschangne et al. (2009)). As the Timoshenko beam theory is reasonably accurate even for high frequencies, it is generally preferred for wave analysis in beams. We will compare the dispersion relation by the Timoshenko beam theory and that by the Euler–Bernoulli theory to see why the former is much preferred over the latter. The fundamental difference between the two theories is that in the Timoshenko beam theory, the surface normal to the neutral axis is not assumed to remain normal and the effect of the rotary inertia is also considered. When the normal does not remain normal, nonzero shear strain γ exists. Because γ = ∂v/∂ x − φ = 0, φ, the rotation of the normal, should be treated as an independent variable. Therefore, the Timoshenko beam theory employs two independent variables v(x, t) and φ(x, t). In deriving the wave equation based on the Timoshenko beam theory, we can use Eqs. (8.1)–(8.5). The normal stress σx is expressed in Eq. (8.4) but the shear stress σx y (x, y, t) can be explicitly related to γ (x, t) as σx y (x, y, t) ≈ τ (x, t) = Gγ (x, t).

(8.43)

In Eq. (8.43), the shear stress over the whole beam cross section is assumed to be the same as the shear stress τ (x, t) in the middle plane (y = 0) for any x and t. The shear modulus is denoted by G. Using τ (x, t), the shearing force V (x, t) can be evaluated as  V (x, t) = τ (x, y, t)d A = G Aγ (x, t). (8.44) A

As the actual shear stress is not uniform over the beam cross section (see, e.g., Crandall et al. (2012)), additional nonuniformity in shearing deformations reduces the shearing stiffness over the beam cross section. To consider this additional shearing effect, Eq. (8.44) is modified as V (x, t) = K G Aγ (x, t),

(8.45)

where K (0 < K ≤ 1) is called the shear correction factor. In static problems, K is typically set as 5/68 for a rectangular cross section and 9/10 for a circular cross section For a rectangular cross section (height: h, width: b), the shear stress is distributed as σx y =  ! " (V /2I ) (h/2)2 − y 2 and V = σx y dA. If Eq. (8.43) is assumed, V is related to τ and γ as V = (5/6)Aτ = (5/6)G Aγ . Generally, a value between 5/6 and 1 is used for K, and the dispersion relation below the first cutoff frequency is not extremely sensitive to the K value. 8

8.3 Wave Analysis by Timoshenko Beam Theory

219

(see, e.g., Crandall et al. (2012)). Alternatively, K can be adjusted comparing the wave phenomena by the Timoshenko beam theory and the exact elasticity theory. This will be discussed later after the dispersion relation obtained based on the Timoshenko theory is studied. To derive the dispersion relation by the Timoshenko theory, the governing equations for the Timoshenko beam theory are summarized as Equations of motion ∂ 2v ∂ 2φ ∂V ∂M = ρ A 2 − q, + V = ρI 2 ∂x ∂t ∂x ∂t

(8.46)

Constitutive relations M(x, t) = E I κ(x, t) = E I



∂v ∂φ , V (x, t) = K G Aγ (x, t) = K G A −φ , ∂x ∂x (8.47)

where E I and K G A are called the bending rigidity and shear stiffness. Substituting Eqs. (8.47), (8.3), and (8.5) into Eq. (8.46) yields the equations of motion in terms of w and φ as

2 ∂ 2v ∂ v ∂φ ρ A 2 − G AK = q, − ∂t ∂x2 ∂x

∂ 2φ ∂ 2φ ∂v ρ I 2 − E I 2 − G AK − φ = 0. ∂t ∂x ∂x

(8.48)

(8.49)

To determine the dispersion relation from the 2nd-order coupled partial differential equations, Eqs. (8.48) and (8.49), q is set as zero and the method of separation of variables can be used. Guided by the form of the general solution to the wave equations for the Euler–Bernoulli beam theory, one may directly assume the harmonic solutions propagating along the x-axis as v(x, t) = Vei(ωt−kx) ,

(8.50)

φ(x, t) = ei(ωt−kx) ,

(8.51)

where ω denotes the given excitation frequency and V and  are complex-valued amplitudes. The wavenumber k is real for propagating waves and imaginary or complex for evanescent waves. Substituting Eqs. (8.50) and (8.51) into Eqs. (8.48) and (8.49) yields

220

8 Flexural Waves in a Beam



−ikG AK G AK k 2 − ρ Aω2 ikG AK E I k 2 + G AK − ρ I ω2



V 



  0 = . 0

To simplify Eq. # (8.52), the following symbols#are introduced:

bar speed: cb =

E , ρ

shear wave speed: c˜s =

KG , ρ

(8.52)

radius of gyration: q =

#

I . A

Note that the shear wave speed in an infinite three-dimensional medium is cs = √ G/ρ, but the correction factor K is added in the definition of c˜s because K is introduced to consider the influence of nonuniform shearing deformations over the beam cross section for beam bending analysis using the Timoshenko beam. Using the above-defined symbols, Eq. (8.52) can be rewritten as 

k2 −

ω2 c˜s2

ik



−ik k 2 cb2 q 2 c˜s2

+1−

ω2 q 2 c˜s2

V 

 =

  0 . 0

(8.53)

To find a nontrivial solution from Eq. (8.53), the determinant of the matrix in Eq. (8.53) should be zero. Accordingly, the following dispersion relation can be obtained k −k ω 4

2

2

1 1 + 2 2 c˜s cb



−

ω cb q

2

ω4 − 2 2 c˜s cb

 = 0.

(8.54)

Solving Eq. (8.54) for k yields k = ±k1 (ω) or k = ±k2 (ω),

(8.55)

where ⎧ ⎫



2 ⎬1/2

2 ⎨1 1 ω 1 1 1 1 k1 (ω) = + 2 ω2 + + − 2 ω4 , ⎩ 2 c˜s2 ⎭ cb q 4 c˜s2 cb cb ⎧ ⎫



2 ⎬1/2

2 ⎨1 1 ω 1 1 1 1 + 2 ω2 − + − 2 ω4 . k2 (ω) = ⎩ 2 c˜s2 ⎭ cb q 4 c˜s2 cb cb

(8.56)

(8.57)

Equations (8.55)–(8.57) indicate that the Timoshenko beam theory predicts two dispersion branches, k = k1 (ω) and k = k2 (ω), resembling the lowest two branches of the exact dispersion relation predicted by the theory of elasticity (see, e.g., Miklowitz (1978)) with substantial accuracy. Note that the theory of elasticity predicts an infinite number of dispersion branches, whereas the Timoshenko theory predicts only two lowest branches. Thus, the Timoshenko beam theory is commonly adopted for the analysis of the bending waves in a beam in which the field variation over its thickness is not significant. For k = ki (ω) (i = 1, 2), the relative magnitude between V and  can be determined from Eq. (8.53) as  = V(k 2 − ω2 /c˜s2 )/ik.

8.3 Wave Analysis by Timoshenko Beam Theory

221

Fig. 8.10 Dispersion relations by the Timoshenko beam theory and the Euler–Bernoulli beam theory for an aluminum beam with a rectangular cross section. Material and geometrid data: ρ = 2700 kg/m3 , E = 68 GPa, G = 25.6 GPa, K = 0.85, t = 1 cm, and b = 1 cm)

As observed in Fig. 8.10, the Timoshenko beam theory predicts two propagating branches above the cutoff frequency; the cutoff frequency is a frequency below which no wave associated with the corresponding the branch (the 2nd branch in this case) can propagate. The 1st and 2nd branches are indicated by k1 (ω) and k2 (ω), respectively. After the 2nd branch starts from k = 0, it enters the zone of purely imaginary wavenumbers corresponding to evanescent waves and passes through the point marked by the cutoff frequency ωcuto f f . Then, it enters the zone of real wavenumbers corresponding to propagating waves. (Here, we only consider the range of k with Re k, Im k > 0.) On the contrary, the Euler–Bernoulli beam theory predicts only a single propagating branch for all frequencies. It is useful to summarize the following observations using Fig. 8.10. 1. For small ω, the dispersion relation by the Euler–Bernoulli beam theory asymptotically approaches that by the Timoshenko beam theory. To demonstrate this, we consider the limits of k1 (ω) and k2 (ω) as ω → 0: ⎛

⎞





2 1/2 1 ω 1 1 ⎠ lim k 1 (ω) ≈ ⎝ + 2 ω2 + ω→0 2 c˜s2 cb q cb



ω 1/2 ρ A 1/4 √ ≈ = ω, cb q EI ⎛ ⎞





2 1/2 1 ω 1 1 ⎠ lim k 2 (ω) ≈ ⎝ + 2 ω2 − ω→0 2 c˜s2 cb q cb



ω 1/2 ρ A 1/4 √ ≈ − =i ω. cb q EI

(8.58)

(8.59)

The results in Eqs. (8.58) and (8.59) are identical to the results obtained from Eqs. (8.21) and (8.22) based on the Euler–Bernoulli beam theory. Therefore, the wave analysis using the Euler–Bernoulli theory can be justified at low frequencies.

222

8 Flexural Waves in a Beam

2. According to the Timoshenko beam theory, there exists a cutoff frequency ωcutoff for the 2nd branch of the dispersion relation:

ωcutoff =

G AK . ρI

(8.60)

Unlike in the Euler–Bernoulli beam theory for which the 2nd branch represents evanescent wave only, waves belonging to the 2nd branch of the dispersion relation by the Timoshenko beam theory can propagate if ω > ωcutoff . 3. The most essential feature in the Timoshenko dispersion relation is that the group velocities for both branches are finite even if ω → ∞. To demonstrate this, we consider the limits of k1 (ω) and k2 (ω) as ω → ∞: ⎧ ⎫



2 ⎬1/2 ⎨1 1 1 1 1 1 lim k 1 (ω) ≈ + 2 ω2 + − 2 ω4 ω→∞ ⎩ 2 c˜s2 ⎭ 4 c˜s2 cb cb

1/2  1 1 1 1 1 1 ω 2 ω ω2 = + + − = , 2 c˜s2 2 c˜s2 c˜s cb2 cb2

1/2  1 1 1 1 1 1 ω 2 ω ω2 lim k 2 (ω) ≈ + − − = . 2 2 2 2 ω→∞ 2 c˜s 2 c˜s cb cb cb

(8.61)

(8.62)

Therefore, the group velocities (vg ) of the 1st and 2nd branches become lim vg = c˜s for the first branch,

ω→∞

lim vg = cb for the second branch.

ω→∞

This result is in contrast with the result by the Euler–Bernoulli beam theory predicting an infinite group velocity as ω → ∞. Because no wave can propagate at the infinite group (energy) velocity, the Euler–Bernoulli theory is invalid when ω → ∞. 4. If the shear correction factor K is adjusted, the limiting group velocity of the 1st branch can be matched with the theoretical value obtained from the exact three-dimensional elasticity solution.9 Refer to Problem 8.6. 5. Alternatively, the value of K can be chosen to make the cutoff frequency by the Timoshenko beam theory equal to the lowest cutoff frequency by the theory of elasticity. The group velocity (vg |ω→∞ ) of the 1st branch by the theory of elasticity is the Rayleigh surface wave √ velocity c R and that of the 2nd branch is the shear wave speed cs in an infinite medium where cs = G/ρ and c R /cs ≈ (0.862 + 1.14ν)/(1 + ν) (ν: Poisson’s ratio). See, e.g., Miklowitz (1978).

9

8.4 Flexural Waves in a Beam with Distributed Resonators

223

Generally, the typical range of K will be 5/6 ≤ K ≤ 1 for most beams if K is adjusted based on the argument above (No. 4). In this book, we will use K = 0.85 because this value yields a dispersion relation comparable with that by the theory of elasticity for most solid beams.

8.4 Flexural Waves in a Beam with Distributed Resonators The dispersion curve in Fig. 8.4 demonstrates that flexural waves excited at any frequency can propagate along a beam. A stopband can be created by installing mass–spring resonators on a beam. Liu and Hussein (2012) investigated the case of a beam with periodically installed resonators. The ideal situation in which distributed mass–spring resonators are attached to a beam is depicted in Fig. 8.11. For this problem, the Euler–Bernoulli beam theory will be utilized to facilitate the analysis. In order to derive the governing equation, we refer v(x, t) to the vertical displacement of the beam in Fig. 8.11 and w(x, t) as the vertical displacement of the distributed mass. The equilibrium conditions (or the equations of motion) for infinitesimal elements ρ Ax and m  x shown in Fig. 8.12 can be stated as for infinitesimal element ρ Ax: force equilibrium: ρ Ax moment equilibrium: M +

∂ 2v = V + S  x(w − v), ∂t 2

x x V ≈ 0, (V + V ) + 2 2

(8.63a) (8.63b)

Fig. 8.11 A beam with uniformly-distributed mass-spring resonators. (S  : distributed stiffness, m  : distributed mass)

Fig. 8.12 Free-body diagrams for a an infinitesimal beam element and b an infinitesimal distributed mass

224

8 Flexural Waves in a Beam

for infinitesimal element m  x: force equilibrium: m  x

∂ 2w = S  x(v − w). ∂t 2

(8.64)

By taking the limits of Eqs. (8.63a, b) as x → d x, dividing them by x, and combining the two resulting equations, one can obtain ρA

∂ 4v ∂ 2v + E I + S  (v − w) = 0. ∂t 2 ∂x4

(8.65)

In deriving Eq. (8.65), V = −E I ∂ 3 v/∂ x 3 is used. Similarly, Eq. (8.64) becomes m

∂ 2w + S  (w − v) = 0. ∂t 2

(8.66)

Assuming the following form of harmonic solutions, v(x, t) = Vei(ωt−kx) ,

(8.67)

w(x, t) = Wei(ωt−kx) ,

(8.68)

and substituting Eqs. (8.67) and (8.68) into Eqs. (8.65) and (8.66), the following result can be obtained      −S  E I k 4 + S  − ρ Aω2 V i(ωt−kx) 0 = e . (8.69) −S  −m  ω2 + S  W 0 Then, the following dispersion relation can be determined by setting the determinant of the matrix appearing in Eq. (8.69):   2 E I k 4 + S  − ρ Aω2 −m  ω2 + S  − S  = 0,

(8.70a)

ρ Am  ω4 − S  (m  + ρ A)ω2 − m  E I k 4 ω2 + E I S  k 4 = 0.

(8.70b)



or

If m  = 0 is inserted in Eq. (8.70) and the resulting equation is divided by S  , Eq. (8.70) reduces to the dispersion relation in Eq. (8.20). The dispersion curve corresponding to Eq. (8.70) is plotted in Fig. 8.13a. It includes a stopband for ω p < ω < ωq , where ω p and ωq are defined as $ ωp =

S , m

(8.71a)

8.4 Flexural Waves in a Beam with Distributed Resonators

225

Fig. 8.13 Dispersion curves for a beam with uniformly distributed mass-spring resonators for a finite m  and b m  → ∞



1 1  > ωp. ωq = S + ρ A m

(8.71b)

Note that ω p is the resonance frequency of the distributed mass–spring resonators and ωq is the non-zero frequency corresponding to k = 0. If the dispersion relation (8.70) is written in terms of the effective density ρeff defined as ρeff = ρ

ωq2 − ω2 ω2p − ω2

,

(8.72)

Eq. (8.70) can be written as10 ρeff A

∂ 2v ∂ 4v + E I = 0. ∂t 2 ∂x4

(8.73)

Equation (8.72) and Fig. 8.13a show that the stopband11 is formed due to the negative effective density induced by the local resonance. At any frequency belonging to the passband (ω < ω p , ω > ωq ), there are two branches (considering a wave propagating or decaying along the +x-axis), one propagating branch covering real k values and the other evanescent branch covering purely imaginary k values. Due to the formed stopband, the propagating branch is not connected unlike the propagating branch shown in Fig. 8.4. When m  → ∞, Eq. (8.70) becomes ρ Aω2 = E I k 4 + S  , 10

(8.74)

The derivations of Eqs. (8.72) and (8.73) are given as Problem 8.4. Due to the periodicity and coupling effects of the periodicity and discrete mass–spring resonators, additional stopbands are formed if discrete mass–spring resonators are installed at an equal distance. See Liu and Hussein (2012) for additional information.

11

226

8 Flexural Waves in a Beam

Fig. 8.14 Periodic discrete system designed to carry flexural waves

which represents the dispersion curve for a beam on an elastic foundation with the foundation stiffness S  . In this case, there is a cutoff frequency ωcutoff below which no wave can propagate. Therefore, a stopband is formed from ω = 0 in this case.

8.5 Flexural Waves in Periodic Discrete System A periodic discrete system designed to carry flexural waves (Oh and Assouar 2016) is presented in Fig. 8.14.12 Each unit cell comprises a rigid block with mass m and rotary inertia I R and two springs of translational (or shear) stiffness α (unit: force/displacement) and rotational stiffness β (unit: moment/rotation). In the figure, the lattice period is denoted by a. Assuming that the lengths of springs α and β are zero, the springs are treated as point springs. In a beam carrying flexural waves, the rotational motion (φ(x, t)) and translation motion (v(x, t)) are primarily governed by (I R and β) and (m and α), in spite that the two motions are coupled. Before deriving the equations of motion for the lattice system, we introduce the symbols vnL (t) and vnR (t) to denote the transverse displacements of the left- and righthand side of the rigid block of the n-th unit cell. For small values of φn , vnL (t) and vnR (t) can be expressed as 1 1 vnL = vn − aφn , vnR = vn + aφn . 2 2

(8.75)

Referring to the free-body diagram depicted in Fig. 8.15, the equations of motion for transverse and rotational movements for the n-th rigid block can be expressed as m

12

 R  L ∂ 2 vn = α vn+1 − vnR + α vn−1 − vnL , 2 ∂t

(8.76)

The transfer matrix method in conjunction with the Bloch–Floquet theorem described in Sects. 7.11 and 7.12 can be utilized to analyze periodic continuum beams carrying flexural waves. The distinction lies in the types of waves being analyzed (Chap. 7: longitudinal waves, Chapter 8: flexural waves). Therefore, we omit the analysis of periodic continuum beams and instead present the wave analysis in recently introduced beam systems with periodic discrete components.

8.5 Flexural Waves in Periodic Discrete System

227

Fig. 8.15 Free-body diagram of the n-th block

IR

1  R ∂ 2 φn 1  L R L aα v aα v = β(φ − φ − v − v − . + ) n+1 n n+1 n n−1 n ∂t 2 2 2

(8.77)

Substituting Eq. (8.75) into Eqs. (8.76) and (8.77) yields the wave equations in terms of vn and φn as m

% % a a & a a & ∂ 2 vn φ φ φ φn , + α v = α v − v − − − v + + n+1 n n+1 n n−1 n n−1 ∂t 2 2 2 2 2 (8.78) IR

∂ 2 φn = β(φn+1 − φn ) + β(φn−1 − φn ) ∂t 2 a a & 1 % + aα vn+1 − vn − φn+1 − φn 2 2 2 a a & 1 % − aα vn−1 − vn + φn−1 + φn . 2 2 2

(8.79)

To derive the dispersion relation from Eqs. (8.78) and (8.79), we can assume the following harmonic solutions: vn (t) = Vei(ωt−kna) ,

(8.80)

φn (t) = ei(ωt−kna) ,

(8.81)

where x = na represents the axial coordinate of the center of the n-th unit cell. Using Eqs. (8.80) and (8.81), we can write vn+1 (t) = e−ika vn (t) and vn−1 (t) = eika vn (t),

(8.82)

φn+1 (t) = e−ika φn (t) and φn−1 (t) = eika φn (t).

(8.83)

Substituting Eqs. (8.80)–(8.83) into Eqs. (8.78) and (8.79) yields

228

8 Flexural Waves in a Beam

 1  −mω2 V = α eika + e−ika − 2 V + aα eika − e−ika , 2  −I R ω2  = β eika + e−ika − 2  2  1  1 a α eika + e−ika + 2  + aα e−ika − eika V. − 2 2

(8.84)

(8.85)

Equations (8.84) and (8.85) can be rewritten in a matrix form as          V V 0 0 D11 D12 D = = . or D21 D22   0 0

(8.86)

The matrix D = [Di j ] is given by 

 −D 1 (k; α) − mω2 −D2 (k; α) D= , −D3 (k; α, β) − I R ω2 −D2∗ (k; α)

(8.87)

where  D1 (k; α) = α eika + e−ika − 2 = 2α(cos ka − 1), 1  D2 (k; α) = aα eika − e−ika = iaα sin ka, 2 2  ika  1 −ika a α eika + e−ika + 2 D3 (k; α, β) = β e + e −2 − 2 1 2 = 2β(cos ka − 1) − a α(cos ka + 1). 2

(8.88)

In Eq. (8.87), D2∗ (k; α) represents the complex conjugate of D2 (k; α). The dispersion relation can be determined by setting the determinant of the matrix D in Eq. (8.87) equal to zero:13 det D(ω, k; α, β) = D11 D22 − D12 D21 = 0

(8.89)

To investigate the dispersion relation given by Eq. (8.89), we consider a specific geometry of the unit cell which can be actually fabricated. Figure 8.16 shows that the finite-sized thin plate (beam) of length l is used to realize the springs α and β. The rigid rectangular parallelepiped block (density: ρ, dimension: a m × b × h) has the following mass and inertia: ( ' 2 mass: m = ρa m bh, mass moment of inertia: I R = m a m + h 2 /12. 13

See Problem 8.7.

(8.90)

8.5 Flexural Waves in Periodic Discrete System

229

Fig. 8.16 Geometry of the unit cell made of aluminum for the specific lattice model in consideration

The stiffnesses α and β can be approximated by α ≈ K G A p /l, β ≈ E I p /l,

(8.91)

where A p = bt and I p = bt 3 /12 represent the area and the moment of inertia of the thin connecting plate with thickness t, respectively. To obtain the second result in Eq. (8.91), we employed the model described in Fig. 8.17a, which represents a long, cantilevered beam (v = 0 and φ = 0 at x = 0) subjected to an applied moment M at its end. As the bending moment and curvature are uniform across all locations in this case, the integration of the first relation in Eq. (8.47) using (8.6) yields l

l kdx = E I p

Ml = E I p 0

0

∂φ dx = E I p (φx=l − φx=0 ) = E I p φx=l . ∂x

Therefore, we can obtain β as β = M/φx=l = E I p /l. To obtain the first result in Eq. (8.91), we use the model described in Fig. 8.17b, which depicts an extremely short cantilevered beam (v = 0 and φ = 0 at x = 0) subjected to an applied shear force V at its end. Assuming uniform shear strain γ at all locations and constant V across all locations, the integration of the second relation in Eq. (8.47) using (8.5) with φ = 0 yields

Fig. 8.17 Approximate estimation of a rotational stiffness β using a long, cantilevered beam subjected to a bending moment M and b translational stiffness α using a short beam subjected to a shear force V

230

8 Flexural Waves in a Beam

Fig. 8.18 Dispersion curve for the lattice system shown in Fig. 8.14 carrying flexural waves. (a = 7 cm, h = 4 cm, am = 5 cm, l = 2 cm, t = 0.1 cm, ρ = 2700 kg/m3 , E = 68 GPa, G = 25.6 GPa, K = 0.85). ω1 = 783.76 rad/s (f 1 = 124.74 Hz) and ω2 = 2.84 × 103 (f 2 = 4.52 kHz)

l V l = K G Ap

l γ dx = K G A p

0

0

∂v dx = K G A p (vx=l − vx=0 ) = K G A p vx=l . ∂x

Therefore, we can determine α as α = V /vx=l = K G A p /l. The dispersion curve for the lattice system with the unit cell in Fig. 8.16 is plotted in Fig. 8.18. It has two branches, namely the acoustic (lower) and optical (upper) branches. In Fig. 8.18, the first Brillouin zone is plotted with Re(ka) ≥ 0. At ka = π , two cutoff frequencies ω1 and ω2 can be calculated. Substituting ka = π into Eq. (8.87) yields 

0 4α − mω2 0 4β − I R ω2



V 



  0 = . 0

(8.92)

Evidently, Eq. (8.92) yields two decoupled equations for V (or vn ) and  (or φn ), implying that the shear motion (associated with vn ) and the rotational motion (associated with φn ) are decoupled at ka = π . Setting the determinant of the matrix in Eq. (8.92) equal to zero, two cutoff frequencies (ω1 and ω2 ) are determined as $ ω1 = 2

$ β α and ω2 = 2 . I m

(8.93)

Figure 8.18 shows that a stop band is formed between ω1 and ω2 . Note that the lowest cutoff frequency should be determined by the √ ratio of r = (β/I )/(α/m) = √ (m/I ) · (β/α); the lowest cutoff frequency becomes 2 β/I if r < 1 and 2 α/m if r > 1. In general, r < 1, but r > 1 can be also possible depending on metamaterial design. The analysis for ka = π suggests that depending on various parameters such as α and β, a stop band of different characteristics can be formed.

8.5 Flexural Waves in Periodic Discrete System

231

To examine the behavior of the dispersion relation for ka → 0, we use a → 0. In this case,14 the stiffnesses α and β are approximated (see Eq. (8.91)) as α=

EI KGA and β = , a a

(8.94)

where A = bh and I = bh 3 /12 denote the area and cross-sectional moment of inertia of the rigid block behaving as a beam. The mass m and rotary moment of inertia I R becomes m = ρabh = ρa A IR =

m[(a)2 +h 2 ] 12

=

ρabh (a 2 +h 2 ) 12

→

ρabh (h 2 ) 12

= ρa I (as a → 0)

.

(8.95)

In the limits of ka → 0, D1 , D2 , and D3 in Eq. (8.88) become

(ka)2 − 1 = −α(ka)2 , D1 (k; α) = 2α(cos ka − 1) ≈ 2α 1 − 2

(8.96)

D2 (k; α) = iaα sin ka ≈ iaα(ka),

(8.97)

1 D3 (k; α, β) = 2β(cos ka − 1) − (a)2 α(cos ka + 1) 2 ≈ −β(ka)2 − αa 2 ,

(8.98)

where cos ka ≈ 1 − (ka)2 /2 and sin ka ≈ ka are used for small ka. Substituting Eqs. (8.94) and (8.95) into Eqs. (8.96)–(8.98) yields D1 = a(G AK )k 2 , D2 = −ia(G AK )k, D3 = a(E I k 2 + G AK ).

(8.99)

If the expressions in Eqs. (8.99) and (8.95) are substituted into Eq. (8.87), the following set of equations is obtained as       −iaG AK k a G AK k 2 − ρ Aω2  V 0 = . iaG AK k a E I k 2 + G AK − ρ Aω2  0

(8.100)

If Eq. (8.100) is divided by a, the resulting equation is precisely the same as Eq. (8.52), which is derived based on the Timoshenko beam theory. This demonstrates that the wave in the lattice system for small ka behaves as the wave in the continuum beam in the low-frequency range. Further interesting wave phenomena of the flexural waves in lattice beam systems, such as abnormal group velocity, are discussed in Lee 14

This limiting case simulates a continuous uniform beam of thickness h and width b, with a → 0.

232

8 Flexural Waves in a Beam

and Oh (2018) and Park and Oh (2019). See also metamaterial-based methods for broadband low-frequency flexural vibration shielding15 using a discrete model (Oh et al. 2017) and a continuum model (Park et al. 2021). As for the absorption of flexural vibrations, one can consider various methods such as acoustic black holes (see references on the related subject in Chap. 1), the bound state in the continuum (Cao et al. 2021), and non-Hermitian elastic metamaterials (Cao et al. 2022).

8.6 Enhanced Actuation or Sensing by Resonators (Advanced Topic) When additional mass and/or springs are attached to a beam, its effective properties can be changed. In this case, the observed phenomena are similar to those in Sect. 7.16. Some examples are considered in Problems 8.5–8.6, where the frequencydependent effective material properties can be defined. Accordingly, flexural waves can be manipulated to suit specific needs. In Sect. 7.16, we demonstrated that installing a resonator pair in a bar carrying longitudinal waves can lower the effective characteristic impedance. Accordingly, the far-field strain output can be substantially increased at a target frequency if an actuating element is located inside the bar region bounded by the resonator pair. The target frequency is a frequency at which the effective impedance of the bounded bar segment is much lowered and is also a Fabry–Pérot resonance frequency. With an expectation that the same output-boosting principle can be applied to the case of bending waves in a beam, we present a method to enhance the actuation or sensing of flexural waves in a beam using a resonator (Kim et al. 2019). At the resonance frequency, the resonators’ motions should interact with the flexural wave motion in the beam. Due to the actuation and sensing reciprocity, we primarily explain the actuation enhancement. As depicted in Fig. 8.19a, we consider a beam to which a U-shaped resonator is installed. The deformed configuration of the U-shaped resonator and the beam at the target frequency is illustrated in Fig. 8.19b. This U-shape resonator actually functions like a pair of point resonators attached to a beam at two points distanced by 2L. Therefore, one can expect from the results in Sect. 7.16 that the region surrounded by the resonator can behave as an equivalent metamaterial beam with frequencydependent effective material properties.

15

If the absorption of flexural vibrations is of interest, using acoustic black holes (see references on the subject in Chap. 1) or the bound state in the continuum (Cao et al. 2021) can be considered.

8.6 Enhanced Actuation or Sensing by Resonators (Advanced Topic)

233

Fig. 8.19 a Aluminum beam (plate) with a resonator attached and b snapshot of the deformed shape of the resonator and the beam (plate) at the target frequency. The x-directional stretch (or compression) of the magnetostrictive patch induced by an applied dynamic magnetic field causes flexural waves in the beam (plate). The mechanism of the generation of the flexural waves in the beam (plate) by the patch can be modeled by two-point bending moments (Kim et al. 2019)

Figure 8.19a also shows that the actuation element is a “magnetostrictive16 ” patch of a magnetostrictive patch transducer (MPT) that is bonded onto a beam. The actuation by a magnetostrictive patch bonded onto a beam can be modeled by two concentrated forces applied on the top surface of the beam (Kim et al. (2019)). Based on the concentrated force model, the force Finp applied on the top surface of the beam can be replaced by a system of a force Finp and a bending moment Minp action on a point on the centerline of the beam, as portrayed in Fig. 8.20b. With interest in flexural waves in a beam, only the bending moment Minp will be considered in the subsequent discussions. With adequate adjustment of the resonance frequency, the vibration of the resonator can interact with the flexural waves in a beam, as illustrated in Fig. 8.19b. As will be discussed later, the resonator’s resonance frequency should be sufficiently close to the actuation frequency to maximize the far-field output by the actuating element. It should be noted, however, that no output wave will be generated if the resonance frequency is the same as the actuation frequency because the resonator will act as a dynamic damper in this case. The geometry of the U-shaped resonator is presented in Fig. 8.21a, whereas the lowest three eigenmodes and eigenfrequencies for t1 = 2.1 mm are depicted in Fig. 8.21b. As the resonator’s motion is supposed to be coupled with the flexural 16

Magnetostriction is a phenomenon coupling the elastic (mechanical) and magnetic fields. Mechanical deformation is generated when a magnetic field is applied to a magnetostrictive material (owing to the Joule effect). In contrast, the magnetization of a magnetostrictive material is changed when it is subjected to mechanical stress (because of the Villari effect). As this section is not intended to provide details of the magnetostrictive phenomenon or the working principle of the transducer based on magnetostriction, i.e., the magnetostrictive patch transducers (MPTs), we refer to Kim and Kwon (2015) for further descriptions of the wave excitation or sensing using MPTs.

234

8 Flexural Waves in a Beam

Fig. 8.20 A simplified model to explain the wave generation by the magnetostrictive patch. a Pin force model (Finp ; time-varying excitation force) and b equivalent force and moment (Minp ) calculated with respect to the neutral axis. (Kim et al. 2019)

Fig. 8.21 a Dimension of the U-shaped resonator and b its first three eigenmodes for t 1 = 2.1 mm. (f eigi : i-th eigenfrequency). The second eigenmode with f eig2 is the desired mode that can couple with flexural waves in the beam in Fig. 8.19a (Kim et al. 2019)

waves of the beam (Fig. 8.19a), the second eigenmode with f eig2 = 64.6 kHz should be used as the desired resonance mode of the resonator. As the resonator is made of a single continuum body, the desired resonance mode interacting with the flexural waves of the beam may not necessarily appear as its first eigenmode. Thus, we select the second eigenmode shown in Fig. 8.21b as the desired resonance mode. The geometry of the U-shaped resonator is selected to have its 2nd eigenfrequency f eig2 near the actuation frequency of f p = 64 kHz. The actuation frequency f p is selected to maximize the far-field strain induced by the MPT. Figure 8.22a shows that the output strain by the MPT is maximized17 at f = f p = 64 kHz. To facilitate the wave analysis in the beam with the attached resonator, we introduce the generalized lumped mass–spring model of the U-shaped continuum resonator, as shown in Fig. 8.23a. The spring is a generalized spring, and the stiffness of which is described by a 3×3 stiffness matrix S. The lumped mass is connected Although the mechanism of the maximum output by the MPT at f = f p is not the subject of this section, it will be briefly explained. To generate the maximum output using the MPT, the patch size (2L = 9 mm) is selected as half the wavelength (λ/2) of a flexural wave in a beam at the target frequency, as illustrated in Fig. 8.22b. (More precisely, λ is the wavelength of the lowest antisymmetric Lamb mode of the plate.) The wavelength is 16.75 mm at 64 kHz, close to 4L = 18 mm. If the magnetostrictive patch is modeled to generate two concentrated moments of opposite signs to the beam at A and B (see Fig. 8.22b), they can efficiently excite the flexural wave mode in the beam having the wavelength λ. Note that the target wave mode of the beam has the peak values of opposite signs at A and B because the distance between A and B is half the wavelength of a target wave mode.

17

8.6 Enhanced Actuation or Sensing by Resonators (Advanced Topic)

235

Fig. 8.22 a Measured far-field strain in a beam excited by the MPT (magnetostrictive patch transducer) plotted as a function of frequency. Detailed model: detailed finite element model considering the mechanical–magnetic coupling phenomenon of the magnetostrictive patch; simplified transduction model: a beam (plate) subjected to two concentrated moments illustrated in Fig. 8.20b. b Illustration of the patch size (2L) selected to be equal to half the wavelength (λ) of a bending wave propagating in a beam at the target frequency (Kim et al. 2019)

Fig. 8.23 a Generalized lumped mass–spring model of the U-shaped resonator; b deformed shape of half of the U-shaped resonator under applied forces (f x and f y ) and moment (mz ) acting on the junction point P (Kim et al. 2019)

to the beam at the points P and P  on the top surface of the beam. To calculate S, we consider the deformed shape of half of the U-shaped resonator, as shown in Fig. 8.23b. Referring to Fig. 8.23b, the stiffness matrix S = [Si j ] can be defined as ⎞ ⎛ ⎞ ⎞⎛ fx uP S11 S12 S13 ⎝ S21 S22 S23 ⎠⎝ v P − v R ⎠ = ⎝ f y ⎠, S31 S32 S33 θ mz ⎛

(8.101)

where u P , v P , and v R denote the horizontal and vertical18 displacements of point P and the vertical displacement of point R, respectively. The symbol θ represents the rotation angle at point P, as defined in Fig. 8.23b. The positive directions of u P , v P , v R , and θ are sketched in Fig. 8.23b. The symbols f x , f y , and m z in Eq. (8.101) denote the horizontal and vertical forces and the moment about the z-axis per unit length in the z-direction at point P, respectively.

Herein, v represents the vertical displacement, not the velocity given by v = ∂u/∂t as used in other chapters. 18

236

8 Flexural Waves in a Beam

To determine the values of the components of the generalized spring matrix S, finite element simulations were conducted using COMSOL Multiphysics by following the procedure described in Oh et al. (2016). Although the wave of interest is a flexural wave related to the vertical displacement v(x, t) of the beam mid-plane and the rotation of the normal φ(x, t) of the beam, we considered the horizontal displacement u p because it is coupled with v and φ. This coupling is inevitable because the resonator is composed of a continuum body. The effective mass of the lumped model of the continuum U-shaped resonator can be accurately estimated from the following equation: 2m =

2S22 , (2π f eig2 )2

(8.102)

where the second eigenfrequency f eig2 of the U-shaped resonator with the clamped ends is numerically calculated using COMSOL Multiphysics. As the resonator is supposed to affect the flexural wave motion in a beam primarily, only the vertical stiffness component S22 is used to evaluate the effective mass m in Eq. (8.102). The estimated parameters by the numerical analysis for various values of t1 are listed in Table 8.1 (taken from Kim et al. (2019)). For the analysis of the wave phenomena in the beam, u P , v P , and θ should be expressed in terms of the primary field variables of the beam, i.e., v(x, t) and φ(x, t), which are the vertical displacement of the neutral axis of the beam and the rotation of its normal. (Refer to Fig. (8.2)) for the positive directions of v(x, t) and φ(x, t). To this end, we consider the deformed shapes of the beam and the U-shaped resonator illustrated in Fig. 8.24, wherein point Pˆ denotes the location of point P in the undeformed configuration. In the illustration, the resonator is detached from the beam. According to the geometrical consideration in Fig. 8.24, the following relations can be obtained: v P = v Q = vx=L , Table. 8.1 Estimated parameters of the U-shaped resonator for various values of t1 (Si j = S ji )

(8.103)

Case 1: t 1 = 2.1 mm

Case 2: t 1 = 2.5 mm

Case 3: t 1 = 2.9 mm

feig2

64.6 kHz

71.1 kHz

77.8 kHz

m

11.2 g

13.6 g

15.9 g

S11

1.74 GN/m

2.61 GN/m

4.05 GN/m

S12

−0.95 GN/m

−1.36 GN/m

−2.00 GN/m

S13

−1.80 kN/rad

−2.15 kN/rad

−2.50 kN/rad

S22

1.85 GN/m

2.72 GN/m

3.81 GN/m

S23

0.44 kN/rad

0.57 kN/rad

0.59 kN/rad

S33

3.49 kN m/rad

3.68 kN m/rad

3.81 kN m/rad

8.6 Enhanced Actuation or Sensing by Resonators (Advanced Topic)

237

Fig. 8.24 Relationship between (up , vp , θ) and (v, φ), where v and φ denote the vertical displacement of the beam neutral axis and the rotation of its normal (Kim et al. 2019)

h u P = − φx=L , 2 )

) 1 ∂v )) ∂v )) . θ= + 2 ∂ x )x=L − ∂ x )x=L +

(8.104) (8.105)

Equation (8.103) states that the vertical displacement v P of point P located on the interface of the U-shaped resonator and the top surface of the beam is assumed as equal to the vertical displacement v Q , which is identical to v at x = L. The rotation φ of the surface normal to the neutral axis causes the horizontal displacement u P of point P. Equation (8.105) expresses that the rotation θ of the resonator about point P can be approximated as the average rotation of the beam neutral axis (corresponding to y = 0). To theoretically analyze the wave field in the beam, we use the free-body diagrams of the resonator and beam presented in Fig. 8.25. Here, the Timoshenko beam theory expressed by Eqs. (8.48) and (8.49) will be used. Considering the symmetry with respect to x = 0 (see Fig. 8.25), the wave field only for x ≥ 0 is considered for subsequent analysis because the wave field for x ≤ 0 is found directly from that for x ≥ 0 by the symmetry. Accordingly, the following forms of solution can be assumed for Sections A and B: Section A: |x| ≤ L v(x, t) = V1 ei(ωt−k1 x) + V2 ei(ωt−k2 x) + V1 ei(ωt+k1 x) + V2 ei(ωt+k2 x) , φ(x, t) = α1 V1 ei(ωt−k1 x) + α2 V2 ei(ωt−k2 x) − α1 V1 ei(ωt+k1 x) − α2 V2 ei(ωt+k2 x) , (8.106) Section B: x ≥ L x ≥ L v(x, t) = V3 ei(ωt−k1 x) + V4 ei(ωt−k2 x) ,

238

8 Flexural Waves in a Beam

Fig. 8.25 Free-body diagrams of a beam (h = 2 mm, shear correction factor K = 1) and the Ushaped resonator. The vertical displacement fields (v(x, t)) in different sections of the beam are indicated. (The beam and resonators are composed of aluminum. (E = 70 GPa, ν = 0.33, ρ = 2700 kg/m3 ) (Kim et al. 2019)

φ(x, t) = α1 V3 ei(ωt−k1 x) + α2 V4 ei(ωt−k2 x) ,

(8.107)

where the wavenumbers k1 and k2 are stated in √ Eqs. (8.56)*and (8.57). For a rectan√ gular beam of thickness h (and width b), q = I /A = (bh 3 /12)/bh = h/ 12 is used in the equations. The cutoff frequency f cutoff = ωcutoff /2π for the flexural waves of the aluminum beam of h = 2 mm is 861 kHz, which is far greater than the frequency range of interest (~100 kHz). Therefore, k1 is real and k2 is purely imaginary with Re(k1 ) > 0 and Im(k2 ) < 0 below the cutoff frequency. The constant αn (n = 1, 2) can be determined from the first equation of Eq. (8.53) as:  2  −i 2 ω αn ≡ k − . kn n cs

(8.108)

At the interface (x = L) between Sections A and B, the following relations must hold:

8.6 Enhanced Actuation or Sensing by Resonators (Advanced Topic)

239

vx=L − = vx=L +

(8.109)

φx=L − = φx=L +

(8.110)

Vx=L − + f y = Vx=L +

(8.111)

Mx=L − − M i np = Mx=L + − m z +

h fx 2

(8.112)

In addition, f y is related to the v R by the equation of motion for the resonator: fy = m

d2 v R = −ω2 mv R , dt 2

(8.113)

where v R is assumed as a harmonic function as v R (t) = V R eiωt .

(8.114)

The next step is to express five equations (Eqs. (8.109)–(8.113)) only in terms of five variables V1 , V2 , V3 , V4 , and V R . To this end, we use Eqs. (8.47), (8.106), and (8.107) to rewrite Eqs. (8.109)–(8.113) as V1 (e−ik1 L + e ik1 L ) + V2 (e−ik2 L + e ik2 L ) = V3 e−ik1 L + V4 e−ik2 L ,

(8.115)

α1 V1 (e−ik1 L − e ik1 L ) + α2 V2 (e−ik2 L − e ik2 L ) = α1 V3 e−ik1 L + α2 V4 e−ik2 L , (8.116) k1 V1 (e−ik1 L − e ik1 L ) + k2 V2 (e−ik2 L − e ik2 L ) −

1 F y = k1 V3 e−ik1 L iKGA + k2 V4 e−ik2 L , (8.117)

M i np α1 k1 V1 (e−ik1 L + e ik1 L ) + α2 k2 V2 (e−ik2 L + e ik2 L ) + iEI

1 h Mz − F x , = α1 k1 V3 e−ik1 L + α2 k2 V4 e−ik2 L + iEI 2 ω2 mVR + F y = 0, where F y , Fx , and Mz are defined as ( f x , f y , m z ) = (Fx , F y , Mz )eiωt .

(8.118) (8.119)

240

8 Flexural Waves in a Beam

Fig. 8.26 Output strain calculated using Eq. (8.121) and two-dimensional finite element simulation a without and b–d with a resonator installed. (Kim et al. 2019)

To eliminate F y , Fx , and Mz from Eqs. (8.115)–(8.119), we use Eq. (8.101) and Eqs. (8.103)–(8.105) to express F y , Fx , and Mz in terms of (V1 –V4 ,V R ) as ⎞ ⎞ ⎛ ⎞⎛  S11 S12 S13 −h α1 V3 e−ik1 L + α2 V4 e−ik2 L /2 Fx ⎟ ⎜ ⎟ ⎜ ⎟⎜ −ik L −ik L V3 e 1 + V4 e 2 − VR ⎠. ⎝ F y ⎠ = ⎝ S21 S22 S23 ⎠⎝ −ik L −ik L 1 2 Mz S31 S32 S33 −k1 V1 sin(k1 L) − k2 V2 sin(k2 L) − ik1 V3 e /2 − ik2 V4 e /2 ⎛

(8.120) Substituting Eq. (8.120) into Eqs. (8.115)–(8.119) finally yields five equations for five unknowns (V1 –V4 and V R ). Therefore, the V3 component governing the far-field output of the flexural wave can be computed if Minp is given. Once V3 is determined, the bending strain εx on the top surface of the beam can be expressed as ) ) ) ) ) ∂u ) ) )h ) ) ) |εx | = ) ) = ) α1 k1 V3 )). ∂ x y=h/2 2

(8.121)

In Fig. 8.26, the normalized theoretical strain field |εx | obtained from Eq. (8.121) is plotted in solid lines as a function of frequency. The theoretical results agree well with those predicted by detailed two-dimensional finite element simulation results plotted in dotted lines. This indicates that the phenomenon of the increased output

8.6 Enhanced Actuation or Sensing by Resonators (Advanced Topic)

With resonator (Exp.) Without resonator (Exp.)

Without resonator (Sim.)

With resonator (Sim.)

Amplitude (mV)

(a)

241

Experimentally measured signals at f = fp ( : with and : without resonator)

Time (μs)

(b) Without resonator (Sim.)

No resonator (Exp.) With resonator (Sim.)

Experimentally measured signals at f = fp ( : with and : without resonator) Amplitude (mV)

With resonator (Exp.)

Time (μs)

Experimentally measured signals at f = fp ( : with and : without resonator) Without resonator (Sim.)

With resonator (Exp.) No resonator (Exp.) With resonator (Sim.)

Amplitude (mV)

(c)

Time (μs)

Fig. 8.27 Comparison of the experimental and simulation results for the strain (| x |) output. a Case 1 (f R = 68 kHz), b Case 2 (f R = 74.6 kHz), and c Case 3 (f R = 81.5 kHz). (Kim et al. 2019)

can be correctly explained by the developed one-dimensional theoretical model based on the Timoshenko beam theory. For the numerical simulations, we conducted harmonic analyzes using COMSOL Multiphysics under a two-dimensional planestrain assumption. The finite element model was based on the configuration in Fig. 8.19a, where the excitation by the magnetostrictive patch is replaced with two concentrated moments. The experimental results for the three cases are presented in Fig. 8.27 and compared with the simulation results. (See Kim et al. (2019) for the detailed experimental setup and procedure.) As observed in Fig. 8.27, the simulation and experimental results of the strain |εx | measured at the sensing location were consistent.19 In addition, the output at the peak frequency ( f p ) increased significantly because 19

The difference in the output magnitudes at the peak frequency between the experimental and simulation results was primarily caused by damping, which is quite significant for narrow bandwidth. In addition, the bonding state of the resonator onto the MPT could influence the results.

242

8 Flexural Waves in a Beam

of the installed resonators. In Case 1, the experimental output increased by 78% at f = f p (in comparison, the simulation output increased by 111%) compared with the maximum output without any installed resonator. In the right-hand side plots of Fig. 8.27, the major wave signals around 400 μs denote the flexural wave mode (corresponding to the antisymmetric (A0 ) Lamb wave mode in a plate). The preceding signals with small amplitudes correspond to the longitudinal wave mode (corresponding to the lowest symmetric (S0 ) Lamb wave mode).

8.7 Problem Set Problem 8.1 Derive the reflection coefficients Cr and Dr for an elastically supported end for transverse motion given by Eq. (8.26d) for Ci = 0 and Di = 0. Examine the solution behavior as K t varies from 0 to ∞. Problem 8.2 Derive the reflection coefficients Cr and Dr for an elastically supported end for rotational motion expressed in Eq. (8.26e) for Ci = 0 and Di = 0. Examine the solution behavior as αr varies from 0 to ∞. Problem 8.3 Consider a semi-infinite beam depicted in the figure below where an incident wave toward x = 0 along the positive x-direction is given vi = Ci e−ikx eiωt . A mass–spring resonator is attached to the free end of the beam. (a) Determine the displacement field of the reflected flexural wave. (b) If the mass m approaches 0 or ∞, determine the corresponding reflected displacement fields. Explain the boundary conditions the limiting cases (m → ∞ and m → 0) represent. (c) Determine the reflected displacement fields when m → ∞ and s → ∞.

(mass: m, spring stiffness: s)

Problem 8.4 Answer the following questions in a beam with distributed mass–spring resonators shown in Fig. 8.12. Use the Euler Bernoulli theory to solve this problem. For numerical calculations, use the following values:

ρ = 2713 kg/m3 , E = 70 GPa, m  = 1 kg/m,

8.7 Problem Set

243

S  = 50 N/m2 , A = (a × a) m2 , a = 0.03 m, Izz = (a 4 /12) m4 (a) Derive Eqs. (8.72) and (8.73). Then plot ρeff (ω) as a function of ω and find the range of ω for which ρeff (ω) is negative. (b) Using Eq. (8.20) with ρ replaced by ρeff (ω) given in Eq. (8.72), plot the dispersion curve shown in Fig. 8.13. Problem 8.5 We revisit Problem 8.4. This time, we additionally consider an elastic foundation for the rotational motion of the beam, as shown in the figure below. The rotation foundation stiffness is denoted by R  .

(a) Show that the dispersion relation for this system can be expressed as R k =− ± 2EI



2

R 2EI

2 +

ρeff (ω)A 2 ω , EI

(8.122)

where ρeff (ω) is given by Eq. (8.72). (b) Among two branches of the dispersion relation, one branch has evanescent waves for all frequencies while the other branch has propagating or evanescent waves depending on frequency. Show that only the parameters that can make the latter branch represent evanescent waves are m  and S  , not R  . (c) Show that a stopband is formed for ω1 ≤ ω ≤ ω2 , where ω1 and ω2 are given by

ω1 = ω0 and ω2 =

ω02 +

S , ρA

S ω02 =  . m

(d) Plot the dispersion curve similar to that shown in Fig. 8.13. For the plot, use the data given in Problem 8.4. Consider the cases when R  = 50, 500, and 5000 N. Problem 8.6 Plot the group velocities for the first and second branches using the Timoshenko beam theory. Use a circular aluminum beam of 1 cm radius with ρ = 2700 kg/m3 , E = 68 GPa, ν = 0.328, G = 25.6 GPa, and K = 5/6. Compare the

244

8 Flexural Waves in a Beam

group velocities against the exact velocities from the theory of elasticity. The exact + ν) velocities are the Rayleigh surface wave velocity c R = cs (0.862 + 1.14ν)/(1 √ for the first branch (lowest branch) and the shear wave velocity cs = G/ρ for the second branch. If we wish to make the limiting group velocity of the first branch by the Timoshenko beam theory equal to c R , what value of K should be used? Problem 8.7 Equation (8.89) representing the dispersion relation for the periodic lattice system can be written as m I ω4 + A(k)ω2 + B(k) = 0.

(8.123)

(a) Derive A(k) and B(k). (b) Eq. (8.123) represents two branches: acoustic and optical branches. Write the dispersion relation (ω − k relation) for each branch in terms of A(k) and B(k). (c) Express the group velocity (∂ω/∂k) in terms of A(k), B(k), A (k), and B  (k), where () = ∂()/∂k. (d) Find the condition that * the optical branch has negative group velocities regardless of ω. (Hint 1: A2 (k) − 4ImB(k) is always real, positive in the optical branch. Hint 2: Consider the condition only for 0 < ka < π .)

References Achenbach JD (1976) Wave propagation in elastic solids. North-Holland Cao L, Zhu Y, Wan S, Zeng Y, Li Y, Assouar B (2021) Perfect absorption of flexural waves induced by bound state in the continuum. Extreme Mech Lett 47:101364 Cao L, Zhu Y, Wan S, Zeng Y, Assouar B (2022) On the design of non-Hermitian elastic metamaterial for broadband absorbers. Int J Eng Sci 181:103768 COMSOL Multiphysics®, www.comsol.com. COMSOL AB, Stockholm, Sweden Crandall SH, Dahl NC, Lardner TJ, Sivakumar MS (2012) An introduction to mechanics of solids, 3rd edn. Tata McGraw-Hill Kim YY, Jang G-W, Choi S (2023) Analysis of thin-walled beams. Springer Kim K, Lee HJ, Park CI, Lee H, Kim YY (2019) Enhanced transduction of MPT for antisymmetric lamb waves using a detuned resonator. Smart Mater Struct 28:075035 Kim YY, Kim EH (2001) Effectiveness of the continuous wavelet transform in the analysis of some dispersive elastic waves. J Acoust Soc Am 110:86–94 Kim YY, Kwon YE (2015) Review of magnetostrictive patch transducers and applications in ultrasonic nondestructive testing of waveguides. Ultrasonics 62:3–19 Labuschagne A, van Rensburg NJ, Van der Merwe A (2009) Comparison of linear beam theories. Math Comput Model 49:20–30 Lee SW, Oh JH (2018) Abnormal stop band behavior induced by rotational resonance in flexural metamaterial. Sci Rep 8:14243 Liu L, Hussein MH (2012) Wave motion in periodic flexural beams and characterization of the transition between Bragg scattering and local resonance. J Appl Mech 79:011003 Miklowitz J (1978) The theory of elastic waves and waveguides. North-Holland Oh JH, Assouar B (2016) Quasi-static stop band with flexural metamaterial having zero rotational stiffness. Sci Rep 6:334410

References

245

Oh JH, Kwon YE, Lee HJ, Kim YY (2016) Elastic metamaterials for independent realization of negativity in density and stiffness. Sci Rep 6:23630 Oh JH, Qi S, Kim YY, Assouar B (2017) Elastic metamaterial insulator for broadband low-frequency flexural vibration shielding. Phys Rev Appl 8(5):054034 Park HW, Oh JH (2019) Study of abnormal group velocities in flexural metamaterials. Sci Rep 9:13973 Park HW, Seung HM, Kim M, Choi W, Oh JH (2021) Continuum flexural metamaterial for broadband low-frequency band gap. Phys Rev Appl 15:024008

Chapter 9

Fundamentals of Elastic Waves in 2D Elastic Media

Before discussing extraordinary two-dimensional (2D) wave phenomena using 2D elastic metamaterials in the next chapter (Chap. 10), we present the fundamentals of elastic wave phenomena in 2D elastic media, such as dispersion relations. According to Sect. 9.1, a 2D infinite isotropic elastic solid can carry two distinct wave modes: longitudinal and transverse. The wave speed varies depending on the mode type but is independent of the wave propagation direction. In the case of an anisotropic solid, wave modes involve coupled longitudinal and transverse motions. Moreover, the wave speed varies depending on the wave propagation direction. This chapter begins with a discussion of these phenomena in terms of dispersion relations. In the second portion of this chapter, we analyze the reflection and transmission of an obliquely incident elastic wave at the interface between two dissimilar isotropic media. Due to the fact that dissimilar media have distinct mechanical impedances, a wave incident from one medium cannot be transmitted in its entirety to the other medium. The well-known quarter-wave impedance-matching element achieves full transmission only for normally incident waves. The next chapter will show that an anisotropic metamaterial matching element enables the complete transmission of an obliquely incident wave across dissimilar media. Before discussing the full transmission method, the later part of this chapter investigates in detail the reflection and transmission of an obliquely incident wave at the interface of two dissimilar isotropic media.

9.1 Governing Field Equation in Elastic Media Particle motions (or movements) in a three-dimensional elastic medium can be described by displacement u(x): u(x, t) = u 1 (x, t)e1 + u 2 (x, t)e2 + u 3 (x, t)e3 , © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_9

(9.1a)

247

248

9 Fundamentals of Elastic Waves in 2D Elastic Media

or u(x, t) = u i (x, t)ei = u i (x1 , x2 , x3 , t)ei ,

(9.1b)

where ei denotes the unit base vector along the Cartesian xi -axis and x =x1 e1 +x2 e2 + x3 e3 represents the position vector in space. Time is denoted by t. In Eq. (9.1b) and subsequent equations, the doubly-repeated index (i for Eq. (9.1b)) is presumably added over i from i = 1 to i = 3 for three-dimensional bodies and from i = 1 to i = 2 for 2D bodies unless stated otherwise. To describe wave motion in terms of u(x) in an elastic medium, three sets of equations are required, such as, equations of motion; stress–strain relation (constitutive relation); strain–displacement relation (kinematic relation). Here, we state these equations without explicit derivations. The readers may find the deviations of these equations elsewhere, including the books by Auld (1973), Achenbach (1976), and Miklowitz (1978). We begin with the strain–displacement relation, expressed as εkl (x, t) =

  1 ∂u k ∂u l , (k, l ∈ {1, 2, 3}) + 2 ∂ xl ∂ xk

(9.2)

where εkl denotes a component of the linear strain tensor ε(x, t). Owing to the definition of εkl , the strain tensor is symmetric such as εkl (x, t) = εlk (x, t), (k, l ∈ {1, 2, 3}).

(9.3)

Subsequently, we consider the equation of motion (corresponding to the linear momentum balance law) in terms of the components σq p of the stress tensor σ (q, p ∈ {1, 2, 3}): ∂σq p (x, t) ∂ 2 u q (x, t) =ρ , ∂xp ∂t 2

(9.4)

where ρ denotes density. As explained earlier, the summation over the repeated index p is assumed in Eq. (9.4). Owing to the law of angular momentum balance, the following symmetry holds σ pq (x, t) = σq p (x, t).

(9.5)

The relationship between the stress field σ(x, t) and the strain field ε(x, t) depends on the properties of the material in consideration. If a linear stress–strain relation is considered, it can be written for a general anisotropic medium as σ pq (x, t) = C pqkl εkl (x, t),

(9.6)

9.1 Governing Field Equation in Elastic Media

249

where C pqkl ( p, q, k, l ∈ {1, 2, 3}) denotes the component of the elasticity tensor C, representing the stiffness coefficients of a linear elastic material. Equation (9.6) is also known as a constitutive relation. The summation over the repeated index is assumed, and the elasticity tensor C emerging in Eq. (9.6) has the following symmetry relations: C pqkl = Cq pkl , C pqkl = C pqlk , C pqkl = Cklpq .

(9.7a, b, c)

Equations (9.7a) and (9.7b) result from the facts that σ pq = σq p and εkl = εlk , respectively. To establish Eq. (9.7c), we introduce a strain energy density function U (ε) that is solely a function of strain from which stress can be defined as σ = ∂U (ε)/∂ε (i.e., σ pq = ∂U (ε)/∂ε pq ). Accordingly, C pqkl can be defined as C pqkl = ∂ 2 U (ε)/∂ε pq ∂εkl . As ∂ 2 U (ε)/∂ε pq ∂εkl = ∂ 2 U (ε)/∂εkl ∂ε pq , the relation C pqkl = Cklpq described in Eq. (9.7c) holds. Owing to the symmetry properties stated in Eq. (9.7), 21 independent coefficients1 exist for general anisotropic media. If a state of plane strain or plane stress2 is assumed in a x − y (or 1–2) plane, only six independent coefficients remain in the elasticity tensor C: C1111 , C1122 (= C2211 ), C1112 (= C1121 , = C1211 = C2111 ), C2222 , C2212 (= C2212 = C1222 = C2122 ), C1212 (= C2112 = C2121 = C1221 ). For isotropic media, the elasticity tensor C can be described in terms of only two independent coefficients, e.g., C1111 and C1122 (or C1212 ). However, it is more convenient to use λ and μ, called the Lamé constants.3 In terms of λ and μ, C pqkl can be compactly expressed as C pqkl = λδ pq δkl + μ(δ pk δql + δ pl δqk ),

(9.8)

where δ pq denotes the Kronecker delta defined as δ pq = 1 if p = q; otherwise, δ pq = 0. The Lamé constants are related to Young’s modulus E and Poisson’s ratio ν, which are preferred in engineering practice, as λ= 1

E νE ,μ = (1 + ν)(1 − 2ν) 2(1 + ν)

(9.9)

Triclinic anisotropy is the most general case with 21 independent material constants in 3D media. Depending on the number of reflection and rotation symmetries, materials can be classified as monoclinic (13 coefficients), orthotropic (9 coefficients), transverse isotropic (5 coefficients), cubic (3 coefficients), and isotropic (2 coefficients) materials (Auld 1973). 2 The plane-stress condition applies to a thin plate under in-plane loading. In contrast, the planestrain condition applies to a thick (infinitely thick in theory) plate under in-plane loading, which does not vary along the thickness direction. 3 The fundamental reason for introducing λ and μ is the compactness of writing the constitutive relation as expressed in Eq. (9.11).

250

9 Fundamentals of Elastic Waves in 2D Elastic Media

Table. 9.1 Index conversion table Ci j

i or j

1

2

3

4

5

6

C pqkl

(pq) or (kl)

11 (xx)

22 (yy)

33 (zz)

23 (yz), 32 (zy)

31 (zx), 13 (xz)

12 (xy), 21 (yx)

or E=

μ(3λ + 2μ) E ,ν = λ+μ 2(λ + μ)

(9.10)

Note that μ is identical to the shear modulus, which is typically denoted as G = E/2(1 + ν). Therefore, substituting Eq. (9.8) into Eq. (9.6) yields σ pq (x, t) = λεkk (x, t)δ pq + 2με pq (x, t),

(9.11)

where εkk = ε11 + ε22 + ε33 denotes the dilation and implies the volume change per unit volume. From now on, we use the shorthand notation Ci j (i, j ∈ {1, 2, 3, 4, 5, 6}) in which only two indices are used instead of 4-index tensor notation C pqkl ( p, q, k, l ∈ {1, 2, 3} = {x, y, z}).4 The conversion rules between C pqkl and Ci j are listed in Table 9.1. For instance, the following shorthand symbols are used for 2D problems defined in the 1–2 (or x−y) plane: C11 = C1111 , C12 = C1122 , C16 = C1112 , C22 = C2222 , C26 = C2212 , C66 = C1212 , Ci j = C ji . If Ci j is utilized, the constitutive relation (9.6) can be placed in the following form: Ti = Ci j S j , (i, j ∈ {1, 2, . . . , 6})

(9.12a)

i.e.,

4

To clearly distinguish the shorthand notation from the 4th-order elasticity tensor, ci j is more preferred instead of Ci j . However, we use the same notation C to avoid the introduction of numerous symbols. Note that Ci j is not a tensor component unlike Ci jkl . Therefore, careful consideration is required to relate the stress and strain components when Ci j is used.

9.1 Governing Field Equation in Elastic Media

⎧ ⎫ ⎡ T1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T2 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎨ T3 ⎬ ⎢ ⎢ =⎢ ⎪ ⎪ T 4 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ T5 ⎪ ⎪ ⎣ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ T6

⎤ ⎧ S1 ⎫ ⎪ ⎪ C11 C12 C13 C14 C15 C16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ S2 ⎪ C21 C22 C23 C24 C25 C26 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎥ C31 C32 C32 C34 C35 C36 ⎥ S3 ⎬ , ⎥= ⎪ C41 C42 C43 C44 C45 C45 ⎥ ⎪ S4 ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ S5 ⎪ ⎪ C51 C52 C53 C54 C55 C56 ⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ C61 C62 C63 C64 C65 C66 S6

251

(9.12b)

where the symbols Ti and S j denote the stress and strain components consistent with Ci j , which are defined as ⎧ ⎫ ⎧ σ11 T ⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ22 ⎪ T2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨σ ⎨T ⎪ ⎬ ⎪ 3 33 = ⎪ ⎪ ⎪ T4 ⎪ ⎪ σ23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪σ ⎪ ⎪ ⎪ T5 ⎪ ⎪ 31 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎪ ⎭ ⎪ T6 σ12

⎧ ⎫ ⎧ ⎫ S ε11 = εx x ⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S2 ⎪ ε22 = ε yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨S ⎪ ⎨ε = ε ⎬ ⎪ ⎬ zz 3 33 zz , = ⎪ ⎪ = σ yz ⎪ S4 ⎪ 2ε23 = 2ε yz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = σzx ⎪ S 2ε = 2ε 5⎪ 31 zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭ ⎪ ⎭ ⎪ ⎭ = σx y S6 2ε12 = 2εx y ⎫ = σx x ⎪ ⎪ ⎪ ⎪ = σ yy ⎪ ⎪ ⎪ ⎪ =σ ⎬

(9.13a,b)

Since C pqkl = Cklpq , the following relation holds as well Ci j = C ji ,

(9.14)

  and there are 21 independent coefficients are available in the matrix Ci j for 3D elastic media. In the case of isotropic media, Ci j (Ci j = C ji ) can be expressed in terms of two coefficients, λ and μ, C11 = C22 = C33 = λ + 2μ, C12 = C13 = C23 = λ, C44 = C55 = C66 = μ, other Ci j = 0,

(9.15)

or in terms of E and ν, E(1 − ν) , (1 + ν)(1 − 2ν) νE , = (1 + ν)(1 − 2ν) = G, other Ci j = 0.

C11 = C22 = C33 = C12 = C13 = C23 C44 = C55 = C66

(9.16)

Alternatively, all nonzero values of Ci j for isotropic media can be expressed in terms of two independent coefficients C11 and C66 : for 2D isotropic media: 2 independent coefficients C11 and C66 , with

252

9 Fundamentals of Elastic Waves in 2D Elastic Media

Table. 9.2 Material properties of some materials under the plane-strain condition Material

Type

Density (kg/m3 )

Stiffness (unit: 10 GPa = 1010 N/m2 )

ρ

C 11

C 22

10.37

10.37

C 12

C 66

C 16

C 26

Aluminum (Al)*

Metal

2700

5.11

2.63

0

0

Polyether ether ketone5 (PEEK)*

Polymer

1320

0.901

0.901

0.599

0.151

0

0

Glass–Pyrex*

Insulator

2320

7.3

7.3

2.3

2.5

0

0

Quartz (SiO2 )**

Insulator

2651

3.988

0

0

Rutile (TiO2 )**

Insulator

4260

26.60

26.60

12.33

18.86

0

0

Calcium Molybdate (CaMoO4 )

Insulator

4255

14.47

14.47

6.64

4.51

1.34

1.34

8.674

8.674

0.699

*Isotropic materials bear two independent coefficients because they satisfy C12 = C11 − 2C66 , C22 = C11 , and C16 = C26 = 0. **Cubic materials bear three independent coefficients because they satisfy the C22 = C11 and C16 = C26 = 0.

C12 = C11 − 2C66 .

(9.17)

Ultimately, the substitution of Eqs. (9.2) and (9.6) into Eq. (9.4) yields the equation of motion expressed only in terms of the displacement u(x, t) in general anisotropic elastic media:   ∂ 2 u q (x, t) ∂ ∂u k (x, t) C pqkl =ρ . (9.18) ∂xp ∂ xl ∂t 2 In the case of isotropic media, one can use the relation listed in Table 9.1 and substitute Eq. (9.15) into Eq. (9.18) to obtain μ

∂ 2 u p (x, t) ∂ 2 u q (x, t) ∂ 2 u q (x, t) + (λ + μ) =ρ . ∂ x p∂ x p ∂ x p ∂ xq ∂t 2

(9.19)

In subsequent discussions, we are primarily concerned with 2D problems with u 3 = 0. In this case, the following notations are often convenient to use: x = x1 , y = x2 ; u x = u 1 , u y = u 2 . Similarly, the unit base vectors in the Cartesian coordinate system, ei (i = 1, 2, 3), may be written as ex , e y , and ez , whenever required. Note that in the 2D problems, the following six nonzero stiffness components are independent for general anisotropic media: for 2D anisotropic media: six independent coefficients C11 , C22 , C12 , C16 , C26 , and C66 .

9.2 Dispersion Relations in 2D Anisotropic–Isotropic Media

253

Table. 9.3 Material properties of aluminum and PEEK under the plane-stress condition Material

Type

Density (kg/m3 )

Stiffness (unit: 10 GPa = 1010 N/m2 )

ρ

C 11

C 22

C 12

C 66

C 16

C 26

Aluminum (Al)*

Metal

2700

7.85

7.85

2.59

2.63

0

0

Polyether ether ketone (PEEK)*

Polymer

1320

0.503

0.503

0.201

0.151

0

0

The values of the stiffness Ci j for certain materials are listed in Tables 9.2 and 9.3.

9.2 Dispersion Relations in 2D Anisotropic–Isotropic Media6 Consider a plane wave propagating along the direction of k, as illustrated in Fig. 9.1. Here, we primarily consider a wave in a thin plate, implying that all analyses are based on the plane-stress condition. The displacement in the plate can be expressed as u(x, t) = Uei(ωt−k·x) ,

(9.20a)

u p (x, t) = U p ei(ωt−k·x) , ( p = x, y).

(9.20b)

or

In Eq. (9.20), k denotes the wave vector defined as T  k = kn = k x ex + k y e y = k x , k y ,

(9.21)

where n represents the unit vector defined as  T n = n x ex + n y e y = n x , n y ,

(9.22a)

n 2x + n 2y = 1.

(9.22b)

with

5

Polyether ether ketone (a plastic material) is commonly used to make a wedge used to convert a longitudinal wave to a transverse wave for nondestructive structural testing or noninvasive ultrasonic flow velocity measurement of a flow inside a pipe. 6 The Bloch–Floquet theorem discussed in Chap. 7 for 1D periodic continuum media can be extended if the dispersion analysis of 2D or 3D solid media is needed. See Hussein (2009) for the reduced Bloch mode expansion.

254

9 Fundamentals of Elastic Waves in 2D Elastic Media

Fig. 9.1 Plane wave propagating in elastic media

In Eqs. (9.21) and (9.22), the unit base vectors along the x- and y-coordinates of the 2D Cartesian coordinate system are denoted by ex and e y . The unit base vector along the z-direction is denoted by ez . Based on Eqs. (9.21) and (9.22), the wavenumber k and the components k x and k y of the wave vector k can be expressed as k x = kn x , k y = kn y ,k =



k x2 + k 2y .

(9.23)

Substituting Eqs. (9.20) and (9.21) into Eq. (9.18) yields the following eigenvalue problem, which is known as the Christoffel equation (See, e.g., Rose (2014)): 

12 k 2

11 k 2 − ρω2 2

21 k

22 k 2 − ρω2



Ux Uy

 = 0.

(9.24)

In Eq. (9.24), i j denotes the component of the Christoffel matrix  = [ i j ]:

11 (n) = C11 n 2x + C66 n 2y + 2C16 n x n y

22 (n) = C66 n 2x + C22 n 2y + 2C26 n x n y

12 (n) = 21 =

C16 n 2x

+

C26 n 2y

.

(9.25)

+ (C12 + C66 )n x n y

To have a nontrivial solution k for a given angular frequency ω, the determinant (ω, k, n) of the matrix in Eq. (9.24) should be zero:   k 2 − ρω2

12 k 2 (ω, k, n) =  11 2

21 k

22 k 2 − ρω2

   = 0. 

(9.26)

9.2 Dispersion Relations in 2D Anisotropic–Isotropic Media

255

Equation (9.26) provides the dispersion relation; it gives k-values in the k-space (k x − k y space) as a function of the wave propagation direction n for a fixed ω. As Eq. (9.26) is a quadratic equation for k 2 , four wavenumbers exist. Consequently, four wave modes describe in-plane elastic wave motions in a 2D elastic medium. • Phase velocity and group velocity After determining the k − ω relationship, the phase velocity v p and group velocity vg 7 can be calculated. The meaning of the phase and group velocities in 2D or 3D (three-dimensional) wave problems is fundamentally the same as that in onedimensional wave problems. Still, in the 2D or 3D cases, the magnitudes and directions of v p and vg should be carefully analyzed. First, we consider the phase velocity v p , defined as, vp =

ω ω ex + e y . kx ky

(9.27)

Instead of v p , the slowness vector s is often preferred, which is defined as: s=

kn k = , ω ω

(9.28a)

or  (sx , s y ) =

kx k y , ω ω



 =

 1 1 , , v px v py

(9.28b)

where v px and v py denote the x- and y-component of the phase velocity v p , respectively. In contrast, the group velocity vg is defined as vg =

∂ω ∂ω ex + ey . ∂k x ∂k y

(9.29)

To evaluate the vg defined in Eq. (9.29), Eq. (9.26) can be used 0 = d =

∂ ∂ dk x + dk y . ∂k x ∂k y

(9.30)

Using Eq. (9.30), one can write ∂ω/∂ki (i = x, y) as   ∂/∂k y ∂ω  ∂/∂k x ∂ω ∂ω  ∂ω , . = =− = =−   ∂k x ∂k x k y - fixed ∂/∂ω ∂k y ∂k y kx - fixed ∂/∂ω

7

(9.31)

The same definitions of the phase and group velocity apply to both isotropic and anisotropic cases. .

256

9 Fundamentals of Elastic Waves in 2D Elastic Media

Substituting Eq. (9.31) into Eq. (9.29) yields an equation for vg as   ∇k  ∂ 1 ∂ , ex + ey ≡ − vg = − ∂/∂ω ∂k x ∂k y ∂/∂ω

(9.32)

where ∇k  denotes the gradient of  in the k-space. • Isotropic media To begin with, we first consider the dispersion relation for isotropic media. In the case of isotropic media, Eq. (9.26) provides two explicit k − ω relationships: k(ω) = ±k L = ±

ω , (2 wavenumbers ), cL

(9.33)

ω , (2 wavenumbers), cT

(9.34)

k(ω) = ±k T = ±

where c L and cT denote the longitudinal (or dilatational) and transverse (or shear)8 wave speeds, defined as  cL =

 λ + 2μ E (1 − ν) = · , ρ ρ (1 + ν)(1 − 2ν)    C66 μ G = = . cT = ρ ρ ρ 

C11 = ρ

(9.35a)

(9.35b)

As k in Eqs. (9.33) and (9.34) is independent of n, the dependence of k on n is neglected. Here and later, we use the sub- or superscripts L and T to denote the longitudinal and transverse waves, respectively. It should be noted that the longitudinal wave speed is always larger than the transverse wave speed in isotropic media because the ratio κ =c L /cT is always larger than unity. The ratio κ is explicitly expressed in terms of ν as  cL κ= = cT

λ + 2μ = μ



2(1 − ν) >1 1 − 2ν

(9.36)

Because c L > cT , k L < k T . Therefore, the wavelength λT of a transverse wave is always shorter than the wavelength λ L . of a longitudinal wave: λT =

8

2π 2π > λL = . kT kL

Throughout this book, dilatational and shear waves are also used to denote longitudinal and transverse waves, respectively.

9.2 Dispersion Relations in 2D Anisotropic–Isotropic Media

257

Fig. 9.2 Snapshot of the displacement field for a longitudinal and b transverse waves in an isotropic elastic medium, aluminum, given in Table 9.3

By substituting Eqs. (9.33) and (9.34) into Eq. (9.24), the eigenvector U can be determined as Ui = APi (i = L or T ),

(9.37)

where P denotes the normalized eigenvector, also known as the polarization vector and A is an undetermined amplitude. For a longitudinal wave propagating in an infinite 2D isotropic solid, P is identified as9 P L = n = {n x , n y }T .

(9.38)

For a transverse wave, P becomes T  PT = ez × n = −n y , n x .

(9.39)

The distinct characteristics of the longitudinal and transverse waves are presented in Fig. 9.2. In the case of longitudinal waves, P L is parallel to k L = k{n x , n y }T . Therefore, the wave motion of a longitudinal wave is parallel to the direction of propagation. Also, the corresponding wave field is irrotational because its curl field ∇ × u (representing rotation) is identically zero: for longitudinal wave: ∇ × u = −i Ak L × P L ei(ωt−k·x) = 0 (∇ · u = 0),

9

(9.40)

The same symbol, T, is used to denote the “transverse wave” and the “transpose” of a matrix; readers may need to pay some attention to the meaning of T.

258

9 Fundamentals of Elastic Waves in 2D Elastic Media

where ∇ · u (representing dilation, i.e., rate of volume variation) and ∇ × u are defined as ∇ ·u=

∂u 1 ∂u 2 ∂u 3 ∂u i = + + , ∂ xi ∂ x1 ∂ x2 ∂ x3

∇ × u = ei εi jk

∂u k , ∂x j

with the permutation symbol εi jk defined as εi jk = 0 if any two of (i, j, k) are equal, i.e., for (i, j, k) = (1, 1, 2), (2, 2, 3), etc., εi jk = 1 for even permutions, i.e., for (i, j, k) = (1, 2, 3), (2, 3, 1), (3, 2, 1), εi jk = −1 for odd permutions, i.e., for (i, j, k) = (1, 3, 2)(2, 1, 3), (3, 1, 2). In contrast, the wave motion for a transverse wave should be normal to the direction of wave propagation because PT · kT = 0 (kT = k{−n y , n x }T ). In this case, the wave field is equi-voluminal because its divergence field, ∇ · u (representing volume change rate), is identically zero: for transverse wave: ∇ · u = −i AkT · PT ei((ω−k·k·x) = 0(∇ × u = 0).

(9.41)

The slowness curves for a few isotropic materials are plotted in Fig. 9.3. In 2D media, two slowness curves exist. All curves, either for longitudinal or transverse waves, are circular because k is independent of the propagation direction n. As the longitudinal wave speed (c L ) is faster than the transverse wave speed (cT ), the slowness curve for the longitudinal wave is always closer to the origin than that for the transverse wave. In addition, the group velocity vg is identical to v p because wave speed is independent of k and n.

Fig. 9.3 Slowness curve for isotropic media under the plane-stress condition. a Aluminum and b PEEK (k x = kn x , k y = kn y )

9.2 Dispersion Relations in 2D Anisotropic–Isotropic Media

259

• Anisotropic media Herein, we revert to the case of general anisotropic media. In this case, we can determine the following four eigenvalues, i.e., wavenumbers k solving the 4th-order Eq. (9.26):

k(ω, n) = ±k Q L

    ρω2 ( 11 + 22 − ( 11 − 22 )2 − 4 2 )  12 , ≡± 2 2( 11 22 − 12 )

(9.42)

and

k(ω, n) = ±k QT

    ρω2 ( 11 + 22 + ( 11 − 22 )2 − 4 2 )  12 . ≡± 2 2( 11 22 − 12 )

(9.43)

The subscripts QL and QT denote “quasi-longitudinal” and “quasi-transverse” because the QL and QT waves are nearly dominated by longitudinal and transverse deformations, respectively. Nevertheless, the wave motions in both QL and QT modes involve coupled longitudinal and shear deformation. The QL and QT waves are reduced to pure longitudinal and transverse waves if isotropic media are considered. normalized polarization vectors, For k = k Q L and k = k QT , the  corresponding  P Q L and P QT (P Q L  = 1 and P QT  = 1), are determined from the Christoffel equation, Eq. (9.24) as 

 i T

Pi (ω, n) = Pxi , Py

=

⎧ ⎨ ⎩

Xi

1

⎫T ⎬

 , (i = Q L, QT ), ⎭ 1 + X i2 1 + X i2

(9.44)

where X i (ω, n) =

12 ki2 (i = Q L, QT ). ρω2 − 11 ki2

(9.45)

Due to the orthogonality between P Q L and P QT , the following relation holds P Q L · P QT = PxQ L PxQT + PyQ L PyQT = 0,

(9.46)

! P Q L × P QT · ez = 1 ↔ PxQ L PyQT − PyQ L PxQT = 1

(9.47)

and

Unlike in isotropic media, none of P Q L and P QT is either parallel or perpendicular to the wave vector k in general. The displacement field of the quasi-longitudinal and quasi-transverse waves in quartz (an anisotropic medium) is presented in Fig. 9.4

260

9 Fundamentals of Elastic Waves in 2D Elastic Media

Fig. 9.4 Snapshot of the displacement field for a quasi-longitudinal and b quasi-transverse waves in an anisotropic medium, quartz, listed in Table 9.2

√ √ for n = {1/ 2, 1/ 2}T . Unlike displacements for an isotropic medium shown in Fig. 9.2, the displacements in Fig. 9.4 are neither exactly parallel nor perpendicular to the wave propagation directions. The slowness curves for some anisotropic media listed in Table 9.2 are plotted in Fig. 9.5. For the plot, we may express n as n = {cos θ, sin θ }T , where θ denotes the angle measured from the horizontal axis (k x /ω axis), and θ is incrementally varied from 0◦ to 360◦ . Thus, two values of k/ω can be determined from Eq. (9.26) for the selected value of θ (i.e., n x and n y ), denoted as k Q L /ω and k QT /ω. Thus, using n = {cos θ, sin θ }T , we can determine two points in the k x /ω–k y /ω plane, {k x /ω, k y /ω}TQ L and {k x /ω, k y /ω}TQT . In particular, the slowness curves in Fig. 9.5 are the loci of these points. Note that the distance from the origin ({k x /ω = 0, k y /ω = 0}T ) to a point (with n) in the slowness curve is proportional to the traveling time required to reach the point at a specified distance in the two-dimensional space from the origin. Another important observation from the slowness curves in Fig. 9.5 is that vg is normal to the slowness curve for all instances. Mathematically, this condition may be stated as δk · vg = 0,

(9.48)

where δk denotes an infinitesimal variation along the slowness curve, i.e., the direct tangent to the curve. This condition can be theoretically proven, as Auld (1973) reported. • Findings from dispersion analysis It is worth summarizing some findings from the dispersion analysis. Among others, the wavenumber k depends on the propagation direction n in anisotropic media, whereas it does not in isotropic media, compare Eqs. (9.33) and (9.34) with

9.2 Dispersion Relations in 2D Anisotropic–Isotropic Media

261

Fig. 9.5 Slowness curves (or diagrams) for a quartz, b rutile, and c calcium molybdate described in Table 9.2. (Symbols s and vg denote the slowness vector and group velocity vector. Symbol P represents the polarization vector (i.e., particle motion direction). The plane-strain condition is used

Eqs. (9.42) and (9.43). Depending on the values of the material property in consideration, the shape of the slowness curve significantly differs, as portrayed in Fig. 9.5. (The properties of the material considered for the plots in Fig. 9.5 are listed in Table 9.2.) The distinct characteristics of the slowness curves can be summarized from Figs. 9.3 and 9.5 as follows: 1. Isotropic case (C12 = C11 − 2C66 , C16 = C26 = 0) (with two independent coefficients) a. The inner and outer slowness curves correspond to the (pure) longitudinal and transverse waves, as confirmed by the polarization vectors P’s that are parallel and orthogonal to k, respectively. b. As the longitudinal wave is associated with the inner curve, the phase speed of the longitudinal wave is larger than that of the transverse wave. c. Two slowness curves are circular, implying that the magnitude of the phase velocity is identical, irrespective of the propagation direction k or n. d. The direction of the group velocity is the same as that of the phase velocity.

262

9 Fundamentals of Elastic Waves in 2D Elastic Media

2. Cubic case: (C12 = C11 − 2C66 , C16 = C26 = 0) (with three independent coefficients) a. As the cubic materials do not exhibit isotropic behavior, the phase velocities depend on the wave propagation direction n. b. As observed from the polarization vector, the wave associated with the inner (outer) slowness curve can be characterized as a quasi-longitudinal (quasitransverse) wave. The phase velocity of the quasi-longitudinal wave mode is larger than that of the quasi-transverse wave mode. c. The group velocity vg is normal to the tangent of the slowness curve. 3. General anisotropic case: (C12 = C11 − 2C66 , C16 = 0, C26 = 0) a. Unlike isotropic or cubic materials that exhibit certain symmetries with respect to the x- and y-axes, the case considered in Fig. 9.5c does not manifest such symmetries. Therefore, the slowness curves are no longer symmetric with respect to the x- and y-axes. b. Neither of the slowness curves corresponds to the pure longitudinal or transverse wave; however, their slowness curves may be characterized by quasi-longitudinal and quasi-transverse wave motions. c. As C16 = 0, the dilatational deformation (εx x ) and shear deformation (εx y ) are coupled. Generally, such coupling may not be desirable if the longitudinal and transverse wave modes are to be separately handled. However, this coupling may provide a unique possibility to fully convert an incident longitudinal wave to a transverse wave (Kweun et al. (2017), Yang and Kim (2018), Yang et al. (2018, 2019), and Lee et al. (2022)). The next chapter will present more details on this subject and the realization of the anisotropy required for full mode conversion using metamaterials. Figure 9.5 shows that anisotropic materials exhibit unique slowness curves (or dispersion relations). However, some extraordinary slowness curves, as may be desired for engineering applications, cannot be realized only with natural materials. Figure 9.6 shows that elaborately designed metamaterials can realize various slowness curves. Because the slits in the metamaterial shown in Fig. 9.6a are symmetrically arranged with respect to the x- and y-axes, the coefficient C16 representing coupling between normal and shear deformations is identically zero. Even with C16 = 0, the adjustment of the slit length (h) allows the realization of various cases, including the case of C11 = C66 at 100 kHz10 for which two slowness curves overlap. Ahn et al. (2017) investigated an interesting wave phenomenon occurring at overlapping points representing conical refraction.

10

Because wave behavior in artificially engineered metamaterials is frequency dependent, the relation C11 = C66 does not generally hold at other frequencies.

9.3 Fundamentals of Wave Reflection and Transmission in Isotropic Media

263

Fig. 9.6 Slowness curves obtained by off-centered double-slit metamaterials. a Metamaterial fabricated on the base aluminum and b slowness curves at 100 kHz for different slit lengths (h). (L = L x = L y = 1.8 mm, w = 0.1 mm, s = 0.3 mm) (from Lee et al. (2017))

9.3 Fundamentals of Wave Reflection and Transmission in Isotropic Media This section considers wave reflection and transmission occurring at the interface of two dissimilar isotropic media. Two representative cases of oblique incidence with either a longitudinal or transverse wave are considered in Fig. 9.7a. Media A and B are assumed to be semi-infinite along the horizontal and infinite along the vertical directions. In either case, both longitudinal and transverse waves appear in the reflected wave field in medium A and in the transmitted wave field in medium B. In Fig. 9.7b, the angles of the reflected longitudinal and transverse waves in medium A are denoted as θ LA and θTA , respectively, and the angles of the transmitted longitudinal and transverse waves in medium B are denoted as θ LB and θTB , respectively. The incident longitudinal wave from medium A is denoted as θinc . Depending on the mode type of the incident wave (see Fig. 9.7b) for the case of the longitudinal wave incidence), we write the displacement of the incident wave field as

264

9 Fundamentals of Elastic Waves in 2D Elastic Media

Fig. 9.7 a Oblique incidence of longitudinal or transverse waves from an isotropic solid medium A to another isotropic solid medium B. b Longitudinal wave incidence case with various symbols indicated. (Medium α is assumed with density ρα and elastic coefficients Ciαj , α = A, B)

uinc (x, t) = u LA+ (x, t) for plane longitudinal wave incidence,

(9.49a)

uinc (x, t) = uTA+ (x, t) for plane transverse wave incidence.

(9.49b)

Likewise, the displacements of the reflected longitudinal and transverse waves in medium A are denoted by u LA− (x, t) and uTA− (x, t), respectively. Similarly, the transmitted longitudinal and transverse waves in medium B can be denoted by u LB+ (x, t) and uTB+ (x, t), respectively. Using Eqs. (9.20a) and (9.37), the displacement uij (x, t) can be expressed as uij (x, t) = U ij Pij exp[i(ωt − kij · x)] ≡ U ij Pij exp(iηij ) (i = A, B; j = L + , L − , T + , T − ),

(9.50)

where the wave vectors kij are given by ! ! kij = k ij nij = k ij n i,x j ex + n i,y j e y i = A, B; j = L + , L − , T + , T − ,

(9.51)

with k iL + = −k iL − ≡ k iL and k Ti + = −k Ti − ≡ k Ti .

(9.52)

In addition, ηij = ωt − k ij nij · x = ωt − k ij (n i,x j x + n i,y j y) =k ij (cij t − n i,x j x − n i,y j y). In Eq. (9.53), wave speed cij is defined as

(9.53)

9.3 Fundamentals of Wave Reflection and Transmission in Isotropic Media

cij =

ω or ω = k ij cij , (i = A, B; j = L + , L − , T + , T − ). k ij

265

(9.54)

To facilitate writing, we use ciL or ciT to denote the wave speeds in Eq. (9.54): ciL + = −ciL − ≡ ciL and ciT + = −ciT − ≡ ciT (i = A, B).

(9.55)

The normal vector nij can be expressed in terms of the angle as. • for j = L + and j = T + (forward propagation waves), nij = {n i,x j , n i,y j }T = {cos θ ij , sin θ ij }T ,

(9.56)

• for j = L − and j = T − (backward propagation waves), nij = {n i,x j , n i,y j }T = {− cos θ ij , sin θ ij }T .

(9.57)

Based on Eqs. (9.38) and (9.39), we can explicitly express the polarization vector Pij as. • for j = L + and j = L − (longitudinal waves), PiL + = {cos θ Li + , sin θ Li + }T , PiL − = {− cos θ Li − , sin θ Li − }T ,

(9.58)

• for j = T + and j = T − (transverse waves), PiT + = {− sin θTi + , cos θTi + }T , PiT − = {− sin θTi − , − cos θTi − }T .

(9.59)

To satisfy the continuities in the field variables of the two dissimilar media at x = 0, the following conditions must be satisfied: " displacement: " traction:

u xA (x = 0− , y, t) = u xB (x = 0+ , y, t), u yA (x = 0− , y, t) = u yB (x = 0+ , y, t),

σxAx (x = 0− , y, t) = σxBx (x = 0+ , y, t), σxAy (x = 0− , y, t) = σxBy (x = 0+ , y, t).

(9.60)

(9.61)

266

9 Fundamentals of Elastic Waves in 2D Elastic Media

Among these, we consider first the displacement continuity condition in Eq. (9.60), which can be written in a compact form as u A (x = 0− , y, t) = u B (x = 0+ , y, t),

(9.62)

where +

u A (x = 0− , y, t) = U LA+ P LA+ exp[i(ωt − k LA n yA,L y)] −

+ U LA− P LA− exp[i(ωt − k LA n yA,L y)] −

+ UTA− PTA− exp[i(ωt − k TA n yA,T y)],

(9.63a)

+

u B (x = 0+ , y, t)=U LB+ P LB+ exp[i(ωt − k LB n yB,L y)] +

+ UTB+ PTB+ exp[i(ωt − k TB n yB,T y)].

(9.63b)

In the right-hand side of Eq. (9.63a), the first term represents the incident longitudinal wave, and the remaining terms denote the reflected waves in medium A. Equation (9.63b) contains the terms denoting the transmitted waves in medium B. Because Eq. (9.62) should be valid for any location y at any time t, the following relations must hold:11 +

−

−

+

+

k y component: k LA n yA,L = k LA n yA,L = k TA n yA,T = k LB n yB,L = k TB n yB,T  σ. (9.64) Equation (9.64) states that all involved waves in the reflection and transmission (refraction) should bear the same y-directional wavenumber ( k y = kn y  σ ) as that of the incident wave. Note that the y-direction denotes the tangential direction relative to the interface x = 0. Thus, the relationship (9.64) is often referred to as the conservation of the tangential momentum. For subsequent analysis, σ can be explicitly written as σ = k LA sin θinc for longitudinal wave incidence.

(9.65)

• Longitudinal wave incidence This is the case described in Fig. 9.7b. In this case, we use (9.49a) and set

This symbol σ representing the y-directional wavenumber should not be confused with σ used to represent stress components σij ; when σ is used without indices, it is used represent the y-directional wavenumber.

11

9.3 Fundamentals of Wave Reflection and Transmission in Isotropic Media

267

θ LA+ = θinc . Using Eqs. (9.54) and (9.55), the relations between the wavenumbers are found as k LA ≡ kinc , k TA = k LB

c LA kinc = κ A kinc , cTA

cA cA = LB kinc , k TB = LB kinc . cL cT

(9.66)

To determine the angles of the reflected and transmitted waves, we substitute the relations in Eq. (9.66), and n y expressed in (9.56). The results are θ LA− = θ LA+ = θinc , sin θTA− = sin θ LB+

cTA 1 sin θinc = A sin θinc , κ c LA

cB cB = LA sin θinc , sin θTB+ = TA sin θinc . cL cL

(9.67)

Equation (9.67) represents Snell’s law involving both longitudinal and transverse waves for the longitudinal wave incidence. Interestingly, only a single or no wave mode is transmitted in medium B in the following two cases: Case 1: c LB /c LA > 1: In this case, there exist an incidence angle θcr such that θcr = sin−1 (c LA /c LB ),

(9.68)

above which no transmitted longitudinal wave propagates into medium B. If θinc = θcr , θ LB becomes π/2.12 Case 2: cTB /c LA > 1: Because c LB > cTB , c LB /c LA > 1 also holds in this case. If cTB /c LA > 1, there exists an incidence angle θcr such that θcr = sin−1 (c LA /cTB ),

(9.69)

above which neither the transmitted transverse wave nor the transmitted longitudinal wave is transmitted into medium B. All waves in medium B are evanescent. • Transverse wave incidence If θinc > θcr , the longitudinal wave in medium B will decay along the x-direction. However, it can propagate along the interface (along y-direction), and therefore, the wave may be called the interface wave. We will not investigate further into the case when θinc > θcr .

12

268

9 Fundamentals of Elastic Waves in 2D Elastic Media

In the case of the transverse wave incidence, we use Eq. (9.49b) to write the displacement field in medium A as +

u A (x = 0− , y, t) = UTA+ PTA+ exp[i(ωt − k TA n yA,T y)] −

+ U LA− P LA− exp[i(ωt − k LA n yA,L y)] −

+ UTA− PTA− exp[i(ωt − k TA n yA,T y)].

(9.70)

The displacement field in medium B is given by Eq. (9.63b). Using Eq. (9.70) and repeating the analysis performed for the longitudinal wave incidence, we obtain the following results: k TA = kinc , k LA = k TB

cTA 1 kinc = A kinc , κ c LA

cA cA = TB kinc , k LB = TB kinc , cT cL

(9.71)

and θTA− = θTA+ = θinc , sin θ LA− = sin θ LB+

c LA sin θinc = κ A sin θinc , cTA

cB cB = LA sin θinc , sin θTB+ = TA sin θinc , cT cT

(9.72)

Equation (9.72) represents Snell’s law involving both longitudinal and transverse waves for the case of transverse wave incidence. Based on the results stated in Eqs. (9.67) and (9.72), the following observations can be made: 1. The angle θTα of a transverse wave is smaller than angle θ Lα of a longitudinal wave, i.e., θTα < θ Lα , (α = A, B) because the transverse wave speed cTα is always smaller than the longitudinal wave cαL (cTα < cαL ) in any isotropic medium. 2. The propagation angle of the wave of the same mode for the same medium is the same regardless of its propagation direction. Therefore, we will drop “+” and “–” in the definition of the angles and use the following notations: θ Lα− = θ Lα−  θ Lα , θTα+ = θTα−  θTα , (α = A, B).

(9.73)

To determine the remaining unknowns U LA− , UTA− , U LB+ , and UTB+ for a prescribed (corresponding to the longitudinal wave incidence), or a prescribed UTA+ (corresponding to the transverse wave incidence), the four equations in Eqs. (9.60) and (9.61) should be solved simultaneously. We will use the state vector formulation explained in the next section for the solving.

U LA+

9.4 State Vector Representation for Transmission and Reflection Analysis

269

9.4 State Vector Representation for Transmission and Reflection Analysis The field-matching conditions at the interface (x = 0) of two dissimilar media in Fig. 9.7b are stated in Eqs. (9.60) and (9.61). Instead of using the displacement components u x and u y , the velocity components v y and v y are more convenient to use because vx · σx x and v y · σx y at x = constant represent the instantaneous power. Therefore, we introduce the following state vector f defined as f = {vx , v y , σx x , σx y }T .

(9.74)

The state variables are the physical quantities that can be prescribed at x = constant. In this respect, σ yy is not the state variable that can be prescribed at x = constant. Therefore, it does not appear as a component of f defined in Eq. (9.74). For the subsequent discussions, we will assume that the media are anisotropic because we need to also deal with anisotropic media in the next chapter. To explicitly write f in terms of u x and u y , we use the following expressions: ∂u y ∂u x , vy = , ∂t ∂t   ∂u y ∂u y ∂u x + C12 + C16 + , ∂y ∂x ∂y   ∂u y ∂u y ∂u x + C26 + C66 + . ∂y ∂x ∂y

vx = ∂u x ∂x ∂u x = C16 ∂x

σx x = C11 σx y

(9.75)

(9.76)

For wave reflection and transmission of a plane wave depicted in Fig. 9.7b, we denote the k y components of all waves by σ ; the same y-directional wavenumber is used for all wave modes because of the reason stated in writing Eq. (9.65). Using σ , the exponential wave form, exp[i(ωt − k · x)] = exp[i(ωt − k x x − k y y)] appearing in the displacement and stress fields can be written in the following form: exp[i(ωt − k x x − k y y)] ≡ exp[i(ωt − τ x − σ y)],

(9.77)

where τ is introduced to denote the x-directional component of the wave vector k.13 If a considered medium is finite in the x-direction, four wave modes can exist. The x-directional wavenumbers of four possible wave modes in a solid medium are denoted as   τ (= k x ) ∈ τα , τβ , τδ , τγ ,

(9.78)

where 13

We write k x and k y , the x- and y-components of the wavevector k, as τ and σ , respectively.

270

9 Fundamentals of Elastic Waves in 2D Elastic Media

τα : forward QL-mode (L-mode) wave in anisotropic (isotropic) media, τβ : backward QL-mode (L-mode) wave in anisotropic (isotropic) media, τγ : forward QT-mode (T-mode) wave in anisotropic (isotropic) media, τδ : backward QT-mode (T-mode) wave in anisotropic (isotropic) media. In the case of isotropic media, τ j ( j = α, β, γ , δ) becomes +

−

+

τα = k L n xL and τβ = k L n xL , τγ = k T n Tx and τδ = k T n Tx

−

(9.79)

Denoting the polarization vector corresponding to the τ j ( j = α, β, γ , δ) as P j , we can express the displacement as ! u(x, y, t) = APα e−iτα x + BPβ e−iτβ x + CPγ e−iτγ x + DPδ e−iτδ x e−iσ y eiωt , (9.80) where A, B, C, and D denote the displacement amplitudes. To explicitly express P j , Eqs. (9.38) and (9.39) can be used in the case of isotropic media and Eq. (9.44), in the case of anisotropic media. As the displacement is assumed to have the specific form stated in Eq. (9.80), the state vector f(x, y, t) can be put into the following form: ⎧ ⎫ ⎧ ⎫ Vx (x, y) ⎪ vx (x, y, t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ v y (x, y, t) Vy (x, y) = eiωt . f= ⎪ ⎪ ⎪ ⎪ (x, y, t) (x, y) σ  ⎪ ⎪ ⎪ ⎩ xx ⎩ xx ⎭ ⎪ ⎭ σx y (x, y, t) x y (x, y)

(9.81)

Because all state variables vary as exp(−iσ y) in the y-coordinate, as in Eq. (9.80), we can write f as ⎧ ⎫ ⎧ ⎫ Vx (x) ⎪ vx (x, y, t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ v y (x, y, t) Vy (x) = e−iσ y eiωt = F(x)e−iσ y eiωt . f= ⎪ ⎪ σx x (x, y, t) ⎪ x x (x) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ σx y (x, y, t) x y (x)

(9.82)

If Eq. (9.80) is substituted into Eqs. (9.75) and (9.76), F(x) can be expressed as ⎧ ⎫ ⎧ ⎫ A⎪ Vx (x) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎨ ⎬ ⎬ B Vy (x) = MN(x) , F(x) = ⎪ ⎪ ⎪C ⎪ ⎪ x x (x) ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ D x y (x) where the matrices M and N are defined as

(9.83)

9.5 Reflection and Transmission across Two Dissimilar Isotropic Media

⎡

iω Pxα ⎢ ⎢ iω Pyα M=⎢ ⎣ α α

β

iω Px β iω Py β β

γ

iω Px γ iω Py γ γ

⎤ iω Pxδ ⎥ iω Pyδ ⎥ ⎥, δ ⎦ δ

271

(9.84)

with  j = −iτ j (C11 Pxj + C16 Pyj ) − iσ (C16 Pxj + C12 Pyj ), j = α, β, γ , δ,

(9.85)

 j = −iτ j (C16 Pxj + C66 Pyj ) − iσ (C66 Pxj + C26 Pyj ), j = α, β, γ , δ,

(9.86)

and ⎤ e−iτα x 0 0 0 ⎢ 0 e−iτβ x 0 0 ⎥ ⎥. N(x) = ⎢ −iτγ x ⎣ 0 0 ⎦ 0 e 0 0 0 e−iτδ x ⎡

(9.87)

We will use F(x) for the analysis of the wave reflection and transmission across two dissimilar media in the subsequent sections of this chapter. It will be also used in the next chapter.

9.5 Reflection and Transmission across Two Dissimilar Isotropic Media In this subsection, we revert to the problem illustrated in Fig. 9.7b to find the reflected and transmitted waves using the state vector representation given in the previous section. As the angles of the reflected and transmitted waves can be determined from Eq. (9.67) for the longitudinal wave incidence and Eq. (9.72) for the transverse wave incidence, this section is mainly focused on calculating the amplitudes of the reflected and transmitted waves. The field-matching conditions at the interface of two dissimilar media can be written in terms of the state vector F as F A (x = 0− ) = F B (x = 0+ ), where the state vectors F A (x = 0− ) are expressed as

(9.88)

272

9 Fundamentals of Elastic Waves in 2D Elastic Media

⎧ ⎫A ⎧ ⎫B A⎪ A⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎨ ⎪ ⎬ ⎬ B B A − A A − B + B B + F (x = 0 ) = M N (x = 0 ) , F (x = 0 ) = M N (x = 0 ) . ⎪ ⎪ C⎪ ⎪C ⎪ ⎪ ⎪ ⎩ ⎩ ⎪ ⎭ ⎭ D D (9.89a,b) For later use, we explicitly write the matrix Mi and its inverse matrix (Mi )−1 for isotropic media (i = A, B) as ⎡

cos θ Li − cos θ Li − sin θTi − sin θTi i i i ⎢ sin θ L sin θ L cos θT − cos θTi ⎢ Mi = iω⎢ −Z i cos 2θ i −Z i cos 2θ i Z i sin 2θ i −Z i sin 2θ i ⎣ L T L T T T T T Z i2 sin 2θ i Z Ti2 sin 2θ Li i i i i − T Zi L −Z cos 2θ −Z cos 2θ i T T T T Z L

i −1

(M )

⎡

=

L

Z Ti sin θTi Z iL Z Ti sin θTi Z iL cos 2θTi 2 cos θTi cos 2θ i i − sin θT − 2 cos θTi T

cos 2θTi 2 cos θ Li ⎢ cos 2θ i T 1⎢ ⎢ − 2 cos θLi ⎢ iω ⎢ − sin θTi ⎣

tan θ Li 2Z iL tan θ Li 2Z iL − 2Z1 i T − 2Z1 i T

− 2Z1 i − L

− 2Z1 i

L tan θTi i 2Z T tan θTi 2Z Ti

−

⎤ ⎥ ⎥ ⎥, ⎦

(9.90)

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

(9.91)

In Eqs. (9.90)–(9.91), Z iL and Z Ti are the mechanical impedances for the longitudinal and transverse waves in medium i, respectively. They are defined as Z iL =



i ρi C11 , Z Ti =



i ρi C66 .

(9.92)

Recall that in isotropic media, Z iL and Z Ti can be also written as (due to Eq. (9.35)): Z iL = ρi ciL , Z Ti = ρi ciT (i = A, B) It is often more convenient to write A, B, C, and D as A = U L + , B = U L − , C = UT + , D = UT − ,

(9.93)

because the subscripts of symbols U L + , etc., denote the wave mode and the propagation direction. • for longitudinal wave incidence As the field quantities in consideration for media A and B are expressed in Eq. (9.63) for the longitudinal wave incidence, the coefficient vector {A, B, C, D}T becomes

9.5 Reflection and Transmission across Two Dissimilar Isotropic Media

273

⎧ ⎫ A ⎧ A ⎫ ⎧ ⎫B ⎧ B ⎫ A⎪ A⎪ U +⎪ ⎪ U +⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎨ ⎪ ⎬ ⎪ ⎬ ⎨ LA ⎪ ⎨ L ⎪ ⎬ ⎬ B B UL− 0 , = = ⎪ ⎪ ⎪ C⎪ 0 ⎪ ⎪ UB ⎪ ⎪C ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭ ⎪ ⎭ ⎩ A ⎪ ⎩ T+ ⎪ ⎭ ⎭ D 0 UT − D where L and T denote the longitudinal and transverse waves, respectively. We introduce the following reflection (r L , r T ) and transmission (t L , tT ) coefficients to quantify the amounts of reflection and transmission: rL =

U LA− UTA− BA DA = , r = = , T AA AA U LA+ U LA+

(9.94)

tL =

U LB+ UTB+ AB CB = , t = = , T AA AA U LA+ U LA+

(9.95)

To facilitate further analysis, we set A A = U LA+ = 1. T Then, {A, B, C, D} can be expressed simply as ⎧ ⎫ A ⎧ ⎫ ⎧ ⎫B ⎧ ⎫ A⎪ 1⎪ ⎪ tL ⎪ A⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ ⎪ ⎬ ⎨ ⎪ ⎨ ⎪ ⎨ ⎪ ⎬ ⎬ B rL B 0 , . = = ⎪ ⎪ ⎪ C⎪ 0⎪ ⎪ C⎪ t ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎪ ⎭ ⎩ ⎪ ⎩ T⎪ ⎩ ⎪ ⎭ ⎭ D 0 rT D

(9.96)

Substituting Eq. (9.96) into Eq. (9.89) and using N A (x = 0− ) = I and N B (x = 0 ) = I, the following equations can be derived: +

⎧ ⎫ ⎧ ⎫A 1⎪ Vx (0− ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ ⎨ ⎬ − Vy (0 ) A rL , = M F A (x = 0− ) = − ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ x x (0 ) ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ x y (0− ) rT ⎧ ⎧ ⎫ ⎫ B tL ⎪ Vx (0+ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎨ ⎬ ⎬ 0 Vy (0+ ) B + B F (x = 0 ) = . = −M + ⎪ ⎪  (0 ) ⎪ t ⎪ ⎪ ⎪ ⎩ T⎪ ⎩ xx + ⎪ ⎭ ⎭ 0 x y (0 )

(9.97)

To find the unknowns (r L , t L , r T , tT ), we can use Eqs. (9.88) and (9.97) to establish the following relation:

274

9 Fundamentals of Elastic Waves in 2D Elastic Media

⎧ ⎫ ⎧ ⎫ tL ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎪ ⎬ r 0 = (M B )−1 M A L . ⎪ ⎪ 0⎪ t ⎪ ⎪ ⎪ ⎩ T⎪ ⎩ ⎪ ⎭ ⎭ 0 rT

(9.98)

Substituting Eqs. (9.90) and (9.91) into Eq. (9.98) yields ⎡

⎤⎧ ⎫ cos θ LB − cos θ LA − sin θTB − sin θTA ⎪ tL ⎪ B A B A ⎢ ⎨ ⎪ ⎥⎪ ⎬ sin θ sin θ cos θ − cos θ ⎢ ⎥ rL L L T T ⎢ −Z B cos 2θ B −Z A cos 2θ A Z B sin 2θ B −Z A sin 2θ A ⎥ ⎣ L T L T T T T T ⎦⎪ ⎪ tT ⎪ ⎪ ⎭ Z B2 sin 2θ B Z TA2 sin 2θ LA B B A A ⎩ − T ZB L −Z cos 2θ −Z cos 2θ r A T T T T T ZL L ⎧ ⎫ cos θ LA ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ sin θ LA (9.99) = −Z A cos 2θ A . ⎪ L L ⎪ ⎪ ⎪ A2 ⎪ ⎪ ⎩ − Z T sin 2θ A ⎭ L ZA L

Equation (9.99) can be solved for four unknowns (r L , r T , t L , and tT ) if the material properties of the involved media and the incidence angle of the incident longitudinal wave are specified. It should be noted that the reflection and transmission coefficients are “independent” of frequency ω because Eq. (9.99) does not involve ω. Instead of the reflection and transmission coefficients (r L , r T , t L , tT ), one can compare the time-averaged powers of the reflected (transmitted) waves with that of the incident wave. The time-averaged power14 P i, j of mode i in medium j is defined as ! ! ! !$ # P i, j = Re σxi,xj Re vxi, j + Re σxi,yj Re v i,y j T =2π/ω  1  = Re σxi,xj × (vxi, j )∗ + σxi,yj × (v i,y j )∗ 2  1  = Re xi,xj × (Vxi, j )∗ + xi,yj × (Vyi, j )∗ , 2

i, j

i, j

i, j

(9.100)

i, j

where vx , v y , σx x , and σx y represent the velocity and stress components associated with the i-wave in the medium j (i = L + , L −%, T + , T− 'and j = A, B). The &T time-averaging operator (•)T is defined as •T = 0 (•)dt /T . (See the paragraph below Eq. (9.36) why the time-averaged power can be written as the last expression in Eq. (9.100).) To calculate P i, j defined in Eq. (9.100), we explicitly express the stresses and velocities using Eqs. (9.82) and (9.97) as: for mode L + in medium A (at x = 0),

14

Note that the same symbol P is used to denote the time-averaged power in Eq. (9.100) and polarization in (9.37) because there is no danger of confusion.

9.5 Reflection and Transmission across Two Dissimilar Isotropic Media

VxL

+

,A

A (0) = M11 · (1), VyL

+

+

,A

275

A (0) = M21 · (1),

+

A A xLx ,A (0) = M31 · (1), xLy ,A (0) = M41 · (1),

for mode L − in medium A (at x = 0), VxL

−

,A

A (0) = M12 · r L , VyL

−

−

,A

A (0) = M22 · rL ,

−

A A xLx ,A (0) = M32 · r L , xLy ,A (0) = M42 · rL ,

for mode T − in medium A (at x = 0), VxT

−

,A

A (0) = M14 · r T , VyT

−

−

,A

A (0) = M24 · rT ,

−

A A xTx ,A (0) = M34 · r T , xTy ,A (0) = M44 · rT ,

for mode L + in medium B (at x = 0), VxL

+

,B

B (0) = M11 · t L , VyL

+

+

,B

B (0) = M21 · tL ,

+

B B xLx ,B (0) = M31 · t L , xLy ,B (0) = M41 · tL ,

for mode T + in medium B (at x = 0), VxL

+

,B

B (0) = M13 · tT , VyL

+

+

,B

B (0) = M23 · tT ,

+

B B xLx ,B (0) = M33 · tT , xLy ,B (0) = M43 · tT ,

Therefore, P i, j can be determined as  +  1 ! !   A  L ,A  A ∗ A A ∗ · M31 + M21 · M41  = M11 , P 2  −  1 ! !   L ,A   A A ∗ A A ∗ · M32 + M22 · M42 P  = M12  · |r L |2 , 2  −  1 ! !   A  T ,A  A ∗ A A ∗ · M34 + M24 · M44  = M14  · |r T |2 , P 2  +  1 ! !   B  L ,B  B ∗ B B ∗ · M31 + M21 · M41 P  = M11  · |t L |2 , 2  +  1 ! !   B  L ,B  B ∗ B B ∗ · M31 + M21 · M41 P  = M11  · |t L |2 . 2

(9.101a) (9.101b) (9.101c) (9.101d) (9.101e)

Using Eq. (9.101), we can define the power-based reflection and transmission coefficients, called the reflectance (R L , RT ) and transmittance (TL , TT ). In the case of the longitudinal wave incidence being considered, they can be expressed as

276

9 Fundamentals of Elastic Waves in 2D Elastic Media

−

P T ,A RT = L + ,A P

 −   P L ,A    R L =  L + ,A  = |r L |2 ,  P  A A A C66 cos θTA Z T cos θT 2 |r T | =  |r T |2 . = A Z L cos θ LA A A C cos θ 11

(9.102a)

(9.102b)

L



L + ,B

B ρ B C11 cos θ LB |t L |2 , =  A A ρ A C11 cos θ L  B ρ B C66 cos θTB Z TB cos θTB 2 |tT | =  |tT |2 . = A Z L cos θ LA A A ρ A C11 cos θ L

P TL = L + ,A = P +

P T ,B TT = L + ,A P

Z LB Z LA

cos θ LB |t L |2 cos θ LA

(9.103a)

(9.103b)

To obtain the explicit results in Eqs. (9.102) and (9.103), we used the following results:  ! !   A ! !   A A ∗ A A ∗ A ∗ A A ∗ + M21 · M41 + M22 · M42  = M12 · M32  M11 · M31   Z A2 sin 2θ A = ω2 cos θ LA · Z LA cos 2θTA + sin θ LA · T A L ZL ! 2 A A A A A = ω Z L cos θ L cos 2θT + Z T sin 2θ L sin θTA = ω2 Z LA cos θ LA cos 2θTA + 2Z TA sin θ LA cos θ LA sin θTA ! = ω2 Z LA cos θ LA cos 2θTA + 2Z LA cos θ LA sin2 θTA ! = ω2 Z LA cos θ LA cos 2θTA + 2 sin2 θTA = ω2 Z LA cos θ LA ,

!

(9.104a)

 ! !  !  A A ∗ A A ∗ + M24 · M44 M14 · M34  = ω2 sin θTA · Z TA sin 2θTA + cos θTA · Z TA cos 2θTA ! = ω2 2Z TA sin2 θTA cos θTA + Z TA cos θTA cos 2θTA ! = ω2 Z TA cos θTA 2 sin2 θTA + cos 2θTA = ω2 Z TA cos θTA .

(9.104b)

  B2 B ! !   B B ∗ B B ∗ 2 B B B B Z T sin 2θ L M = ω cos θ · M + M · M · Z cos 2θ + sin θ ·  11 31 21 41  L L T L Z LB ! = ω2 Z LB cos θ LB cos 2θTB + Z TB sin 2θ LB sin θTB = ω2 Z LB cos θ LB cos 2θTB + 2Z TB sin θ LB cos θ LB sin θTB ! = ω2 Z LB cos θ LB cos 2θTB + 2Z LB cos θ LB sin2 θTB ! = ω2 Z LB cos θ LB cos 2θTB + 2 sin2 θTB

!

9.5 Reflection and Transmission across Two Dissimilar Isotropic Media

= ω2 Z LB cos θ LB ,

277

(9.104c)

 ! !  !  B B ∗ B B ∗ + M23 · M43 M13 · M33  = ω2 sin θTB · Z TB sin 2θTB + cos θTB · Z TB cos 2θTB ! = ω2 2Z TB sin2 θTB cos θTB + Z TB cos θTB cos 2θTB ! = ω2 Z TB cos θTB 2 sin2 θTB + cos 2θTB = ω2 Z TB cos θTB ,

(9.104d)

Equation (9.104a) can be obtained if Eq. (9.90) and the relation Z LA sin θTA = Z TA sin θ LA are substituted into Eqs. (9.101a) and (9.101b), and Eqs. (9.104b–d) can be obtained if Eq. (9.90) and the relation Z LB sin θTB = Z TB sin θ LB are substituted into Eqs. (9.101c)–(9.101e). One can check that the following relation holds R L + RT + TL + TT = 1.

(9.105)

In fact, Eq. (9.105) implies the law of energy conservation. • for transverse wave incidence To investigate wave reflection and transmission for transverse wave incidence, we can use the procedure used for the case of longitudinal wave incidence discussed above. Therefore, the detailed analysis will not be repeated here. The expressions for the reflectance and transmittance can be found as:   −  A  P L ,A  A A C11 cos θ LA Z cos θ L   2 |r | |r L |2  R L =  T + ,A  = LA = (9.106a) L  P Z T cos θTA A A C cos θ 66

T

 −   P T ,A    RT =  T + ,A  = |r T |2  P   +  B  P L ,B  B B ρ B C11 cos θ LB Z L cos θ L   2 |t L | =  |t L |2 TL =  T + ,A  = A  P Z T cos θTA A A ρ C cos θ A

66

(9.106b)

(9.107a)

T

  +  B  P T ,B  B B ρ B C66 cos θTB Z T cos θT   2 |tT | =  |tT |2 TT =  T + ,A  = A  P Z T cos θTA A A ρ A C66 cos θT

(9.107b)

278

9 Fundamentals of Elastic Waves in 2D Elastic Media

Fig. 9.8 Reflectance (R L , RT ) and transmittance (TL , TT ) for waves incident from a low-impedance PEEK plate to a high-impedance aluminum plate with material properties taken from Table 9.3. a Longitudinal wave incidence and b transverse wave incidence. (Results are frequency-independent)

• Some examples15 As specific examples, we consider the wave incidences from a PEEK plate to an aluminum plate or vice versa under the plane-stress condition. The reflectance and transmittance coefficients are plotted in Fig. 9.8 for the PEEK-to-aluminum incidence and in Fig. 9.9 for the aluminum-to-PEEK incidence, respectively. Based on the material properties listed in Table 9.3 (for plane-stress condition), the phase velocities and mechanical impedances are calculated as = 1952.1 m/s, cTPEEK = 1069.6 m/s, cPEEK L Z LPEEK = 2.57 × 106 kg/m2 s, Z TPEEK = 1.41 × 106 kg/m2 s, c LAl = 5392.0 m/s, cTAl = 3121.0 m/s, Z LAl = 14.6 × 106 kg/m2 s, Z TPEEK = 8.43 × 106 kg/m2 s. Because the mechanical impedances of aluminum are larger than those of PEEK, there exists a nontrivial critical angle θcr discussed in Sect. 9.3 if a wave is incident from the PEEK plate and aluminum plate. We will examine Fig. 9.8 first. It plots the reflectance and transmittance for the PEEK-to-aluminum wave incidence. Note that results similar to those plotted in Fig. 9.8 can be obtained as long as a wave is incident from a low-impedance material 15

Here, we consider natural isotropic materials. The wave phenomena of reflection and transmission across isotropic media, possibly with artificial negative effective properties, were studied by Zhu et al. (2015).

9.5 Reflection and Transmission across Two Dissimilar Isotropic Media

279

Fig. 9.9 Reflectance (R L , RT ) and transmittance (TL , TT ) for waves incident from a highimpedance aluminum plate to a low-impedance PEEK plate with material properties taken from Table 9.3. a Longitudinal wave incidence and b transverse wave incidence. (Results are frequency-independent)

to a high-impedance material. Figure 9.8a shows the results for the longitudinal wave incidence from a low-impedance PEEK plate to a high-impedance aluminum plate. The following observations can be made from the results in Fig. 9.8a. • At normal incidence, i.e., θinc = 0, no mode conversion occurs; the incident longitudinal wave is transmitted and reflected solely in the longitudinal wave mode without generating any transverse wave; RTA = 0 and RTB = 0. • If θinc > θcr = sin−1 (cPEEK /c LAl ) ≈ 21.2◦ , only a transverse wave can be L transmitted to the aluminum plate (i.e., TLB = 0). The analysis is related to Eq. (9.68). • If the incidence angle θinc is further increased to be larger than another critical angle /cTAl ) ≈ 38.7◦ , no longitudinal or transverse can be transmitted θcr = sin−1 (cPEEK L to the aluminum plate. According to the analysis on Eq. (9.69), R L + RT = 1 implies that the incident power is completely reflected. The phenomenon of zerowave transmission occurs because the phase velocities of the longitudinal (c LAl ) and transverse (cTAl ) waves in aluminum are simultaneously larger than the phase velocity (cPEEK ) of the incident longitudinal wave in PEEK. L • At any incidence angle, the complete transmission of the incident longitudinal wave from the PEEK plate either to the longitudinal or transverse wave in the aluminum plate is not possible due to impedance mismatch. Figure 9.8b shows the wave phenomena for the case of transverse wave incidence from a PEEK plate to an aluminum plate. The following observations can be summarized • The wave phenomenon at normal incidence is identical to the case of the PEEKto-aluminum longitudinal wave incidence: the incident transverse wave is transmitted and reflected solely in the transverse wave mode without generating any longitudinal wave.

280

9 Fundamentals of Elastic Waves in 2D Elastic Media

• If θinc > θcr = sin−1 (cTPEEK /c LAl ) ≈ 11.4◦ , no longitudinal wave can be transmitted to the aluminum plate. • If θinc > θcr = sin−1 (cTPEEK /cTAl )≈ 20.0◦ , no longitudinal or transverse wave can be transmitted to the aluminum plate. However, the reflected waves include both the longitudinal and transverse wave modes. • If θinc is further increased as θinc > θcr = sin−1 (cTPEEK /cPEEK )≈ 33.2◦ (see L A ◦ Eq. (9.72) with θ L − = 90 and θcr = θinc ), the incident transverse wave is completely and solely reflected as a transverse wave in the PEEK plate. This phenomenon is called the total reflection without mode conversion. In this instance, RT = 1. • No complete transmission of the incident transverse wave is possible from the PEEK plate either to the longitudinal or transverse wave in the aluminum plate at any incidence angle due to impedance mismatch. Figure 9.9 presents the wave phenomena occurring when waves are incident from a high-impedance aluminum plate to a low-impedance PEEK plate. Unlike the wave incidence from PEEK to aluminum, no critical angle is observed in the wave incidence from aluminum to PEEK. Except at normal incidence, an incident wave, either longitudinal or transverse, generates two reflected and two transmitted waves at all instances. This phenomenon occurs because the phase velocities of both the longitudinal and transverse waves in aluminum are larger than any phase velocity in PEEK. As observed in Fig. 9.9, no complete transmission is possible for the incidence of ◦ any wave mode at any incidence angle θinc (0 ≤ θinc < 90 ).

9.6 Reflections from Traction-free Boundary In this section, we consider wave reflections from a traction-free boundary. This case corresponds to the wave system depicted in Fig. 9.7b, where medium B is considered as vacuum. The traction-free boundary condition at x = 0 is expressed as "

σxAx σxAy

( = 0.

(9.108)

x=0

• Longitudinal wave incidence In this case, we use u A stated in Eq. (9.63a). From Eq. (9.67), the angles of the reflected longitudinal (θrL ) and transverse (θrT ) are determined as θrL = θiL , sin θrT =

cT sin θiL , cL

(9.109)

where we introduced new symbols to facilitate writing because there is only a single medium:

9.6 Reflections from Traction-free Boundary

281

incident longitudinal wave: θiL = θinc ; reflected longitudinal and transverse waves: θrL = θ LA− and θrT = θTA− . To calculate the reflection coefficients r L and r T , we use Eq. (9.89a) and Eq. (9.108): )

A A A A M31 M32 M33 M34 A A A A M41 M42 M43 M44

⎫ ⎧ 1 ⎪ ⎪   *⎪ ⎪ ⎬ ⎨ 0 rL = , ⎪ ⎪ 0 ⎪ 0 ⎪ ⎭ ⎩ rT

(9.110)

which can be rearranged as 

A A M32 M34 A A M42 M44



rL rT



 =

 A −M31 A . −M41

(9.111)

In writing the expression in Eqs. (9.110) or (9.111), we set A = U LA+ = 1, C = 0, B = U LA− , D = UTA− , ∴ B = r L U LA+ = r L A = r L , D = r T U LA+ = r T A = r T . The components MiAj appearing in Eqs. (9.110) or (9.111) can be explicitly obtained from Eq. (9.90). Once r L and r T are calculated from Eq. (9.111), the reflectances R L and RT can be evaluated using Eq. (9.102). For the longitudinal wave incidence in an aluminum plate under the plane-stress condition, R L and RT are plotted in Fig. 9.10 as a function of the incidence angle θiL . At the normal incidence (θiL = 0), R L = 1 and RT = 0, implying that the incident longitudinal wave is entirely reflected to a longitudinal wave. If θrL = 90◦ ,16 we obtain R L = 1 and RT = 0. In this case, the total longitudinal wave field vanishes because the incident and reflected longitudinal waves propagate along the same direction with opposite signs. Therefore, the resulting wave field is trivial, indicating that the assumption of plane waves does not remain valid in this case. Therefore, a nontrivial solution is expected to involve nonplane waves. We will not consider this case here as this limiting case is not generally of practical interest.17 If a longitudinal wave is incident at a certain angle, the incident longitudinal wave can be completely converted to a reflected transverse wave in certain media without being accompanied by the reflected longitudinal wave. This phenomenon is called total mode conversion (R L = 0). It occurs when the following condition is 16

Generally, the incidence at this limiting angle is called the grazing incidence. Goodier and Bishop (1952) reported that if an incident longitudinal plane wave propagates along the positive y-direction (i.e., θrL = 90◦ ), the reflected longitudinal wave propagates along the positive y-direction, but its amplitude linearly increases with x. Therefore, the reflected wave is no longer a plane wave and cannot be used in an infinite medium. In contrast, the reflected transverse √ wave is an ordinary plane wave that propagates at the angle of θrT for which tan θrL = κ 2 − 1. Refer to Miklowitz (1978).

17

282

9 Fundamentals of Elastic Waves in 2D Elastic Media

Fig. 9.10 Reflectances R L (of the longitudinal wave mode) and RT (of the transverse wave mode) for a “longitudinal” wave incident on the traction-free boundary of an aluminum plate

satisfied18 : sin 2θrL sin 2θrT = κ 2 cos2 2θrT , (κ = c L /cT ).

(9.112)

If this condition is satisfied, r T = κ cot 2θrT . However, Eq. 9.112 yields a nontrivial θrL for Poisson’s ratio ν smaller than a certain value (ν ≈ 0.265). For instance, √ consider a medium with ν = 0.25, which corresponds to λ = μ and κ = 3. In this case, the total mode conversion occurs at two angles, θiL = 60◦ (θrT = 30◦ ) ◦ and θiL = 77.24◦ (θrT = 34.27 ), where the angle θrT is calculated from the 2nd A T expression (with θT = θr ) in Eq. (9.67). • Transverse wave incidence In this case, we use u A expressed in Eq. (9.70). From Eq. (9.72), the angles of the reflected longitudinal (θrL ) and transverse (θrT ) are determined as θrT = θiT , sin θrL =

cL sin θiT , cT

(9.113)

where we introduced a new symbol, θiT to denote the incident transverse wave: incident transverse wave: θiT = θinc . The symbols θrL and θrL denote the angles of the reflected longitudinal and transverse waves, respectively, for the case of transverse wave incidence. Following the procedure used to derive Eq. (9.111), we can obtain the following equation: 

A A M34 M32 A A M42 M44



rL rT



 =

 A −M33 A . −M43

(9.114)

The reflectances R L and RT for waves reflected from the traction-free boundary in aluminum are presented in Fig. 9.11. At the normal incidence (θiL = 0), R L = 0 and 18

The detailed derivation is skipped here (see Achenbach (1976)).

9.6 Reflections from Traction-free Boundary

283

Fig. 9.11 Reflectance R L (of longitudinal wave mode) and RT (of transverse wave mode) for a “transverse” wave incident on the traction-free boundary of an aluminum plate

RT = 1. This result implies that the incident transverse wave is completely reflected ◦ to a transverse wave. If θiT = 90 , we can obtain RT = 1 and R L = 0. As in the case of longitudinal wave incidence based on the plane-wave assumption, the total field vanishes. Therefore, nonplane waves should be considered to find a nontrivial solution. We will not present the detailed analysis19 because this limiting case is not of practical interest in general. As observed in Fig. 9.11, RT = 1 and R L = 0 at θiT = π/4, implying total reflection without mode conversion. This angle of incidence is known to be independent of medium properties. Let us consider a special case when θrL = 90o and try to find the corresponding angle of incidence denoted by θcrTR . It is called the critical angle, which can be determined from Eq. (9.72) if θ LA− = θrL = 90◦ and θinc = θiT = θcr are set: ! θcrTR = sin−1 cT /c L = sin−1 (1/κ ).

(9.115)

The total reflection occurs for an aluminum plate (under the plane-stress condition) ◦ if θiT ≥ θcrTR = 35.37 . If θcrTR ≤ θiT < π2 , the incident transverse wave is completely reflected to the plane transverse wave, and no propagating longitudinal wave can be reflected. Based on the above discussion, both reflected longitudinal and transverse waves can exist only if θiT < θcrTR . Under the condition θiT < θcrTR , Achenbach (1976) showed that an incident transverse wave can be entirely converted to the reflected longitudinal wave upon satisfying the following criterion: sin 2θiT sin 2θrL = κ 2 cos2 2θiT .

(9.116)

The value of θiT satisfying Eq. (9.116) will be denoted by θcrMC . For an aluminum plate (under the plane-stress condition), it is found as ◦ θcrMC = 29.9 or θcrMC = 34.4◦ . 19

Further detailed analysis of this case is reported in Miklowitz (1978), based on Goodier and Bishop’s investigation (1952).

284

9 Fundamentals of Elastic Waves in 2D Elastic Media

Fig. 9.12 Reflections of longitudinal waves in a plate from a the traction-free boundary (θrL = θiL ) L

and b the metasurface-embedded boundary (θ r = θiL ). (L: longitudinal wave, T: transverse wave)

9.7 Anomalous Reflection from Traction-free Boundary: Metasurfaces Figure 9.12a shows wave reflections when a longitudinal wave is incident. The angles of reflected waves from a traction-free surface are uniquely determined from Eq. (9.109) once the medium, the incident angle, and the mode type are specified. This section presents a method to alter the reflected angles by adding an extra passive element to the traction-free boundary. Figure 9.12b illustrates a periodic structure attached to the traction-free boundary, called a metasurface. It is supposed to direct the reflected wave at any desired angle different from the reflection angle predicted by Snell’s law. The metasurface consists of repeatedly arrange supercells, each of which consists of several unit cells of varying lengths (L d in Fig. 9.13) along the y-direction. Thus, the waves reflected from unit cells of different lengths will experience different phase delays. If the size of each unit cell is infinitely small, the phase delay20 can be defined as a continuous function of y. If the phase delay due to the unit cell located at y is denoted by (y), d(y)/dy = 0. In this case, the reflected angle becomes different from that predicted by Snell’s law. If d(y)/dy = 0, the law predicting the reflection angle is different from the conventional Snell’s law, and it is called the generalized Snell’s law21 (Yu et al. 2011). It is given by

The phase delay  will appear in the reflected wave as |B| exp[i(ωt + kx + )] if the incident wave behaves as |A| exp[i(ωt − kx)] where |A| and |B| are the magnitudes of the incident and reflected waves. 21 The generalized Snell’s law was initially established for optics (Yu and Capasso (2014)), but it can be also extended in the field of elastic waves (see, e.g., Kim et al. (2018)). 20

9.7 Anomalous Reflection from Traction-free Boundary: Metasurfaces

285

Fig. 9.13 Thin plate with finite-sized rods attached at its free end to delay the phase of the reflected longitudinal wave. Rods function as one-dimensional resonators. (Kim et al. 2018)

Generalized Snell’s law (for longitudinal wave incidence): k L sin θiL +

d L T = k L sin θ r = k T sin θ r , dy

(9.117a)

or L

T

sin θiL sin θ r 1 d sin θ r = + = , cL ω dy cL cT

(9.117b)

In Eq. (9.117), k L and k T denote the wavenumbers for longitudinal and transverse L T waves, respectively. Symbols θ r and θ r represent the reflected angles of longitudinal and transverse waves from a free boundary, respectively, as portrayed in Fig. 9.12b. L T Note that θ r > θ r because c L > cT . When d(y)/dy = 0, Eq. (9.117) reduces to L T the conventional Snell’s law where θ r and θ r become θrL and θrT , respectively. If the incident wave is a transverse wave, k L sin θiL in Eq. (9.117a) and sin θiL /c L in Eq. (9.117b) should be replaced by k T sin θiT and sin θiT /cT , respectively. Following Kim et al. (2018), we assume that the gradient of the phase delay, d(y)/dy is constant along the y-direction over the whole interface (x = 0) between the medium and the metasurface: d(y)/dy = h 1 ,or (y) = h 1 y + h 0 (h 0 and h 1 : some constants)

(9.118) L

T

the reflected wave forms a plane wave with the reflected angles of θ r and θ r , as predicted by Eq. (9.117). We will assume h 0 = 0 without loss of generality. If the interface is infinitely long in the y-direction, the additional phase delay (y) will

286

9 Fundamentals of Elastic Waves in 2D Elastic Media

grow indefinitely as y increases. As remarked in footnote 20, only the remainder of the division of (y) by 2π (i.e., (y) modulo 2π ) affects the wave field. Therefore, the phase delay (y) can be a periodic function of y, covering 2π over a supercell. In the metasurface described in Fig. 9.12b, (y) should be (y) =

2π y for y ∈ [nd, (n + 1)d] (n = 0, 1, 2, . . .), d

(9.119)

where d is the supercell size. In this case, the value of d(y)/dy = h 1 affecting the reflected angle can be adjusted by d. We will present the method of using a set of finite-sized bars (Kim et al. 2018) to realize the constant phase delay gradient. The lengths of the bars shown in Fig. 9.12b vary along the y-direction. We will consider a bar of length L d and investigate the phase delay due to the attached bar. This case will be compared with the case when a wave is reflected directly from the free surface of the plate. For this analysis, we will use a plane-wave assumption where an infinite number of bars of the same length, L d , are attached to the plate as in the analysis model shown in Fig. 9.13. This assumption can be also valid when the bar lengths vary slowly along the y-direction, as sketched in Fig. 9.12b. Because the incident longitudinal wave is assumed to be a plane wave, the horizontal displacement (u x ) in the plate is regarded as independent of the y-coordinate. In this case, the longitudinal wave motions in the plate and bar can be described by a one-dimensional model, respectively, as ⎛ ⎞  p p Ep ∂ 2u x 1 ∂ 2u x ⎝ ⎠ = 2 , cp = ∂x2 c p ∂t 2 ρ  0 / Eb 1 ∂ 2 u bx ∂ 2 u bx = 2 , cb = ∂x2 ρ cb ∂t 2

(9.120a)

(9.120b)

where subscripts p and b refer to the plate and bar. Note that the plane-wave longitudinal motion along the x-direction in a plate (with large or infinite length in the y-direction) can be described using the plane-strain assumption,22 which results in σx = E p εx for a plate in Figure 9.13,

(9.121)

with E p = E p /(1 − ν 2 ) (ν: Poisson’s ratio). As given by Eq. (7.20), the wave motions in the plate and the bar can be stated as u αx (x, t) = Aα ei(ωt−k

α

x)

+ Bα ei(ωt+k

α

x)

, (α = p or b)

(9.122)

In a 2D isotropic elastic medium, the stress–strain relations are expressed as εx = (σx − νσ y ) and ε y = (σ y − νσx ). If ε y = 0 is assumed because of the plane-strain assumption (i.e., ∂(·)/∂ y = 0 and u y = 0), we can obtain εx = σx (1 − ν 2 )/E.

22

9.7 Anomalous Reflection from Traction-free Boundary: Metasurfaces

287

where k α denotes the wavenumber in the plate (α = p) or the bar (α = b). It is related to frequency ω as ω = k α cα where cα is the longitudinal wave speed defined in Eq. (9.120). Note that A p and B p correspond to the incident and reflected wave, respectively, in the plate. To find B p and Bb for nonzero A p (Ab = 0), we impose the following continuity relations at the interface (x = 0) of the plate and the bar: force: ar σxp (0, t) = ad σxb (0, t),

(9.123a)

velocity: vxp (0, t) = vxb (0, t).

(9.123b)

Using Eq. (9.122), the velocity vxα and stress σxα (α = p or b) can be expressed as vxα (x, t) = iω Aα ei(ωt−i

α

x)

+ Bα ei(ωt+i

α

x)

! , (α = p or b)

(9.124)

and α

σxα (x, t) = −i E k α Aα ei(ωt−i 

p

p

α

x)

− Bα ei(ωt+i

α

x)

! , (α = p or b)

(9.125)

b

where E = E for the plate and E = E b for the bar. In the subsequent analysis, it is assumed that E p = E b  E and ν p  ν. Substituting Eqs. (9.124) and (9.125) into Eq. (9.123) yields ! ! ar k p A p − B p = ad k b Ab − Bb 2 1−ν

(9.126a)

A p + B p = Ab + Bb

(9.126b)

The traction boundary condition at x = L d requires that σxb (L d , t) = 0,

(9.127)

which results in Bb = Ab e−i2k

b

Ld

.

(9.128)

The reflection coefficient r defined as the ratio of B p to A p can be obtained by solving Eqs. (9.126a, b) and (9.128) as:

288

9 Fundamentals of Elastic Waves in 2D Elastic Media

 p  b ar k + ad k b (1 − ν 2 ) (1 + e−i2k L d ) − 2ad k b (1 − ν 2 ) Bp  r= p = p A ar k − ad k b (1 − ν 2 ) (1 + e−i2k b L d ) + 2ad k b (1 − ν 2 )

(9.129)

Because the reflection coefficient r in Eq. (9.129) is complex-valued, its phase  can be evaluated as  = Im(ln r ),

(9.130)

where “Im” represents the imaginary component, and “ln” denotes the natural logarithmic function. Physically,  represents the phase delay due to the attached rod, which functions as a one-dimensional resonator. Figure 9.14 shows how  varies as a function of L d , the bar length. The result in the figure is presented when both the plate and the bar are made of aluminum (E = 70 GPa, ν = 0.33, and ρ = 2700 kg/m3 ). To plot  in Fig. 9.14, we added 2π when  calculated by Eq. (9.130) is negative because adding 2π to the calculated phase delay does not affect the relation given by Eq. (9.117). This figure shows that the method of using the attached bars can cover the complete range (0 ~ 2π) of phase delay of a longitudinal wave only with simple bars of varying bar length (L d ). Therefore, any phase delay can be realized using a bar whose length is shorter than or equal to 0.0253 m at 100 kHz (see Fig. 9.14.). If a different frequency is used, different L d values are needed to realize the same phase delays; see Fig. 9.14. Using the bars of different lengths yielding different phase delays as given in Fig. 9.14, we can cover some phase delays in the 2π span. We construct a supercell consisting of eight uniformly spaced rods along the y-direction (the number of rod: n r = 8) as in Kim et al. (2018). In this case, the change in the phase delay between two consecutive rods should be Fig. 9.14 Phase delay  of the reflected wave in the thin aluminum plate shown in Fig. 9.13 due to the attached finite-sized aluminum bar of length L d (ar = 0.01 m and ad = 0.005 m). Theo.: calculated using Eq. (9.130); FEM: calculated using finite element analysis. (Kim et al. 2018)

9.7 Anomalous Reflection from Traction-free Boundary: Metasurfaces

 =

289

2π 2π π = = nr 8 4

If the supercell size d is chosen to be 0.08 m, the distance ar between two consecutive rods shown in the inset of Fig. 9.15a is calculated as ar =

d 0.08 m = 0.01 m. = nr 8

Therefore, the phase delay gradient d/dy can be estimated as d 2π  ≈ = = 25π rad/m = 78.54 rad/m. dy d ar

(9.131)

If d/dy in Eq. (9.131) is used for an incident longitudinal plane wave at f = ω/2π = 100 kHz, Eq. (9.117b) can be explicitly written as L

T

sin θiL 25π sin θ r sin θ r + = = , 3 cL 2π × 100 × 10 cL cT

(9.132)

where c L = 5395.13 m/s and cT = 3122.66 were calculated using Eq. (9.35) for aluminum (aluminum 6061 with E = 70 GPa, ρ = 2700 kg/m3 , ν = 0.33). L T Table 9.4 lists θ r ’s and θ r ’s for different angles θiL of the incident longitudinal wave. The simulated wave fields corresponding to the incident angles given in Fig. 9.16. The results in Table 9.4 and Fig. 9.16 show that the reflected L T angles do not follow the conventional Snell’s law predicting θ r = θiL and θ r = sin−1 [(cT /c L ) sin θiL ] due to the presence of the metasurface. Furthermore, no reflected longitudinal wave exists for θiL larger than 25°. To find the minimum critical incidence angle θc for the mode-conversion L reflection, we set θ r = 90◦ in Eq. (9.117b) and solve for θiL = θc :

Fig. 9.15 a Phase delay by rods of varying lengths (L d ) and b reflected longitudinal wave fields at 100 kHz due to attached rods of different lengths. (Kim et al. 2018)

290 Table. 9.4 The reflected angles based on Eq. (9.132)

9 Fundamentals of Elastic Waves in 2D Elastic Media L

T

θiL

θr

θr

10°

58.0°

29.4°

25°

–

39.4°

45°

–

53.1°

60°

–

63.1°

Fig. 9.16 Effects of a metasurface comprising 50 supercells on wave reflections for longitudinal wave incidence at 100 kHz in an aluminum plate. The incident angles are a θiL = 10◦ , b θiL = 25◦ , ◦ c θiL = 45◦ , and d θiL = 63◦ . If θiL > θCL = 19 , no reflected longitudinal wave appears. The reflection angles in the plot are estimated using the numerical simulation results (Kim et al. 2018)

  c L d − +1 . θc = sin ω dy −1

Using the data used to write Eq. (9.132), we can find θc = 19◦ . Therefore, an incident longitudinal wave is mode-converted solely to a reflected transverse wave if θiL ≥ θc = 19◦ . This can be a useful application of a metasurface. In addition to the application presented in Fig. 9.16, various applications of elastic metasurfaces and metasurface design methods are reported in Liu et al. (2017), Ahn et al. (2019), Colombi et al. (2017), Su et al. (2018), Zheng et al. (2020), Zhu et al. (2018), Lee et al. (2018), and Kim et al. (2020). Flexural waves can be also manipulated using elastic metasurfaces (Cao et al. 2018a, b). Recently, metagratings,

9.8 Problem Set

291

which can also manipulate the angles of reflected (and refracted) waves, have been used to control the angles of reflected elastic waves Kim et al (2021, 2023). The relation between metasurfaces and metagratings can be found in Larouche and Smith (2012).

9.8 Problem Set Problem 9.1 Express wavenumber k as a function of ω for two-dimensional isotropic media, i.e., derive Eqs. (9.33) and (9.34). Problem 9.2 Derive the polarization vector P L and PT in Eqs. (9.38) and (9.39). Problem 9.3 Consider an anisotropic medium the material properties of which are given by ρ = 2070 kg/m3 , C16 = C26 = 0, C11 = C66 = 10.99 × 1010 N/m, C12 = 6.230 × 109 N/m, C22 = 5.066 × 1010 N/m. (a) Plot the slowness curves in the k x /ω and k y /ω plane and mark the points where two curves intersect. (b) Draw the directions of the group velocities vg at the intersecting points. Utilize the fact vg is always normal to the slowness curves. (If a wave is incident to this material to excite waves at one of the intersection points, the so-called conical refraction phenomenon can take place. (Please see Ahn et al. (2017) for more detailed accounts of the phenomenon and its realization using a metamaterial.) Problem 9.4 Derive Eqs. (9.106) and (9.107). Problem 9.5 We wish to find the reflectances (R L , RT ) and transmittances (TL and TT ) for oblique wave incidence from an aluminum plate to a steel plate (use the material data in Table 9.3) using Eq. (9.99) and Eqs. (9.102)–(9.103). (a) Plot the reflectances and transmittances for longitudinal wave incidence. Calculate the critical incidence angle above which only a transverse wave can be transmitted to the steel plate. Also calculate the critical incident angle above which no longitudinal or transverse wave is transmitted to the steel plate. (b) Repeat (a) for transverse wave incidence. Problem 9.6 Consider wave reflection from a traction-free boundary of a PEEK plate. Plot the reflectances (R L , RT ) for both longitudinal and transverse wave incidences. Use the material properties of PEEK in Table 9.3 which is given under the plane-stress condition.

292

9 Fundamentals of Elastic Waves in 2D Elastic Media

References Achenbach JD (1976) Wave propagation in elastic solids. North-Holland Ahn B, Lee H, Lee JS, Kim YY (2019) Topology optimization of metasurfaces for anomalous reflection of longitudinal elastic waves. Comput Methods Appl Mech Eng 357:112582 Ahn YK, Lee HJ, Kim YY (2017) Conical refraction of elastic waves by anisotropic metamaterials and application for parallel translation of elastic waves. Sci Rep 7:10072 Auld BA (1973) Acoustic fields and waves in solids. vol 1 & 2. Wiley Cao L, Xu Y, Assouar B, Yang Z (2018a) Asymmetric flexural wave transmission based on dual-layer elastic gradient metasurfaces. Appl Phys Lett 113:183506 Cao L, Yang Z, Xu Y, Assouar B (2018b) Deflecting flexural wave with high transmission by using pillared elastic metasurface. Smart Mater Struct 27:075051 Colombi A, Ageeva V, Smith RJ, Clare A, Patel R, Clark M, Colquitt D, Roux P, Guenneau S, Craster R (2017) Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces. Sci Rep 7:6750 Goodier JN, Bishop RED (1952) A note on critical reflections of elastic waves at free surfaces. J Appl Phys 23:124–126 Hussein MI (2009) Reduced Bloch mode expansion for periodic media band structure calculations. Proc Royal Soc: Math Phys Eng Sci 465:2825–2848 Kim MS, Lee WR, Kim YY, Oh JH (2018) Transmodal elastic metasurface for broad angle total mode conversion. Appl Phys Lett 112:241905 Kim MS, Lee W, Park CI, Oh JH (2020) Elastic wave energy entrapment for reflectionless metasurface. Phys Rev Appl 13:054036 Kim SY, Lee W, Lee JS, Kim YY (2021) Longitudinal wave steering using beam-type elastic metagratings. Mech Syst Sig Process 156:107688 Kim SY, Oh YB, Lee JS, Kim YY (2023) Anomalous mode -converting reflection of elastic waves using strip-type metagratings. Mech Syst Sig Process 186:109867 Kweun JM, Lee HJ, Oh JH, Seung HM, Kim YY (2017) Transmodal Fabry-Pérot resonance: theory and realization with elastic metamaterials. Phys Rev Lett 118:205901 Larouche S, Smith DR (2012) Reconciliation of generalized refraction with diffraction theory. Opt Lett 37:2391–2393 Lee H, Lee JK, Seoung HM, Kim YY (2018) Mass-stiffness substructuring of an elastic metasurface for full transmission beam steering. J Mech Phys Solids 112:577–593 Lee HJ, Lee JR, Moon SH, Je TJ, Jeon EC, Kim K, Kim YY (2017) Off-centered double-slit metamaterial for elastic wave polarization anomaly. Sci Rep 7:15378 Lee J, Kweun M, Lee W, Park CI, Kim YY (2022) Perfect transmission of elastic waves obliquely incident at solid–solid interfaces. Extreme Mech Lett 51:101606 Liu Y, Liang Z, Liu F, Diba O, Lamb A, Li J (2017) Source illusion devices for flexural lamb waves using elastic metasurfaces. Phys Rev Lett 119:034301 Miklowitz J (1978) The theory of elastic waves and waveguides. North-Holland Rose JL (2014) Ultrasonic guided waves in solid media. Cambridge University Press Su X, Lu Z, Norris AN (2018) Elastic metasurfaces for splitting SV-and P-waves in elastic solids. J Appl Phys 123:091701 Yang X, Kim YY (2018) Asymptotic theory of bimodal quarter-wave impedance matching for full mode-converting transmission. Phys Rev B 98:144110 Yang X, Kweun JM, Kim YY (2018) Theory for perfect transmodal Fabry-Perot interferometer. Sci Rep 8:69 Yang X, Kweun M, Kim YY (2019) Monolayer metamaterial for full mode-converting transmission of elastic waves. Appl Phys Lett 115:071901 Yu N, Capasso F (2014) Flat optics with designer metasurfaces. Nat Mater 13:139–150 Zheng M, Park CI, Liu X, Zhu R, Hu G, Kim YY (2020) Non-resonant metasurface for broadband elastic wave mode splitting. Appl Phys Lett 116:171903

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Zhu H, Walsh TF, Semperlotti F (2018) Total-internal-reflection elastic metasurfaces: design and application to structural vibration isolation. Appl Phys Lett 113:221903 Zhu R, Liu XN, Huang GL (2015) Study of anomalous wave propagation and reflection in semiinfinite elastic metamaterials. Wave Motion 55:73–83

Chapter 10

Perfect Transmission Across 2D Different Media Using Metamaterials

As demonstrated by the wave analysis in Chap. 9, perfect transmission between two media with different impedances is not generally possible. Perfect transmission1 refers to the transmission of a wave with 100% power efficiency from one medium to another. Figure 10.1a demonstrates that a longitudinal wave incident on PEEK from aluminum cannot be fully transmitted without reflection. If a wave is normally incident on the interface of two dissimilar solids, a well-known quarterwave impedance-matching layer can be utilized. However, an obliquely incident wave cannot be fully transmitted across dissimilar media, and no traditional quarterwave impedance-matching element enables full transmission in this situation. The primary reason is that longitudinal and transverse wave modes are typically coupled in solids. Consequently, complete transmission across dissimilar solids is impossible without a novel type of matching element. We present a method for fully transmitting an obliquely incident wave of a particular wave mode from one medium to another wave of any desired wave mode in the other medium using an “anisotropic” metamaterial matching layer. As in Chap. 9, we will focus primarily on two-dimensional (2D) wave phenomena. Herein, we consider metamaterials with anisotropic stiffness, and it may be possible to envision metamaterials with anisotropic density (Zhu et al. 2012, 2016). First, we will examine the special case in which a normally incident longitudinal (transverse) wave from a medium is mode-converted to a transverse (longitudinal) wave with full-power transmission across different media or within a single medium. The possibility of perfect mode conversion from a normally incident longitudinal to a normally-transmitted transverse wave (or vice versa) was first investigated by Kweun

1

Instead of “perfect transmission,” one may also use “full transmission.” As we consider both mode-preserving and mode-converting transmissions, we prefer to use the “perfect transmission” because the full-power transmission mainly deals with single-mode wave phenomena. Nevertheless, we will use both “perfect” and “full” interchangeably.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7_10

295

296

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.1 Simulations of longitudinal wave incidences from aluminum to PEEK at 100 kHz. a Direct incidence without any matching layer, b transmission of a longitudinal wave to a longitudinal wave through an anisotropic L-to-L metamaterial layer realizing full L-to-L transmission, and c transmission of a longitudinal wave to a transverse wave through an anisotropic L-to-T metamaterial layer realizing full L-to-T transmission (L: longitudinal, T: transverse) (Lee et al. 2022a)

et al. (2017) in the name of the transmodal Fabry–Pérot resonance. Mode conversion is possible through a particular anisotropic layer. Anisotropic metamaterials are required for the anisotropic layer to exhibit its unique material properties. Yang et al. (2018) elaborated on the theory of transmodal Fabry–Pérot resonance, and Yang and Kim (2018) investigated a method for designing the unit cell of the desired metamaterial via topology optimization. Although the transmodal Fabry–Pérot resonance converts the incident longitudinal (transverse) mode to a single-mode transverse (longitudinal) wave, the entire incident wave power is not transferred to the converted mode because of the presence of the reflected longitudinal (transverse) wave. For full-power mode-converting transmission, i.e., perfect transmodal transmission, Yang and Kim (2018) established an asymptotic theory of full mode-converting transmission of a normally incident wave, and Yang et al. (2019) realized the required anisotropic layer using single-phase monolayer metamaterials. Lee et al. (2022a) established a general theory of full transmission across dissimilar media for an obliquely incident single-mode wave of any type (longitudinal or transverse2 ). The details of the related wave phenomena will be explained in this chapter. Piao et al. (2020) used the full-power mode-converting metamaterial to develop a high-transmission noninvasive ultrasonic flowmeter in which high-power transverse waves are crucially advantageous. The related results are presented in the latter part of this chapter. An interesting application of the theory by Lee et al. (2022a) was made for the retroreflection of an obliquely incident longitudinal wave (Lee et al. 2022b).

2

In the case of transverse waves, Zheng et al. (2019) investigated anisotropic singly-polarized metamaterials carrying transverse waves only.

10.1 Wave Analysis Model

297

Central to this chapter is the development of a generalized impedance-matching theory (Lee et al. 2022a) for the full transmission of an obliquely-incident longitudinal or transverse wave across dissimilar media with mode-preserving or modeconverting. Figure 10.1b and c depict the transmission simulation results for full mode-preserving and mode-converting, respectively. The matching layers inserted between aluminum and PEEK can be manufactured using anisotropic metamaterials satisfying the to-be-derived general matching conditions below. For subsequent analysis, we assume that the two base media across which wave transmission occurs are isotropic, mainly because isotropic media are commonly used to construct the majority of structural components.

10.1 Wave Analysis Model A general situation with an anisotropic layer inserted between the two base isotropic media is presented in Fig. 10.2. Compared with Fig. 9.7, an additional matching layer is considered in Fig. 10.2. The material properties of the isotropic media and the anisotropic matching layer are denoted as isotropic medium A (semi-infinite space): density ρ A , stiffness C A , isotropic medium B (semi-infinite space): density ρ B , stiffness C B , anisotropic matching layer of width d: density ρani , stiffness Cani . Although the case of longitudinal wave incidence is illustrated in Fig. 10.2, the case of transverse wave incidence can be equally treated. Under longitudinal wave incidence from medium A to B (through a matching layer, the properties of which are yet unknown), three waves exist in medium A, and two waves exist in medium B. They are denoted as isotropic medium A: L + mode (incident longitudinal), L − mode (reflected longitudinal), T − mode (reflected transverse) isotropic medium B: L + mode (transmitted longitudinal), T + mode (transmitted transverse). Note that all waves are assumed as planar waves. The forward and backward waves along the x-direction are indicated by the superscripts + and −, respectively. In the anisotropic matching layer of finite length d, the following four waves exist: anisotropic matching layer: Q L + mode (forward propagating quasi-longitudinal), Q L − mode (backward propagating quasi-longitudinal), QT + mode (forward propagating quasi-transverse), QT − mode (backward propagating quasi-transverse). As defined in Sect. 9.2, QL and QT represent the “quasi-longitudinal” and “quasitransverse” wave modes, respectively. Before deriving the conditions, the anisotropic layer must satisfy for full transmission, we note that the relationship between the incidence angle θinc in medium A and the transmitted angles θ LB and θTB in medium B follows Snell’s law, regardless of the existence of an inserted matching layer between

298

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.2 Schematic of wave phenomena occurring in an anisotropic layer inserted between two different isotropic media A and B. A time-harmonic plane longitudinal (L) wave is assumed to be incident. In the isotropic media, the reflected and transmitted fields generally have both longitudinal (L) and transverse (T) waves. Inside the anisotropic matching layer, four longitudinal–transverse coupled wave modes denoted as QL ± and QT ± exist, where Q denotes “quasi,” and ± indicates forward and backward propagation. The oblique arrows represent the wave vectors

the two media. In general, Eq. (9.67) can be applied for longitudinal wave incidence, and Eq. (9.72) can be applied for transverse wave incidence. Depending on the types of the incident and transmitted wave modes, different conditions must be fulfilled: for mode-preserving perfect transmission TL = 1, TT = 0, R L = RT = 0 for longitudinal wave incidence, TL = 0, TT = 1, R L = RT = 0 for transverse wave incidence,

(10.1)

for mode-converting perfect transmission TL = 0, TT = 1, R L = RT = 0 for longitudinal wave incidence, TL = 1, TT = 0, R L = RT = 0 for transverse wave incidence.

(10.2)

10.2 Scattering Parameters for Anisotropic Media

299

The definitions of Ti and Ri (i = L or T ) are expressed in Eqs. (9.102) and (9.103) or Eqs. (9.106) and (9.107). In the following section, we derive the conditions that the anisotropic matching layer must satisfy for the full mode-preserving or mode-converting perfect transmissions. To impose the requirements given by Eqs. (10.1) and (10.2), the scattering parameters which will be discussed in the following subsection will be utilized.

10.2 Scattering Parameters for Anisotropic Media Referring to Fig. 10.2, the displacement field in isotropic medium i (i = A,B) can be explicitly expressed as (from Sect. 9.3):  i i ui (x, y, t) = Ai PiL + e−ik L ,x x + B i PiL − e+ik L ,x x

 i i +C i PiT + e−ikT,x x + D i PiT − e+ikT,x x e−iσ y eiωt ,

(10.3)

with PiL + = {cos θ Li , sin θ Li }T , PiL − = {− cos θ Li , sin θ Li }T ,

(10.4a)

PiT + = {− sin θTi , cos θTi }T , PiT − = {− sin θTi , − cos θTi }T ,

(10.4b)

i k iL ,x = k iL cos θ Li , k T,x = k Ti cos θTi , k iL = ω/ciL , k Ti = ω/ciT ,

(10.4c)

σ = k iL sin θ Li = k Ti sin θTi ,

(10.4d)

sin θTi /ciT = sin θ Li /ciL ,

(10.4e)

where i = A, B. For longitudinal wave incidence from medium A, we set θinc = θ LA and C A = 0 (A A = 0),

(10.5)

and for transverse wave incidence from medium A, we set θinc = θTA and A A = 0 (C A = 0).

(10.6)

Although C B = 0 and D B = 0 when a wave is incident from medium A, we maintain these terms to ensure the generality of the subsequent analysis.

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

The displacement field for the anisotropic layer (density: ρ; stiffness Ci j )3 inserted between the two isotropic media can be expressed as (see Eq. (9.80)):   u(x, y, t) = APα e−iτα x + BPβ e−iτβ x + CPγ e−iτγ x + DPδ e−iτδ x e−iσ y eiωt , (10.7) where τi (i = α, β, γ , δ) denotes the x-components of the quasi-longitudinal (Q L + and Q L − ) and quasi-transverse (QT + and QT − ) wave vectors. The wave modes denoted by Q L + , Q L − , QT + , and QT − modes correspond to τα , τβ , τγ , and τδ , respectively. The symbol σ denotes the y-component of the wave vectors of all wave modes existing in the anisotropic layer. Note that σ is the same for all wave modes because of the field-matching condition at x = 0 and x = d. In practice, σ is calculated from the incident wave from medium A, and it is given by (using Eqs. (10.4)–(10.6)) σ = k LA sin θinc (for longitudinal wave incidence), σ = k TA sin θinc (for transverse wave incidence). The polarization vector Pi is defined as (see Eq. (9.44))

Pi =

⎧ ⎨ ⎩



Xi 1 + X i2

1

⎫T ⎬

, (i = α, β, γ , δ) ⎭ 1 + X i2

where X i is explicitly stated as Xi =

C16 τi2 + C26 σ 2 + (C12 + C66 )τi σ , ρω2 − (C11 τi2 + C66 σ 2 + 2C16 τi σ )

and τi denotes the x-component of the wave vector: τi ∈ {τα , τβ , τγ , τδ }. Substituting the following relationships nx =

τi

σ , ny = ,k= 2 2 2 τi + σ τi + σ 2

τi2 + σ 2 ,

into Eqs. (9.24) and (9.25) yields the dispersion relation in a quartic equation for the x-directional wavenumber τi : aτi4 + bτi3 + cτi2 + dτi + e = 0, 3

Unless there is a danger of confusion, the symbols without any reference to a specific medium denote the anisotropic matching layer. Therefore, ρ and Ci j refer to the anisotropic matching layer.

10.2 Scattering Parameters for Anisotropic Media

301

where 2 , a = C11 C66 − C16

b = 2σ (C11 C26 − C12 C16 ), 2 c = σ 2 (C11 C22 + 2C16 C26 − C12 − 2C26 C66 )−

ρω2 (C11 + C66 ), d = 2σ 3 (C16 C22 − C12 C26 ) − 2ρω2 σ (C16 + C26 ), 2 e = σ 4 (C66 C22 − C26 ) − ρω2 σ 2 (C22 + C66 ) + ρ 2 ω4 .

To analyze the wave phenomena in the system shown in Fig. 10.2, the following field-matching conditions are used at x = 0 and x = d: F(x = 0− ) = F(x = 0+ ), F(x = d + ) = F(x = d − ),

(10.8a,b)

where F(x = 0− ) and F(x = d + ) are calculated in media A and B, respectively, as (see Eq. (9.83)) ⎧ ⎫A ⎧ ⎫A A⎪ A⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎬

⎨B ⎬ B − A A A =M , F(x = 0 ) = M N x=0 ⎪ ⎪ C⎪ C⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ D D ⎫B ⎧ ⎧ ⎫B B ⎪ e−ik L ,x A ⎪ A⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ik LB,x

⎨B ⎬ B + B B B e F(x = d ) = M N x=d =M . B ⎪ ⎪ C⎪ e−ikT,x C ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ B ⎭ ⎩ ikT,x D e D

(10.9a)

(10.9b)

The state vectors F(x = 0+ ) and F(x = d − ) in the anisotropic matching layer4 are evaluated as (see Eq. (9.83)): ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ A⎪ A⎪ A⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ ⎨ ⎪ ⎬ B B B + − F(x = 0 ) = MN|x=0 =M , F(x = d ) = MN|x=d . ⎪ ⎪ ⎪ C⎪ ⎪C ⎪ ⎪C ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎩ ⎪ ⎭ ⎭ ⎭ D D D (10.10a,b) 4

The coefficients A and B appearing in Eqs. (10.7) and (10.10) should not be confused with the symbols denoting media A and B.

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

Note that in Eqs. (10.9a, b) and (10.10a, b), the following symbols are used: M A and N A : for isotropic medium A, M B and N B : for isotropic medium B, M and N(without superscript): for anisotropic matching layer. The matrices M and N defined in Eqs. (9.84)–(9.87) are rewritten below for the sake of convenience: ⎡ ⎤ γ β iω Pxα iω Px iω Px iω Pxδ ⎢ ⎥ γ β ⎢ iω Pyα iω Py iω Py iω Pyδ ⎥ M=⎢ ⎥ ⎣ α β γ δ ⎦ α β γ δ with  j = −iτ j (C11 Pxj + C16 Pyj ) − iσ (C16 Pxj + C12 Pyj ), j = α, β, γ , δ j = −iτ j (C16 Pxj + C66 Pyj ) − iσ (C66 Pxj + C26 Pyj ), j = α, β, γ , δ and ⎡

e−iτα x ⎢0 N(x) = ⎢ ⎣0 0

0 e−iτβ x 0 0

0 0 e−iτγ x 0

⎤ 0 ⎥ 0 ⎥. ⎦ 0 −iτδ x e

Because the matrices M j and N j ( j = A, B) for isotropic media have exactly the same form as M and N, they can be immediately found if the material properties and wave vectors of the corresponding isotropic medium are used. Next, we eliminate the coefficients {A, B, C, D}T from Eq. (10.10a, b) to obtain F(x = d − ) = MN|x=d M−1 F(x = 0+ ).

(10.11)

Substituting Eqs. (10.8a, b) and (10.9a, b) into Eq. (10.11) yields ⎧ ⎫B ⎧ ⎫A B ⎪ e−ik L ,x d A ⎪ A⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B ⎨ ik L ,x d ⎨ ⎪ ⎬ ⎬   B e B B −1 −1 = (M ) MN|x=d M M A , B −ik d ⎪ ⎪ C⎪ e T,x C ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ B ⎩ ikT,x ⎭ D e dD or

(10.12)

10.2 Scattering Parameters for Anisotropic Media

303

⎧ ⎫B ⎧ ⎫A B ⎪ e−ik L ,x d A ⎪ A⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B ⎨ ik L ,x d ⎬ ⎨ ⎪ ⎬ e B B =S , B −ik d ⎪ ⎪ C⎪ e T,x C ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ B ⎩ ikT,x ⎭ D e dD

(10.13)

where S is called the scattering matrix defined as ⎡

S11 ⎢ S21 S=⎢ ⎣ S31 S41

S12 S22 S32 S42

S13 S23 S33 S43

⎤ S14 S24 ⎥ ⎥ = (M B )−1 TM A . S34 ⎦ S44

(10.14)

The matrix T appearing in Eq. (10.14) is the transfer matrix in the anisotropic media relating F(x = d − ) to F(x = 0+ ) as F(x = d − ) = TF(x = 0+ ), where ⎤ T11 T12 T13 T14 ⎢ T21 T22 T23 T24 ⎥ −1 ⎥ T=⎢ ⎣ T31 T32 T33 T34 ⎦ = M N|x=d M T41 T42 T43 T44 ⎤ ⎡ −iτα d 0 0 0 e ⎥ −1 ⎢0 0 e−iτβ d 0 ⎥M . = M⎢ −iτγ d ⎦ ⎣0 0 0 e −iτδ d 0 0 0 e ⎡

(10.15)

The components of the matrix T can be expressed as T pq = a pq e−iτα d + b pq e−iτβ d + c pq e−iτγ d + d pq e−iτδ d , ( p, q ∈ {1, 2, 3, 4}) (10.16) where a pq = c pq =

 q1 (−1)q+1 M p1 det M det M  q3 (−1)q+1 M p3 det M det M

, b pq = , d pq =

 q2 (−1)q M p2 det M det M  q4 (−1)q M p4 det M det M

, .

(10.17)

 pq is a 3 by 3 In Eq. (10.17), M pq is the (p, q) component of the matrix M, and M matrix obtained from M by removing its pth row and qth column. Note that M pq for

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

˜ is purely any p and q is purely imaginary. Therefore, det M is real-valued, and det M imaginary. Therefore, apq , bpq , cpq , and d pq appearing in Eq. (10.17) are real-valued. Substituting Eq. (10.17) into Eq. (10.14) yields S pq = A pq e−iτα d + B pq e−iτβ d + C pq e−iτγ d + D pq e−iτδ d ,( p, q ∈ {1, 2, 3, 4}) (10.18) where A pq = C pq =

4  4 

A M −B pr ar s Msq , B pq =

4  4 

r =1 s=1

r =1 s=1

4  4 

4  4 

A M −B pr cr s Msq , D pq =

r =1 s=1

A M −B pr br s Msq ,

A M −B pr dr s Msq .

(10.19)

r =1 s=1

A A In Eq. (10.19), M pq and M −B pq are the (p, q) components of the matrix M and B −1 A (M ) , respectively. It is noted that like M, all components of M and (M B )−1 are purely imaginary. Therefore, A pq , B pq , C pq , and D pq are real-valued; this realvaluedness will be critically useful in deriving the explicit conditions for perfect transmission below. For later use, some components of the scattering matrix S are explicitly written in terms of Ti j :













1 B B 2 B 2 ωC66 k LA,x k T,x − σ 2 T11 + σ k T,x − σ 2 T12 B A B B 2ωC11 k L k L k L ,x  +2k LA,x k LB,x σ T21 + 2k LB,x σ 2 T22  A 2  B 2    B 2 A B C66 k T,x − σ 2 k T,x − σ 2 T13 + 2k LA,x σ k T,x − σ 2 T14 − C66    A 2 +2k LB,x σ k T,x − σ 2 T23 + 4k LA,x k LB,x σ 2 T24

S11 =

  − ω2 k LA,x k LB,x T31 + k LB,x σ T32 + k LA,x σ T41 + σ 2 T42  B  B 2  A k L ,x k T,x − σ 2 T33 + 2k LA,x k LB,x σ T34 + ωC66    B 2 +σ k T,x − σ 2 T43 + 2k LA,x σ 2 T44 











1 B B 2 B 2 ωC66 −k LA,x k T,x − σ 2 T11 − σ k T,x − σ 2 T12 B A B B 2ωC11 k L k L k L ,x  +2k LA,x k LB,x σ T21 + 2k LB,x σ 2 T22   A 2  B 2    B 2 A B C66 − k T,x − σ 2 k T,x − σ 2 T13 − 2k LA,x σ k T,x − σ 2 T14 − C66    A 2 +2k LB,x σ k T,x − σ 2 T23 + 4k LA,x k LB,x σ 2 T24   − ω2 k LA,x k LB,x T31 + k LB,x σ T32 − k LA,x σ T41 − σ 2 T42

S21 =

 B  B 2  A k L ,x k T,x − σ 2 T33 + 2k LA,x k LB,x σ T34 + ωC66   B 2 − σ k T,x − σ 2 T43 − 2k LA,x det σ 2 T44

10.2 Scattering Parameters for Anisotropic Media



305



1 B B B ωC66 −2k LA,x k T,x σ T11 − 2k T,x σ 2 T12 B A B B 2ωC66 k L k T k T,x  B 2     B 2 +k LA,x k T,x − σ 2 T21 + σ k T,x − σ 2 T22    A 2 A B B B C66 −2k T,x σ k T,x − σ 2 T13 − 4k LA,x k T,x σ 2 T14 − C66  B 2     A 2  B 2 + k T,x − σ 2 k T,x − σ 2 T23 + 2k LA,x σ k T,x − σ 2 T24   B B T41 + k T,x σ T42 − ω2 −k LA,x σ T31 − σ 2 T32 + k LA,x k T,x

S31 =

  A 2  A −σ k T,x − σ 2 T33 − 2k LA,x σ 2 T34 + ωC66  A 2   B B k T,x − σ 2 T43 + 2k LA,x k T,x +k T,x σ T44 



1 B B B ωC66 −2k LA,x k T,x σ T11 − 2k T,x σ 2 T12 B A B B 2ωC66 k L k T k T,x  B 2  B 2    −k LA,x k T,x − σ 2 T21 − σ k T,x − σ 2 T22  A 2   A B B B C66 −2k T,x σ k T,x − σ 2 T13 − 4k LA,x k T,x σ 2 T14 − C66      A 2  B 2 B 2 − k T,x − σ 2 k T,x − σ 2 T23 − 2k LA,x σ k T,x − σ 2 T24   B B T41 + k T,x σ T42 − ω2 k LA,x σ T31 + σ 2 T32 + k LA,x k T,x

S41 =

  A 2  A σ k T,x − σ 2 T33 + 2k LA,x σ 2 T34 + ωC66  A 2   B B k T,x − σ 2 T43 + 2k LA,x k T,x +k T,x σ T44

• Longitudinal wave incidence from medium A In the case of longitudinal wave incidence from medium A, we can set (A, B, C, D) A = (1, r L , 0, r T ) and (A, B, C, D) B = (t L , 0, tT , 0) in Eq. (10.13) to obtain ⎧ ⎫ ⎡ ⎧ −ik B d ⎫ 1 ⎪ S11 e L ,x t L ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ ⎢ ⎬ ⎨ rL 0 S 21 =⎢ =S B ⎣ S31 ⎪ ⎪ e−ikT,x d tT ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ S41 rT 0

S12 S22 S32 S42

S13 S23 S33 S43

⎤⎧ ⎫ 1 ⎪ S14 ⎪ ⎪ ⎨ ⎪ ⎬ r S24 ⎥ ⎥ L . ⎪0 ⎪ S34 ⎦⎪ ⎩ ⎪ ⎭ S44 rT

(10.20)

Note that r L (r T ) is the reflection coefficient of the longitudinal (transverse) wave in medium A, whereas t L (tT ) denotes the transmission coefficient of the longitudinal (transverse) wave in medium B, respectively. They are defined in Eqs. (9.94) and (9.95). Using Eq. (10.20), we can express the reflection and transmission coefficients as rL = B

S41 S24 − S21 S44 S42 S21 − S22 S41 , rT = , S22 S44 − S42 S24 S22 S44 − S42 S24 B

t L = eik L ,x d (S11 + S12 r L + S14 r T ), tT = eikT,x d (S31 + S32 r L + S34 r T ).

(10.21a) (10.21b)

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

For the perfect transmission from medium A to medium B, all reflection coefficients should identically vanish. For full mode selectivity, the transmission coefficient of the undesired wave mode should be zero as well. These conditions can be expressed as r L = 0, r T = 0, tT = 0 (L-to-L perfect transmission)

(10.22a,b,c)

r L = 0, r T = 0, t L = 0 (L-to-T perfect transmission),

(10.23a,b,c)

where “L-to-L” and “L-to-T” stand for “longitudinal wave to longitudinal wave” and “longitudinal wave to transverse wave,” respectively. Equation (10.22a, b, c), representing the conditions for perfect L-to-L (longitudinal wave to longitudinal wave) transmission, can be satisfied if the following conditions are satisfied (see Eqs. (10.21a, b)): S21 = 0, S31 = 0, S41 = 0 (L-to-L perfect transmission).

(10.24a,b,c)

Similarly, Eq. (10.23a, b, c), i.e., the conditions for perfect L-to-T (longitudinal wave to transverse wave) transmission, can be satisfied if the following conditions hold (see Eqs. (10.21a, b)): S11 = 0, S21 = 0, S41 = 0 (L-to-T perfect transmission).

(10.25a,b,c)

Note that Eq. (10.24a, b, c) (or Eq. (10.25a, b, c)) has three equations, but they actually represent six equations because S pq ’s are complex-valued. Therefore, we can examine each of Eq. (10.24a, b, c) (or Eq. (10.25a, b, c)) in terms of its phase and magnitude. • Transverse wave incidence from medium A In the case of transverse wave incidence, we can set (A, B, C, D) A = (0, r L , 1, r T ) and (A, B, C, D) B = (t L , 0, tT , 0) in Eq. (10.13) and find the following relation: ⎧ jk B d ⎫ ⎡ L ,x t S11 L⎪ ⎪ ⎪e ⎪ ⎨ ⎬ ⎢ 0 S 21 =⎢ B jk T,x d ⎣ ⎪ ⎪ S e t 31 T ⎪ ⎪ ⎩ ⎭ S41 0

S12 S22 S32 S42

S13 S23 S33 S43

⎤⎧ ⎫ 0 ⎪ S14 ⎪ ⎪ ⎨ ⎪ ⎬ r S24 ⎥ ⎥ L . S34 ⎦⎪ ⎪1 ⎪ ⎪ ⎩ ⎭ S44 rT

(10.26)

Following the procedure used for establishing Eqs. (10.24a, b, c) and (10.25a, b, c), we can derive the conditions for perfect transmission expressed in terms of Si j : S13 = 0, S23 = 0, S43 = 0 (for T-to-T perfect transmission),

(10.27a,b,c)

S23 = 0, S33 = 0, S43 = 0 (for T-to-L perfect transmission),

(10.28a,b,c)

10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching …

307

where “T-to-T” and “T-to-L” stand for “transverse wave to transverse wave” and “transverse wave to longitudinal wave,” respectively.

10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching Layer5 10.3.1 Theory Herein, we derive the conditions that the material properties of the anisotropic matching layer must satisfy for realizing perfect transmission from one isotropic medium (medium A) to the other (medium B), illustrated in Figure 10.1b, c. We begin with the case of longitudinal wave incidence (Lee et al. 2022a). • Perfect L-to-L transmission: Using S pq expressed in Eqs. (10.18), (10.24a,b,c) can be put into the following form ⎧ −i(τ −τ )d ⎫ ⎡ ⎫ ⎤−1 ⎧ B21 C21 D21 ⎨e β α ⎬ ⎨ −A21 ⎬ e−i(τγ −τα )d = ⎣ B31 C31 D31 ⎦ −A31 , ⎩ −i(τδ −τα )d ⎭ ⎩ ⎭ e B41 C41 D41 −A41

(10.29)

where τi (i = α, β, γ , δ) is explained below Eq. (10.7). The reason why we put Eq. (10.24a, b, c) in the form of Eq. (10.29) is that it is convenient to extract the phase information from Eq. (10.29). Because all components on the right-hand side are real-valued (refer to the remark under Eq. (10.19)), all the components on the  − τα d, left-hand side must be real-valued. Accordingly, it is concluded that τ β   τγ − τα d, and (τδ − τα )d must be integer multiples of π. Therefore, the following conditions, called the generalized phase-matching conditions, must be satisfied for perfect transmission: Generalized phase-matching conditions ⎧ ⎨ φ Q L + ,Q L − ≡ (τβ − τα )d = lπ, φ Q L + ,QT + ≡ (τγ − τα )d = mπ, ⎩ φ Q L + ,QT − ≡ (τδ − τα )d = nπ, (l, m, n : integers)

(10.30)

where φi, j denotes the phase difference between wave modes i and j over the anisotropic matching layer of thickness d. 5

Readers may skip this section and move directly to Sect. 10.4 dealing with the normal incidence case after reading 10.3.1, if they prefer to understand only the essence of perfect mode-converting transmission unique in elastic waves.

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

As the next step, we consider the magnitude information from Eq. (10.24a, b, c). To this end, we first combine Eqs. (10.24b, c) to obtain S31 − S41 = 0, S31 + S41 = 0,

(10.31a,b)

which can be explicitly written as  B 2     A 2   B A k T,x − σ 2 ω k LA,x T21 + σ T22 − C66 k T,x − σ 2 T23 + 2k LA,x σ T24 C66   A 2     A k T,x − σ 2 T33 + 2k LA,x σ T34 = 0, (10.32) + ωσ ω k LA,x T31 + σ T32 − C66     A 2   B A 2C66 σ ω k LA,x T11 + σ T12 − C66 k T,x − σ 2 T13 + 2k LA,x σ T14   A 2     A k T,x − σ 2 T43 + 2k LA,x σ T44 = 0. (10.33) + ω ω k LA,x T41 + σ T42 − C66 Substituting Eqs. (10.32) and (10.33) into Eq. (10.24a, b, c) yields     A 2   A k T,x − σ 2 T13 + 2k LA,x σ T14 2σ k LB,x ω k LA,x T11 + σ T12 − C66     A 2    B 2 A k T,x − σ 2 T43 + 2k LA,x σ T44 = 0. − σ 2 ω k LA,x T41 + σ T42 − C66 − k T,x (10.34) Note that Eqs. (10.32)–(10.34) are real-valued if the phase-matching conditions given by Eq. (10.30) are satisfied. To write Eqs. (10.32)–(10.34) in a compact form, we introduce the following parameter ϕm defined as ϕm ≡ Tm1 + tan θ LA Tm2 − Z LA cos 2θTA Tm3 − Z TA sin 2θTA Tm4 (m = 1, 2, 3, 4),

(10.35)

j

where Z i denotes the normal impedance of wave mode i in medium j, j

j

Zi =

ρ j ci

cos θi

j

( j = A, B, i = L , T ),

j

j

j

j

and ci denotes the wave speed of wave mode i in medium j. Using ki,x = ki cos θi ( j = A, B, i = L , T ), σ = k LA sin θinc , and the definition of ϕm (m = 1, 2, 3, 4), Eqs. (10.32)–(10.34) reduce to Z TB cos 2θTB ϕ2 + tan θTB ϕ3 = 0,

(10.36a)

Z LB sin 2θTB tan θTB ϕ1 + tan θ LB ϕ4 = 0,

(10.36b)

Z TB tan 2θTB ϕ3 − Z LB ϕ4 = 0.

(10.36c)

10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching …

309

To interpret the results in Eqs. (10.36a–10.36c), we introduce the “generalized bimodal impedances” defined as  Z 0L-to-L

=  =

−Z LB ϕ1 L-to-L ,Z 1 = T13 −Z LB ϕ2 L-to-L ,Z 3 = T13

 

ϕ3 , Z 2L-to-L T13 ϕ4 . T13

(10.37)

If Z iL-to-L (i = 0, 1, 2, 3) is used to rewrite Eqs. (10.36a–10.36c), the resulting equations can be put into the following forms, which will be called the generalized impedance-matching conditions: Generalized impedance-matching conditions ⎧ L-to-L ⎪ ⎪ Z = cos2θTB Z 0L-to-L , 1 ⎪ ⎨ Z L-to-L = tan θTB Z 0L-to-L , ⎪ 2 ⎪ ⎪ ⎩ tanθ B Z L-to-L = sin 2θ B tanθ B Z L-to-L . L

3

T

T

(10.38)

0

j

For normal incidence (θinc = 0), the angles of the reflected and transmitted waves, θi ( j = A, B, i = L , T ), are identically zero. Furthermore, certain components of the transfer matrix vanish. Specifically, T11 = T12 = T21 = T22 = T33 = T34 = T43 = T44 = 0 (detailed in Sect. 10.4). Accordingly, the three conditions in Eq. (10.38) can be reduced to a single nontrivial condition that becomes the well-known quarter-wave impedance-matching condition described in Eq. (7.88): 

ρC11 =



Z LA Z LB with C16 = 0,

√ where ρC11 represents the characteristic impedance of the matching layer for longitudinal waves. To summarize the findings from the above analysis, the matching layer must satisfy three phase-matching conditions expressed in Eq. (10.30) and three generalized impedance conditions in Eq. (10.38) for perfect L-to-L transmission for an obliquely incident longitudinal wave from medium A. If the target frequency ω and incidence angle θinc are specified, perfect L-to-L transmission requires the determination of seven effective material parameters (ρ, C11 , C12 , C16 , C22 , C26 , C66 ) of the anisotropic matching layer satisfying the six conditions given by Eqs. (10.30) and (10.38). Because more parameters (seven parameters) are available than the required conditions (six conditions), there is the freedom to “design” the anisotropic material properties. This means that more than a single set of parameters can realize the perfect transmission. Note that the layer thickness d can be also considered as a

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

design parameter. In this case, we have eight design parameters. The actual calculations of the desired (ρ, C11 , C12 , C16 , C22 , C26 , C66 ) values and the realization of the anisotropic layer using a single-phase metamaterial are presented in the following section. • perfect L-to-T transmission In this case, we use Eq. (10.18) to write Eq. (10.25a, b, c) as ⎫ ⎧ −i(τ −τ )d ⎫ ⎡ ⎤−1 ⎧ B11 C11 D11 ⎨ −A11 ⎬ ⎨e β α ⎬ e−i(τγ −τα )d = ⎣ B21 C21 D21 ⎦ −A21 . ⎩ ⎭ ⎩ −i(τδ −τα )d ⎭ e B41 C41 D41 −A41

(10.39)

Based on the arguments employed to derive Eqs. (10.30) and (10.38), we can derive the flowing two sets of conditions for perfect L-to-T transmission: Generalized phase-matching conditions ⎧ ⎨ φ Q L + ,Q L − ≡ (τβ − τα )d = lπ, φ Q L + ,QT + ≡ (τγ − τα )d = mπ, ⎩ φ Q L + ,QT − ≡ (τδ − τα )d = nπ, (l, m, n : integers)

(10.40)

which is the same as Eq. (10.30). To consider the magnitude information from Eq. (10.25a, b, c), we rewrite Eq. (10.25b, c) as S21 − S41 = 0, S21 + S41 = 0,

(10.41a,b)

which can be explicitly written as   A 2     B A k T,x − σ 2 T23 + 2k LA,x σ T24 σ ω k LA,x T21 + σ T22 − C66 2C66   A 2     A k T,x − σ 2 T33 + 2k LA,x σ T34 = 0, (10.42) − ω ω k LA,x T31 + σ T32 − C66  B 2     A 2   B A C66 k T,x − σ 2 ω k LA,x T11 + σ T12 − C66 k T,x − σ 2 T13 + 2k LA,x σ T14   A 2     A k T,x − σ 2 T43 + 2k LA,x σ T44 = 0. (10.43) − ωσ ω k LA,x T41 + σ T42 − C66 Substituting Eqs. (10.42) and (10.43) into Eq. (10.25a) yields   A   A 2   B A ω k L ,x T11 + σ T12 − C66 k T,x − σ 2 T13 + 2k LA,x σ T14 k T,x   A 2     A k T,x − σ 2 T23 + 2k LA,x σ T24 = 0. (10.44) + σ ω k LA,x T21 + σ T22 − C66 j

j

j

Using ki,x = ki cos θi ( j = A, B, i = L , T ), σ = k LA sin θinc , and the definition of ϕm (m = 1, 2, 3, 4) in Eq. (10.35), Eqs. (10.42)–(10.44) reduce to

10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching …

311

Z TB sin 2θTB ϕ2 − ϕ3 = 0,

(10.45a)

Z TB cos 2θTB ϕ1 − tan θTB ϕ4 = 0, and

(10.45b)

ϕ1 + tan θTB ϕ2 = 0.

(10.45c)

Equations (10.45a, b, c) can be put into the following form: Generalized impedance-matching conditions ⎧ ⎪ ⎪ Z 1L-to-T = cos2θTB Z 0L-to-T , ⎪ ⎨ Z 2L-to-T = tan θTB Z 0L-to-T , ⎪ ⎪ ⎪ ⎩ Z L-to-T = sin 2θ B Z L-to-T , 3

T

(10.46)

0

where the bimodal impedances Z iL-to-T (i = 0, 1, 2, 3) are defined as  Z 0L-to-T

≡

−Z TB ϕ2 , Z 1L-to-T ≡ T23



ϕ4 , Z 2L-to-T ≡ T23



Z TB ϕ1 , Z 3L-to-T ≡ T23



−ϕ3 . T23 (10.47)

For the perfect L-to-T transmission, we should find seven effective material parameters (ρ, C11 , C12 , C16 , C22 , C26 , C66 ) to satisfy the required six conditions expressed in Eqs. (10.40) and (10.46). For the case of transverse wave incidence, the conditions for perfect transmission to a longitudinal wave or a transverse in a different medium can be similarly obtained using the procedure used to derive the case of longitudinal wave incidence. See Lee et al. (2022a) for more details.

10.3.2 Design of Anisotropic Metamaterial Layers As a specific example for designing an anisotropic matching layer that realizes perfect transmission across different media, the following data are considered: Medium A: aluminum ρ = 2700 kg m-3 , C 11 = 78.6 GPa, and C 66 = 26.3 GPa, Medium B: PEEK, ρ = 1320 kg m-3 , C 11 = 5.03 GPa, and C 66 = 1.51 GPa, Harmonic excitation frequency: f = ω/2π = 100 kHz, Thickness of matching layer: d = 50 mm, Incident mode and angle: longitudinal wave and θinc = 60°. Based on these data, we can determine the effective material parameters (ρ, C11 , C12 , C16 , C22 , C26 , C66 ) of the metamaterial matching layer that satisfy the

312

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.3 Sets of effective material parameters of anisotropic metamaterial layers that can realize perfect transmission from an aluminum plate to a PEEK plate for the case of longitudinal wave incidence (θinc = 60° at 100 kHz) (Lee et al. 2022a)

generalized phase-matching and impedance-matching conditions (Eqs. (10.30) and (10.38) for perfect L-to-L transmission and Eqs. (10.40) and (10.46) for perfect L-toT transmission). We can find the material parameters by trial and error or by solving an optimization problem. Because the set of the material parameters satisfying the matching conditions is not unique, several sets can be found. The results are plotted in Fig. 10.3. The existence of multiple sets satisfying the desired conditions provides design flexibility in the actual realization of anisotropic layers by metamaterials. For instance, the following set of material properties can be selected among the several sets of results presented in Fig. 10.3: L-to-L matching layer:ρ = 1358.0 kg m−3 , C11 = 34.162, C12 = 4.7172, C16 = −5.0390, C22 = 24.173, C26 = −1.9141, and C66 = 12.359 (unit of Ci j : GPa), (10.48) L-to-T matching layer:ρ = 2177.7 kg m−3 , C11 = 33.571, C12 = −4.4789, C16 = 0.45270, C22 = 62.255, C26 = 7.5424, and C66 = 6.3240 (unit of Ci j : GPa). (10.49) ◦

For the incidence angle of θinc = 60 , the wave propagation direction in medium B (PEEK) can be calculated using the Snell’s law expressed in Eq. (9.67) as

10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching …

313

◦

L − to − L transmission : θ LB+ = 18.3 (sin θ LB+ = (c LB /c LA ) sin θinc ), ◦

L − to − T transmission : θTB+ = 9.89 (sin θTB+ = (cTB /c LA ) sin θinc ). To demonstrate that the anisotropic matching layers with the material properties expressed in Eqs. (10.48) and (10.49) satisfy the required conditions for perfect transmissions, we calculate the wavenumbers, phase differences, and impedances in the matching layer at 100 kHz as follows. L-to-L matching layer: wavenumbers: τ α = 115.3 m–1 , τ β = –73.25 m–1 , τ γ = 178.1 m–1 , τ δ = –198.9 m–1 , phase differences: φ Q L + ,Q L − = –3 π, φ Q L + ,QT + = π, φ Q L + ,QT − = –5 π , impedances: Z 0L-to-L = 7.827 × 106 kg·m–2 ·s–1 , Z 1L-to-L = 7.595 × 106 kg·m–2 ·s–1 , Z 2L-to-L = 4.494 × 106 kg·m–2 ·s–1 , Z 3L-to-L = 3.312 × 106 kg·m–2 s–1 . L-to-T matching layer: wavenumbers:τα = 97.10 m–1 , τβ = –154.2 m–1 , τγ = 159.9 m–1 , τδ = –342.7 m–1 , phase differences: φ Q L + ,Q L − = –4 π, φ Q L + ,QT + = π, φ Q L + ,QT − = –7π , impedances: Z 0L-to-T = 1.031 × 107 kg·m–2 ·s–1 , Z 1L-to-T = 1.001 × 107 kg·m–2 ·s–1 , Z 2L-to-T = 4.293 × 106 kg·m–2 ·s−1 , Z 3L-to-T = 6.025 × 106 kg·m–2 ·s–1 . The calculated phase differences for both L-to-L and L-to-T cases satisfy the three conditions stated by Eqs. (10.30) and (10.40). In addition, the calculated impedance values satisfy the three generalized impedance-matching conditions stated by Eqs. (10.38) and (10.46). It should be noted that there are no natural materials having the above properties or those presented in Fig. 10.3. However, the densities and stiffnesses of the desired anisotropic layers given in (10.48) and (10.49) are of the same order as those of the aluminum plate. Therefore, elaborately designed aluminum-based nonresonant anisotropic metamaterials can realize the desired material properties. While certain actual realizations with metamaterials are provided in the following subsection, this subsection investigates wave phenomena occurring with the anisotropic layers having the desired material properties. Figures 10.4 and 10.5 show the simulation results for the L-to-L and L-to-T perfect transmissions, respectively, where the designed anisotropic materials are used as the matching layer. For the simulation, a longitudinal plane wave at 100 kHz is assumed to be incident at 60° from medium A (thin aluminum plate) to medium B (thin PEEK plate). Based on the results in these figures, the following observations can be made. After the incident longitudinal waves pass through the designed anisotropic matching layers, they are transmitted as an ideal longitudinal wave (Fig. 10.4a) or a transverse wave (Fig. 10.5a) in medium B; the wave polarization and propagation directions are ideally parallel as shown in Fig. 10.4a and perpendicular as shown in Fig. 10.5a. Inside the anisotropic layers, the wave polarization changes from the polarization of the incident wave at the interface with aluminum to that of the desired

314

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.4 Finite element simulation for the wave transmission from medium A (aluminum) to medium B (PEEK) through a perfect L-to-L anisotropic matching layer having the material properties given in Eq. (10.48). a Snapshots of the deformed aluminum-matching layer-PEEK plate system. The Floquet periodic conditions were used on the top and bottom boundaries. Black arrows: displacement vectors. Color contours: displacement phases. b Slowness curves for aluminum (solid curves) and the anisotropic material (dashed curves) of Eq. (10.48) at 100 kHz. Dots on dashed curves: four wave modes in the anisotropic layer; dot on the inner solid curve: incident longitudinal wave; horizontal solid line: the y-directional wavenumber (σ ) of all wave modes (Lee et al. 2022a)

Fig. 10.5 Finite element simulation for the wave transmission from medium A (aluminum) to medium B (PEEK) through a perfect L-to-T anisotropic matching layer having the material properties stated in Eq. (10.49). a Snapshots of the deformed aluminum-matching layer-PEEK plate system. b Slowness curves for aluminum (solid curves) and the anisotropic material (dashed curves) of Eq. (10.48) at 100 kHz. (For the meaning of various symbols, see the captions of Fig. 10.4) (Lee et al. 2022a)

10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching …

315

Fig. 10.6 Decomposition of the standing waves formed inside the anisotropic layer under the incidence of a 100 kHz longitudinal wave at 60° for a perfect L-to-L transmission and b perfect L-to-T transmission. (Solid A: aluminum, Solid B: PEEK) (Lee et al. 2022a)

transmitted wave at the interface with PEEK. This drastic polarization change can be realized because the standing wave pattern inside the anisotropic layer is decomposed by a specific combination of four wave modes existing in its inside. The modal decompositions are further illustrated in Fig. 10.6.

10.3.3 Metamaterial Realization and Experimental Verification To realize perfect transmission across different elastic solids in practice, the matching layers satisfying the required conditions must be fabricated by metamaterials because no real materials possess the desired anisotropic material properties. The excitation frequencies are selected as6 79.5 kHz for perfect L-to-L transmission,

6

(10.50a)

The target frequency was chosen considering the actual fabrication of metamaterials because too small features (such as holes) cannot be machined. That is why relative low frequencies are considered here.

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

59.5 kHz for perfect L-to-T transmission.

(10.50b) ◦

The remaining conditions are identical to those considered in Sect. 10.3.2: θinc = 60 , medium A: aluminum, and medium B: PEEK. To fabricate anisotropic metamaterial matching layers, a few voids were drilled in a base aluminum plate to produce a single-phased7 metamaterial, as depicted in Fig. 10.7a. The unit cell of the proposed metamaterial in Fig. 10.7a can be characterized by nine parameters (a, w, N, l 1 , r 1 , θ 1 , l 2 , r 2 , and θ 2 ) controlling the locations and shapes of the voids.8 The designed metamaterial layers are sketched in Fig. 10.7b, where four unit cells9 (N = 4) were used along the x -direction (wave propagation direction) to realize the matching layers. The unit cell size a was selected as a = 8 mm, which is shorter than the smallest wavelengths of interest.10 Therefore, one can treat the metamaterial layer as a homogenized anisotropic layer that can be characterized by its effective material properties. The remaining eight geometric parameters other than a can be determined by trials and errors or by solving an optimization problem, as remarked earlier. As depicted in Fig. 10.7c, an incident longitudinal wave is completely converted to a transmitted longitudinal wave through the designed L-to-L metamaterial layer.11 The perfect transmission from a longitudinal wave in an aluminum plate to a transverse wave in a PEEK plate is depicted in Fig. 10.7d. In the PEEK plate, the wave field is nearly dominated by the divergence (∇ · u) field as shown in Fig. 10.7c and by the curl (∇ × u) field as shown in Fig. 10.7d. As explained in Eqs. (9.40) and (9.41), the divergence (∇ · u) and curl (∇ × u) fields correspond to longitudinal and transverse wave fields, respectively. Figure 10.8 compares the wave fields in the PEEK plate with and without the designed metamaterial matching layers. Figure 10.8a shows that the longitudinal wave dominates the wave field in the PEEK plate, whereas the transverse wave is nearly suppressed with the insertion of the metamaterial in the case of L-to-L transmission. Similar phenomena occur in the case of L-to-T transmission in Fig. 10.8b. The photo image of the fabricated metamaterial inserted between an aluminum and a PEEK plate is illustrated in Fig. 10.9a. The experimental results are compared with the numerical results in Fig. 10.9b. The L-to-L and L-to-T metamaterial layers 7

A single-phased metamaterial is a metamaterial made of a single material and voids. Therefore, the design of a single-phased metamaterial becomes to determine the number, location, and shape of voids in the base material. 8 The number of voids was determined after some trials and errors. 9 In theory, N should be ∞ because the unit cell design is based on the assumption that the unit cells are infinitely repeated. It was found, however, that using 4 unit cells gives wave simulation results similar to those performed with many unit cells. If N is too small, the effects of the boundaries of the metamaterial facing adjacent media can be significant. Therefore, a metamaterial made of a too-small number of unit cells may not exhibit the desired effective properties. 10 A metamaterial is characterized by its effective medium properties. For the evaluation of the effective properties, the feature size is typically assumed to be smaller than the shortest wavelength of interest. 11 For the further details of numerical simulations, see Appendix B of Lee et al. (2022a).

10.3 Perfect Transmission at Oblique Incidence by Metamaterial Matching …

317

Fig. 10.7 Realization of the anisotropic matching layers using a single-phase metamaterial (base material: aluminum) for the cases in Eqs. (10.50a, b). a Microstructure of the solid-void metamaterial used to make the metamaterial layer. b Designed L-to-L metamaterial layer: l 1 = 0.325a, r 1 = 0.212a, θ 1 = 32.7°, l 2 = 0.325a, r 2 = 0.201a, θ 2 = 89.3°, and w = 0.500a with a = 8 mm. Designed L-to-T metamaterial layers: l 1 = 0.723a, r 1 = 0.098a, θ 1 = 12.8°, l 2 = 0.698a, r 2 = 0.106a, θ 2 = 68.3°, and w = 0.277a with a = 8 mm. c, d Simulation results with the designed L-to-L and L-to-T metamaterial layers, respectively (Lee et al. 2022a)

318

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.8 Transmitted wave fields in a PEEK plate with (“w/”) and without (“w/o”) full transmission metamaterial matching layers. a L-to-L case at 79.5 kHz and b L-to-T case at 59.5 kHz. For both cases, a longitudinal wave is incident at 60° from an aluminum plate (Lee et al. 2022a)

(designed in Fig. 10.7b) composed of 4 × 60 unit cells were fabricated by laserbeam machining on a 5-mm-thick aluminum plate, as portrayed in Fig. 10.9a. A piezoceramic transducer installed on the aluminum plate was used as a transmitter of the incident longitudinal wave. The piezoceramic transducers and magnetostrictive patch transducers (Kim and Kwon 2015) installed on the PEEK plate were used as sensors to capture the transmitted longitudinal and transverse waves, respectively. The actual waves considered in the experiment were the lowest symmetric mode Lamb wave and the fundamental shear-horizontal guided wave, which have been known to correspond well to the in-plane longitudinal and transverse waves in solids. Experimentally measured wave signals were recorded with varying radiation angles (defined as θ rad in Fig. 10.9a) with and without the L-to-L (L-to-T) metamaterial layer. The use of the L-to-L and L-to-T metamaterial layers amplifies the maximum displacement amplitudes of the transmitted longitudinal and transverse waves by 2.59 and 4.55 times, respectively, compared with those without the metamaterial layer, as depicted in Fig. 10.9b. The measured results presented in Fig. 10.9b are consistent with the numerical simulation results. The experimental analysis confirmed that the transmission of the desired wave mode can be drastically enhanced by utilizing the fabricated metamaterial layers.

10.4 Perfect Mode-Converting Transmission at Normal Incidence by Metamaterial Matching Layer The normal incidence from medium A to medium B is a special case of the oblique incidence considered in the previous section. In particular, we look into the perfect

10.4 Perfect Mode-Converting Transmission at Normal Incidence …

319

Fig. 10.9 a Top view image of the test aluminum-metamaterial-PEEK plate systems for perfect L-to-L transmission at 79.5 kHz (above) and L-to-T transmission at 59.5 kHz (below). Incidence angle: 60°. Black regions in photos: voids. b Comparison of the measured (circles) and simulated (lines) displacement amplitudes of the transmitted L (above) and T (below) waves as the function of the radiation angle (θ rad ) with and without L-to-L (above) and L-to-T (below) metamaterial layers. Light lines and circles: with metamaterial layers, dark lines and circles: without them. (a.u.: arbitrary unit) (Lee et al. 2022a)

mode-converting transmission from a normally incident longitudinal wave to a transverse wave, as illustrated in Figure 10.10. Because the perfect mode-preserving transmission between dissimilar media can be easily achieved with well-known isotropic quarter-wave impedance-matching layers, we will primarily focus on the case of perfect mode-converting transmission of a normally incident longitudinal wave.12 The displacement fields in media A and B can be written as Eq. (10.3), and the displacement field in the matching layer can be given by Eq. (10.7). When a normally ◦ incident wave (θinc = 0 ) from medium A is transmitted to medium B through an anisotropic layer of width d (refer to Fig. (10.2)), we have j

θi = 0(i = L , T ; j = A, B) (angles of reflected and transmitted waves) (10.51a) τα = +k Q L , τβ = −k Q L , τγ = +k QT , τδ = −k QT , σ = 0, Pα = Pβ = P Q L , Pγ = Pδ = P QT , 12

(10.51b) (10.51c)

Because the normal incidence of a transverse wave can be analyzed equally as the normal incidence of a longitudinal wave, only the longitudinal wave incidence will be discussed below.

320

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.10 Illustration of the perfect mode-converting transmission from one medium to the same medium (or another medium) through an anisotropic matching layer

where

kQL

k QT

   ρω2 (C11 + C66 ) − ρω2 (C11 − C66 )2 + 4C 2  16 , = 2 2(C11 C66 − C16 )    ρω2 (C11 + C66 ) + ρω2 (C11 − C66 )2 + 4C 2  16 , = 2 2(C11 C66 − C16 )

(10.52)

and PQ L = P QT =

(C16 k 2Q L , ρω2 − C11 k 2Q L ) (C16 k 2Q L )2 + (ρω2 − C11 k 2Q L )2 (C16 k 2QT , ρω2 − C11 k 2QT ) (C16 k 2QT )2 + (ρω2 − C11 k 2QT )2

, .

(10.53)

Let us consider the case of perfect L-to-T transmission. In the case of normal incidence, the reflectances (R L , RT ) and transmittances (TL , TT ) become (see Eqs. (9.102a–d))

A 2 A A A 2 C C66

D

B 66 2 2

|r | R L = |r L | = A , RT = = (10.54a,b) T

A , A A A A C11 C11 B B B 2 B B B 2 ρ B C11 ρ B C11 ρ C ρ B C66

C B 66 2

A , TT =

|t L |2 = |t | TL = = T

A . A ρ CA ρ CA A ρ CA ρ CA A A

11

A

11

A

11

A

11

(10.55a,b)

10.4 Perfect Mode-Converting Transmission at Normal Incidence …

321

Equations (10.54a, b) and (10.55a, b) become simpler when media A and B are the same. When we define the parameter ξ in terms of Poisson’s ratio (ν) of the background isotropic medium as 

1 − 2ν for the plane strain condition 2(1 − ν)  1−ν for the plane stress condition ξ= 2

ξ=

(10.56a)

(10.56b)

Equations (10.54a, b) and (10.55a, b) can be simplified as

A 2

A 2

B

D 2

R L = |r L | = A , RT = ξ |r T | = ξ

A

, A A

B 2

B 2

A

C TL = |t L |2 =

A

, TT = ξ |tT |2 = ξ

A

. A A 2

(10.57a)

(10.57b)

In the case of normal incidence, the phase-matching conditions in Eq. (10.40) become     2k Q L d, k Q L − k QT d and k Q L + k QT d : integer multiples of π.

(10.58)

The results in Eq. (10.58) may be interpreted as 2k Q L d and 2k QT d : integer multiples of π.

(10.59)

There are several possible solution sets that satisfy Eq. (10.58) or Eq. (10.59). To select appropriate solution sets, we recall from Sect. 7.6 that the phase condition that an impedance-matching element (medium M) of thickness d should satisfy is cos k M d = 0 (sin k M d = ±1) for the full transmission of a normally incident longitudinal wave across two dissimilar media where k M denotes the wavenumber in the matching layer. Guided by this observation, the following conditions must be satisfied13 :           sin k Q L d = ±1, sin k QT d = ∓1, cos k Q L d = 0, cos k Q S d = 0 . (10.60) (double signs in the same order) Using the condition in Eq. (10.60), the following statements can be made:     If sin k Q L d = ±1and sin k QT d = ±1, no meaningful results were obtained; refer to the supplementary materials in Yang et al. (2018).

13

322

10 Perfect Transmission Across 2D Different Media Using Metamaterials

(i) if we select sin k Q L d = 1 and sin(k QT ) = −1,14 kQL d =

π 5π 9π 3π 7π 11π , , , . . . , k QT d = , , , . . . , with k QT > k Q L , 2 2 2 2 2 2 (10.61a)

(ii) if we select sin k Q L d = −1 and sin(k QT ) = 1, kQL d =

3π 7π 11π 5π 9π 13π , , , . . . ; k QT d = , , , . . . ; with k QT > k Q L . 2 2 2 2 2 2 (10.61b)

Equations (10.61a, b) can be combined as kQL d = m ·

nQL n QT π, k QT d = m · π, (m = 1, 3, 5, . . .) 2 2

(10.62a)

subject to nQL n QT − = odd integers; 2 2 n QT and n Q L (n QT > n Q L ) : coprime integers.

(10.62b)

It may be more convenient to write the conditions in Eqs. (10.62a, b) in terms of wavelengths λ Q L and λ QT using the relations, k Q L λ Q L = 2π and k QT λ QT = 2π as. Generalized phase-matching conditions d = m · nQL

λQ L λ QT and d = m · n QT (m = 1, 3, 5, . . .), 4 4

(10.63a,b)

subject to nQL n QT − = odd integers; n QT > n Q L : coprime integers. 2 2

(10.63c)

Equations (10.63a, b, c) resemble Eq. (7.90) representing the phase-matching. While Eq. (7.90) applies to the case of the mode-preserving transmission (dealing with a single wave mode) across dissimilar media, Eqs. (10.63a, b, c) deals with two modes, i.e., a longitudinal wave mode in one medium and a transverse wave mode in the other medium. In this respect, we can interpret the generalized phase-matching conditions stated by Eqs. (10.63a, b, c) as the bimodal phase-matching conditions. Using Eqs. (10.52), (10.62a, b), one can find the required material properties of the anisotropic layer that satisfies the bimodal phase-matching condition as

14

Without the loss of generality, k QT > k Q L k is assumed.

10.4 Perfect Mode-Converting Transmission at Normal Incidence …

 C11 + C66 = 16ρ f d

2 2

1 n 2Q L

+

1 n 2QT

323

,

(10.64)

16ρ f 2 d 2 2 C11 C66 − C16 = , n Q L n QT

(10.65)

where f denotes the target frequency for the perfect mode-converting (L-to-T) transmission. The remaining conditions are the generalized impedance-matching conditions stated by Eq. (10.46). If θTB = 0 is inserted in Eq. (10.46), it reduces to Z 1L-to-T = Z 0L-to-T , Z 2L-to-T = 0, Z 3L-to-T = 0.

(10.66a,b,c)

To calculate the bimodal impedances Z iL− to - T defined in Eq. (10.47), we place θ LA = θTA = 0 in ϕm defined in Eq. (10.35): ϕm ≡ Tm1 − Z LA Tm3 (m = 1, 2, 3, 4).

(10.67)

Furthermore, certain components of the transfer matrix T become zero: T11 = T12 = T21 = T22 = T33 = T34 = T43 = T44 = 0.

(10.68)

These equations can be obtained by substituting Eqs. (10.51a, b, c) and (10.60) into Eq. (10.15). Substituting Eqs. (10.67) and (10.68) into Eq. (10.47), one can rewrite Z iL-to-T (i = 0, 1, 2, 3) as

(10.69)

j

where Z α (α = L , T , j = A, B) is the mechanical impedance of mode α of j j medium j defined as Z α = ρ j cα . The components T13 , T23 , T41 , and T31 appearing in Eq. (10.69) are calculated as follows (we can obtain these equations by substituting Eqs. (10.51a, b, c) and (10.60) into Eq. (10.15))

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

T13 = T23 = T31 = T41 =

−ω[k QT PxQ L (C16 PxQT + C66 PyQT ) + k Q L PxQT (C16 PxQ L + C66 PyQ L )] 2 k Q L k QT (C11 C66 − C16 )

−ω[k QT PyQ L (C16 PxQT + C66 PyQT ) + k Q L PyQT (C16 PxQ L + C66 PyQ L )] 2 k Q L k QT (C11 C66 − C16 )

, ,

−[k QT PyQ L (C11 PxQT + C16 PyQT ) + k Q L PyQT (C11 PxQ L + C16 PyQ L )] −ω[k QT PyQ L (C16 PxQT

, ω + C66 PyQT ) + k Q L PyQT (C16 PxQ L + C66 PyQ L )] ω

.

(10.70)

Substituting Eq. (10.70) into Eq. (10.69) yields 

 Z 1L-to-T =

T41 = T23

2 k Q L k QT (C11 C66 − C16 ) . 2 ω

(10.71)

2 The term k Q L k QT (C11 C66 − C16 ) appearing in Eq. (10.71) can be expressed as (using Eq. (10.52)): 2 ) k Q L k QT (C11 C66 − C16 # !" 1 2 ρω2 (C11 + C66 ) − ρω2 (C11 − C66 )2 + 4C16 = 2 #$1/2 " 2 2 2 2 · ρω (C11 + C66 ) + ρω (C11 − C66 ) + 4C16

% $2 & 21 ! 2 1  2 2 2 2 = ρω (C11 + C66 ) − ρω (C11 − C66 ) + 4C16 2 2 =ρω2 C11 C66 − C16 .

(10.72)

Substituting Eq. (10.72) into Eq. (10.71) yields  Z 1L-to-T

=

" # 21 2 k Q L k QT (C11 C66 − C16 ) 2 = ρ (C11 C66 − C16 ) . ω2

(10.73)

If we introduce a symbol Z˜ , which is called the bimodal impedance of an anisotropic medium, 2 Z˜ = ρ (C11 C66 − C16 ),

(10.74)

and Z˜ 0 , which may be called the bimodal impedance of the base media, Z˜ 0 = Z LA Z TB =



A B ρA C11 ρB C66 ,

(10.75)

10.4 Perfect Mode-Converting Transmission at Normal Incidence …

325

Eq. (10.66a) can be expressed as Z˜ = Z˜ 0 .

(10.76)

The relation Eq. (10.76) may be referred to as the bimodal impedance-matching condition for the perfect mode-converting (L-to-T) transmission at the normal incidence. Let us consider Eqs. (10.66b, c). Using Eq. (10.69), we observe that Eqs. (10.66b, c) can be identically satisfied if T13 = 0, T31 = 0.

(10.77)

Using T13 and T31 expressed in Eq. (10.70), we can write Eq. (10.77) explicitly as     PxQ L PxQT k Q L + k QT C16 + k Q L PyQ L PxQT + k QT PxQ L PyQT C66 = 0, (10.78a)     PyQ L PyQT k Q L + k QT C16 + k Q L PxQ L PyQT + k QT PyQ L PxQT C11 = 0, (10.78b) where the polarization vectors (P Q L and P QT ) are related to each other as (see Eqs. (9.46) and (9.47)). PxQ L PyQT − PyQ L PxQT = 1, PxQ L PxQT + PyQ L PyQT = 0.

(10.79)

Adding Eqs. (10.78a) and (10.78b) yields PyQ L PxQT = − 

k QT C66 +k Q L C11  , k Q L + k QT C11 + C66

(10.80)

where the relations given by Eq. (10.79) are used. If we use Eq. (10.53), we can also write PyQ L PxQT as C11 − C66 1 PyQ L PxQT =  − . 2 2 2 2 C11 − C66 + 4C16

(10.81)

Ultimately, equating Eqs. (10.80) and (10.81) yields C11 = C66 and C16 = 0. Based on the derivations above, we can establish the following conditions: Generalized impedance-matching conditions 2 ˜ ˜ Z = Z 0 ↔ ρ (C11 C66 − C16 ) = ρ A ρ B Z LA Z TB ,

(10.82a)

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

C11 = C66 ,

(10.82b)

C16 = 0.

(10.82c)

The above analysis reveals that the full mode-converting transmission of a normally incident longitudinal wave requires an anisotropic matching layer having specific material properties satisfying Eqs. (10.64), (10.65), and (10.82a, b, c). However, there is no possible solution of (ρ, C11 , C66 , C16 ) that can simultaneously satisfy these five equations, because the number (four) of the adjustable material parameters is smaller than the number (five) of the matching conditions. Accordingly, at least one condition must be relaxed to obtain nontrivial material properties. Thus, we relax Eq. (10.82c), allowing nonzero C 16 . If C16 , which expresses the coupling between longitudinal and shear deformations, identically vanishes, mode conversion from a longitudinal to a transverse wave, or vice versa, cannot be possible. Therefore, the matching element should have weak longitudinal-shear coupling (small but nonzero C16 ). If only four equations, Eqs. (10.64), (10.65), (10.82a, b), are used to determine (ρ, C11 , C66 , C16 ),15 the following results can be obtained:  ρ=

4fd

C11 =C66 = C16 =

Z LA Z TB n Q L n QT

Z LA Z TB " n QT 2ρ

nQL

Z LA Z TB " n QT 2ρ

nQL

+

, # nQL , n QT

# nQL . − n QT

(10.83a)

(10.83b, c)

(10.83d)

Because Eq. (10.82c) is not explicitly imposed, C16 will not be exactly zero. However, Eq. (10.83d) shows that Eq. (10.82c) can be asymptotically satisfied if n QL and n QT approach ∞. Because n QL and n QT can be somewhat arbitrarily selected as long as they satisfy Eq. (10.62b), one can quantify the degree of the relaxation of the condition in Eq. (10.82c) using a parameter  defined as =

n 2QT − n 2Q L 2C16 = 2 . C11 + C66 n QT + n 2Q L

The smaller  is, the better the condition in Eq. (10.82c) is satisfied. 15

In the case of normal incidence, C22 can be arbitrary.

(10.84)

10.4 Perfect Mode-Converting Transmission at Normal Incidence …

327

Case Study 1 As a specific example of perfect mode-converting transmission, we consider the following case (refer to Fig. (10.10)): • media A and B: aluminum (ρ = 2700 kg/m3 , ν = 0.33, E = 71 GPa), • incident mode from medium A: longitudinal, • transmitted mode into medium B: transverse, • target frequency f = f T = 100 kHz, • size of the mode-converting matching layer: d = 10 cm.

We may select m = 1, n Q L = 17, and n QT = 19 ( = 0.11) satisfying Eq. (10.63c) and solve Eqs. (10.64), (10.65), (10.82a, b) to find ρ, C11 , C66 , and C16 of the anisotropic full mode-converting matching layer. The results are as follows: ρ = 5014 kg/m3 , C11 = C66 = 24.99 GPa, C16 = 2.77 GPa.

(10.85)

When the anisotropic matching layer having the material property given by Eq. (10.85) is inserted into a base aluminum medium, the resulting transmittances through the matching layer can be found as those plotted in Fig. 10.11a; the transmittance TT from the longitudinal to transverse wave is nearly unity, confirming the realization of full mode-converting transmission. Figure 10.11b shows the displacements in the aluminum medium on the left and right sides which are dominated by the horizontal (u x ) and vertical (u y ) displacements, respectively, whereas the displacement in the matching layer has both horizontal and vertical displacements. Figure 10.11c shows how the displacement inside the anisotropic matching layer is decomposed into its two wave modes, such as the QL and QT modes. Referring to Eq. (10.63a, b), the wavelengths λ Q L and λ QT are related to the matching layer size d (for m = 1, n Q L = 17 and n QT = 19) as # # " " 3 1 λQ L , d = 4 + λ QT . d = 4+ 4 4 From Fig. 10.11c, one can see that the displacement inside the anisotropic matching layer is decomposed by two wave modes which are 17 λ Q L /4 and 19λ QT /4 long, respectively; the matching layer size must be multiples of quarter waves of the quasi-longitudinal and quasi-transverse waves, but their multiple integers should satisfy the relation stated by Eq. (10.63c). This figure also shows how the incident longitudinal wave governed by the horizontal displacement (u x ) at the entrance of the matching layer is perfectly mode-converted to the transmitted transverse wave governed by the vertical displacement (u y ) at its exit. Case Study 2 This case considers the mode-converting transmission between two different media, as stated below. • medium A: aluminum (ρ = 2700 kg/m3 , E = 71 GPa, ν = 0.33), • medium B: lead (ρ = 11340 kg/m3 , E = 16 GPa, ν = 0.444),

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10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.11 Simulation results when an incident longitudinal wave is converted to a transverse wave in aluminum via a full L-to-T mode-converting matching layer having the material properties of Eq. (10.85) at 100 kHz. a Transmittances, b displacement fields in the base medium and matching element, and c the displacement inside the matching layer decomposed into two modes (QL and QT modes) • incident mode from medium A: longitudinal, • transmitted mode into medium B: transverse, • target frequency f = f T = 100 kHz, • size of the mode-converting matching layer: d = 10 cm.

We may select m = 1, n Q L = 17, and n QT = 19 ( = 0.11) satisfying Eq. (10.63c) and solve Eqs. (10.64), (10.65), (10.82a, b) to find ρ, C11 , C66 , and C16 of the anisotropic full mode-converting matching layer. The results are as follows: ρ = 4831 kg/m3 , C11 = C66 = 24.10 GPa, C16 = 2.67 GPa.

(10.86)

When the anisotropic matching layer having the material property given by Eq. (10.86) is inserted into a base aluminum medium, the results shown in Fig. 10.12

10.4 Perfect Mode-Converting Transmission at Normal Incidence …

329

Fig. 10.12 Simulation results when an incident longitudinal wave from aluminum is converted to a transverse wave in lead via a full L-to-T mode-converting anisotropic matching layer having the material properties of Eq. (10.86) at 100 kHz. a Displacement fields and b transmittances in the base medium and matching element

can be obtained. As in Case Study 1, Fig. 10.12 shows that nearly full moveconverting transmission occurs across two different media if the designed anisotropic metamaterial is used. Case Study 3 In this case, we consider the actual realization of the matching layer using a metamaterial (Yang and Kim 2018). The data for this problem are given as follows: • media A and B: aluminum (ρ = 2700 kg/m3 , ν = 0.33, E = 71 GPa), • incident mode from medium A: longitudinal, • transmitted mode into medium B: transverse, • target frequency f = f T = 169.9 kHz, • size of the mode-converting matching layer: d = 10 cm, • base medium of the matching layer: steel (ρFe = 7900 kg/m3 , νFe = 0.286, E Fe = 213 GPa), • the size l of the square unit cell: l = d/300.

To facilitate the metamaterial unit cell design, we can directly design the unit cell that realizes the full mode-converting transmission by a parametric study or an optimization method. The design procedure with the selected parameters of n Q L = 25 and n QT = 27 is presented in Yang and Kim (2018). The designed single-phase square unit cell in a steel base medium is shown in Fig. 10.13a along with its geometric data and the effective material properties estimated by the homogenization method16 (Guedes and Kikuchi 1990). The estimated effective materials can be shown to satisfy all of the required conditions, Eqs. (10.64), (10.65), and (10.82a, b, c) only with small errors. For instance, Z˜ = 0.992 Z˜ 0 can be found, confirming that Eq. (10.82a) is nearly satisfied. Figure 10.13a also shows that the u x -dominant incident longitudinal 16

Alternatively, the material properties can be retrieved by the method developed by Lee et al. (2016).

330

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.13 Full mode-converting transmission for Case Study 2. a The unit cell configuration, the displacement fields in the base medium and the anisotropic metamaterial matching layer. The 2D plane-strain assumption is used

wave field is converted to the u y -dominant transmitted transverse wave field. For the conversion, both u x and u y components inside the metamaterial contribute to the mode-conversion. The conversion transmission efficiency at the target frequency ( f T ) is nearly perfect as evidenced by TT = 0.99 shown in Fig. 10.13b. The magnitude of the input displacement |u x | ≈ 1 is converted to |u y | ≈ 1.407.17

10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter As a practical application of the perfect wave mode conversion, we consider a noninvasive nonintrusive ultrasonic flowmeter consisting of a pair of ultrasonic wedge transducers shown in Fig. 10.14. This is a device to measure the velocity of a fluid flowing inside a pipe without direct contact with the fluid and without pipe drilling in a pitch-catch mode.18 As illustrated in Fig. 10.15, noninvasive methods are preferred Referring to Eq. (10.55b), one can see that |C B |2 , the magnitude of u y in medium B, should A )|A A |2 / (ρ C B ) for T = 1 where |A A | = |u | in medium A. Because be equal to (ρ A C11 B 66 T x

A )/ (ρ C B ) > 1, in this case, u in medium B should be larger than |A A | = |u | in (ρ A C11 B 66 y x medium A. 18 The pitch-catch mode refers to a measurement mode using two transducers where one transducer transmits while the other receives the ultrasonic signal. 17

10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter

331

Fig. 10.14 Illustration of the conventional noninvasive and nonintrusive ultrasonic flowmeter. The difference in the upstream and downstream travel times of an ultrasonic wave is used to estimate the flow speed (Piao et al. 2020)

because they do not interfere with fluid flow. Nonintrusive methods are preferred because they do not require the contact of transducers with a fluid. The key elements in the flowmeter transducers in Fig. 10.14 are piezoelectric elements and wedges. Here, we consider the piezoelectric elements made of PZT19 and the wedges made of PEEK. As depicted in Fig. 10.14, two transducers are installed on the outer wall surface of a pipe inside which a fluid flows at a mean velocity of V f . We will assume that the pipe is made of steel. Before carrying out the detailed wave analysis, the principle of the flow velocity measurement will be briefly explained. If the velocity of the fluid flowing inside the pipe is denoted by V f , one can show that V f is given by (see, e.g., Dell’Isola, et al. 1997) Vf =

TB−A − T A−B D , 2 sin α2 cos α2 TB−A T A−B

(10.87)

where D: inner diameter of the pipe, α2 : refraction angle of the longitudinal wave L 2 in the fluid, T A−B (TB−A ) : traveling time of the longitudinal wave L 2 in the fluid from Wedge A to Wedge B (Wedge B to Wedge A). It should be noted that the mode of the wave in the pipe entering the fluid (see Fig. 10.14) should be a transverse wave mode (T1 ). In the conventional transducer configuration shown in Fig. 10.14, therefore, the incident angle α0 from the wedge 19

It stands for piezoelectric ceramic material based on lead (Pb) Zirconate Titanate.

332

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.15 Comparison of the noninvasive and nonintrusive method and other methods

to the pipe should be selected to be larger than α0cr to ensure that only a transverse wave can be transmitted to the pipe. The critical angle α0cr can be calculated from Eq. (9.68) as   = sin−1 (2612/5860) = 26.5◦ . /csteel α0cr = sin−1 cPEEK L L If α0 < α0cr , there would also exist a transmitted longitudinal wave L 1 of the transmitted angle of α1 > β1 , as shown in Fig. 10.16. In this instance, both longitudinal and transverse waves are transmitted to the pipe, which are then converted to two longitudinal waves in the fluid. The formation of multiple waves under the incidence of a single wave from a transducer makes subsequent flow velocity estimation difficult. Therefore, the condition of α0 > α0cr should be satisfied. To look further into the wave transmission in the conventional flowmeter in Fig. 10.14, we consider the wave phenomena occurring at the interface between the PEEK wedge and the steel pipe shown in Fig. 10.16a. A longitudinal wave (L 0 ) is generated by a PZT element and transmitted through a PEEK wedge having the wedge angle of α0 . As the longitudinal wave is normally incident on an isotropic material, PEEK, the wave mode in PEEK is longitudinal. The incidence angle of a longitudinal wave from the PEEK wedge to the steel pipe is the same as the wedge angle α0 . Assuming that α0 > α0cr , only a single transverse wave T1 is generated in

10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter

333

Fig. 10.16 Wave transmission across dissimilar media. a Longitudinal wave incidence fromPEEK  to steel (α0cr = 26.5o ), and b transverse wave incidence from aluminum to steel β0cr = 32.2o (Piao et al. 2020)

the steel pipe. The transmitted angle of T1 is denoted by β1 . The transverse wave T1 in the pipe is then converted to a longitudinal wave L 2 in the fluid. Because fluid cannot carry transverse waves, only longitudinal waves can be transmitted to the fluid. The converted L 2 wave travels through the fluid and reaches the PZT element of the receiving transducer by the reversed transmission process discussed for the transmitting transducer. For an arbitrary angle α0 , Fig. 10.16a examines how the transmitted angles α1 of the longitudinal wave and β1 of the transverse wave in the steel pipe will vary as a function of the incidence angle α0 . When α0 ≥ α0cr = 26.5◦ , α1 = 90◦ , indicating that there is no propagating longitudinal wave in the transmitted wave field in the steel pipe. The transmitted power of the transverse wave (TL−T ) relative to the power of the incident longitudinal wave from the PZT element is also shown in the figure. Here, we used the subscript L − T to indicate the mode conversion from a longitudinal wave in the PEEK wedge to a transverse wave in the steel pipe. To calculate the

334

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.17 Comparison of the methods to transmit a transverse wave in a steel pipe using a longitudinal wave generated by the PZT element. a Critical-angle-based conventional method, and b a method using a metamaterial slab realizing the full L-to-T mode-converting transmission. The base material of the wedge is aluminum (Piao et al. 2020)

angle β1 of the transmitted transverse wave in the steel pipe and the angle α2 of the transmitted longitudinal wave in the fluid for an incident longitudinal wave L 0 from the PEEK wedge (incidence angle: α0 ), we use Snell’s law: sin α0 sin β1 sin α2 = = . c L |PEEK cT |steel c L |fluid

(10.88)

A critical issue in this conventional ultrasonic flowmeter is that the efficiency of the conversion transmission of the incident longitudinal wave in the PEEK wedge to the transmitted transverse wave in the steel pipe is very low—it is only 28% when ◦ α0 = 31 , as depicted in Fig. 10.16a. The low transmission efficiency is primarily caused by the high impedance mismatch between PEEK and steel: Z Lsteel /Z TPEEK = 32.6; however, the full-power conversion of a longitudinal wave incident from the PEEK wedge solely to a transverse wave20 in the steel pipe is impossible by any means due to the large impedance mismatch and different mode types. Because the incident wave is obliquely incident and the mode types before and after transmission differ, the impedance-matching method discussed in Sect. 7.6 cannot be used in this case. To overcome the low-efficiency issue of the conventional flowmeter, we can employ the principle of perfect L-to-T transmission given in Sect. 10.4. Figure 10.17b shows a new ultrasonic flowmeter employing the mode-converting metamaterial layer. It is compared with the conventional flowmeter illustrated in Fig. 10.17a. The detailed configuration of the metamaterial matching layer made of a single unit cell is illustrated in Fig. 10.18a. It also depicts the simulation model for finite element analysis. Figure 10.18b shows how the metamaterial slab is embedded in the aluminum wedge. To denote the wave transmission from an incident transverse wave (T0 ) from the wedge to the transverse wave (T1 ) in the pipe, we will use the subscript T − T . 20

One may consider the use of a PZT element directly generating a transverse wave to avoid transmission loss due to the conversion, but this approach is not preferred in practice because of its high cost and low-power output.

10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter

335

Fig. 10.18 a Monolayer metamaterial layer (or slab) realizing perfect L-to-T mode-converting transmission (L: longitudinal, T: transverse). Symbols a, b, etc., denote the design parameters used to shape voids. The base material is aluminum. b Metaslab embedded in an aluminum-based wedge and illustration of wave mode change (from L 0 to T 0 ) along the wave passage (Piao et al. 2020)

336

10 Perfect Transmission Across 2D Different Media Using Metamaterials

In the suggested configuration in Fig. 10.18b, the longitudinal wave generated by a PZT element is normally incident to the aluminum wedge. Then, the transmitted longitudinal wave (L 0 ) in the aluminum wedge is perfectly converted to a transverse wave (T0 ) when it passes through the mode-converting matching slab made of an anisotropic metamaterial, in short, metaslab, embedded in the aluminum wedge. The metaslab is so designed to realize perfect L-to-T mode-converting transmission at the target frequency. In the present case, only the wave mode is converted within the same medium; the media carrying the longitudinal and transverse waves are the same. Once the T0 wave is formed right after the metaslab, it can be nearly fully transmitted as a transmitted wave (T1 ) into the steel pipe without generating a transmitted or reflected longitudinal wave. The main reasons to use aluminum as the wedge base material are that (1) the power transmission from the aluminum wedge to the steel pipe is nearly full ◦ (97.5%) at the selected value of the aluminum wedge angle β0 = 34 because the characteristic impedance of aluminum is nearly equal to that of steel, and (2) it is relatively easy to machine voids needed to make the metaslab. Note that wave into the steel pipe is larger than the incident angle β0 of the  transverse −1 = sin 32.2o . Therefore, the transverse β0cr = sin−1 cTaluminum /csteel (3120/5860)= L wave T0 incident from the aluminum wedge is solely transmitted to the transverse wave T1 without being accompanied by a longitudinal wave in the steel pipe. To minimize any adverse effects from manufacturing errors, the actual wedge angle β0 was chosen to be 35◦ , which is slightly larger than the selected value of 34◦ at which the maximum power transmission to the steel pipe occurs; if β0 is somewhat smaller than 34◦ , the transmission efficiency can significantly drop as can be seen in Fig. 10.16b. As a specific case, we consider the target frequency f T = 440 kHz. The effective material properties of the metaslab should be anisotropic satisfying the conditions stated by Eqs. (10.83a, b, c, d). Instead of finding the desired anisotropic material properties realizing the perfect mode-converting transmission at f T = 440 kHz, it is easier to directly configure the metaslab that satisfies the conditions in Eq. (10.2) (for the case of L-to-T transmission), as done in Sect. 10.3.3. For the metaslab design, we use the wave analysis model shown in Fig. 10.18a where the selected topology of the metaslab is shown. The void configuration was based on the study in Yang and Kim (2018). A parameter study was performed to determine the desired parameter values using the model in Fig. 10.18a where a plane harmonic longitudinal wave at a target frequency is set to be incident from the left, and the corresponding wave field everywhere is calculated. After some trials and errors, a set of parameter values (nearly) satisfying the conditions stated by Eq. (10.2) are determined. The followings are the results: target frequency: f T = 440 kHz a = 6 mm, r1 = 0.29 mm, r2 = 0.29 mm, ◦

l1 = 0.96 mm, l2 = 5.70 mm, θ = 55.5

(10.89)

10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter

337

The power transmission ratio TL−T from the incident longitudinal wave to the transmitted transverse wave through the designed metaslab is plotted in Fig. 10.19a. It shows that TL−T at the target frequency ( f T = 440 kHz) is nearly 100%. For the simulations, COMSOL Multiphysics software was used. Figure 10.19b shows the snapshots of the displacement field (u = {u x , u y }T ) where u x and u y denote the longitudinal and transverse displacements at some frequencies near or at f = f T = 440 kHz. These plots clearly demonstrate nearly full-power mode conversion from a longitudinal wave to a transverse wave at the target frequency. It is apparent from Fig. 10.19a that nearly full L-to-T mode-converting transmission occurs when the designed metamaterial is used. We also check if the generalized phase-matching conditions, Eq. (10.63a, b), and the generalized impedance-matching

Fig. 10.19 Finite element simulation results for the metaslab designed for the perfect L-to-T transmission at 440 kHz. a TL-to-T as the function of frequency, and b the divergence (∇·u) and curl (∇ × u) fields (Piao et al. 2020)

338

10 Perfect Transmission Across 2D Different Media Using Metamaterials

conditions, Eq. (10.82a, b, c), are satisfied for the desired full L-to-T mode-converting transmission. To this end, we should estimate the effective material properties of the monolayer metaslab in Fig. 10.18a. Using the estimation scheme given by Yang et al. (2019) and Lee et al. (2016) for the designed metamaterial in Fig. 10.18a with the geometric values given in Eq. (10.89), we can estimate d,21 and its effective material properties ρ, and Ci j of the effective anisotropic material corresponding at f = 440 kHz as22 : d = 11.46 mm, ρ = 13, 144 kg/m3 , C11 = 11.066 GPa, C66 = 11.147 GPa, C16 = 1.028 GPa.

(10.90)

al = 103.72 GPa, Compared with the base aluminum plate (ρ al = 2700 kg/m3 , C11 al = 26.32 GPa, C16 = 0), the metamaterial has considerably different density and stiffness values. Also, it is anisotropic because it has a nonzero longitudinal-shear coupling coefficient C16 . To check if the generalized phase-matching condition in Eq. (10.63a, b) is satisfied, we use the estimated values given in Eq. (10.90) to calculate λ Q L = 2π/k Q L and λ QT = 2π/k QT from Eq. (10.52) and estimate n QL and n QT by solving Eqs. (10.64) and (10.65) as

al C66

λ Q L ≈ 2.18 mm and λ QT ≈ 1.99 mm at f = 440 kHz,

(10.91)

n Q L ≈ 21 and n QT ≈ 23.

(10.92)

The values in Eqs. (10.91) and (10.92) satisfy the phase-matching condition stated by Eqs. (10.63a, b, c) (with m = 1). To see if the generalized impedance-matching condition in Eq. (10.82a, b, c) is satisfied, we calculate impedances Z˜ in Eq. (10.74) and Z˜ 0 in Eq. (10.74) where A and B denote aluminum, Z˜ = 1.45 × 105 Mrayl; Z˜ 0 = 1.41 × 105 Mrayl. Because Z˜ / Z˜ 0 ≈ 1.03, the condition in Eq. (10.82a) is nearly satisfied. We can also obtain C16 C66 = 1.007, = 0.092. C11 C11

21

Note that because the metaslab consists only of a single unit cell, the effective width d cannot be known in advance. Therefore, it should be estimated. Please see Yang et al. (2019) for the details of the estimation scheme. 22 The effective density and stiffness may not be uniquely determined for a given metamaterial unit cell when estimated from the scattering matrix considering only the normal incidence. If one also considers an oblique incidence in addition to the normal incident using the retrieval method of Lee et al. (2016), they can be uniquely determined.

10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter (a)

339

(b)

Fig. 10.20 a Flow velocity measurement experimental setup, and b zoomed view of the installed flowmeter using the designed metaslab wedge shown in Fig. 10.18a (f T = 440 kHz) (Piao el al. 2020)

This result shows that Eq. (10.82b) is nearly satisfied, and the violation of condition in Eq. (10.82c) is marginal. Therefore, it is concluded that the designed monolayer metamaterial slab shown in Fig. 10.18 fulfills the necessary conditions for nearly full L-to-T mode-converting transmission. The actual application of the designed metaslab wave for flow velocity measurement is shown in Fig. 10.20. The dimension of the inner side of the square pipe is 60 mm, and its wall thickness is 3 mm. The pipe is filled with water circulated by a fan installed inside the pipe, and the fan is driven by a pump. Two sets of experiments were performed using the conventional PEEK wedge-based transducers and the metaslab wedge-based transducers for comparison. The flow velocity measured by a laser-based method was also compared with that measured by the ultrasound method using the metaslab wedge transducer. For the experiments, 20 cycles of a sine wave were used. Figure 10.21 shows the multiple paths used for the experiment depicted in Fig. 10.20. The use of a single wave path, as illustrated in Fig. 10.14, is not effective when the diameter of the pipe is small because the propagation time difference between the upstream and the downstream wave propagations for a single path may be only tens of nanoseconds. To overcome this difficulty, a multireflection V-path measurement scheme illustrated in Fig. 10.21 was adopted. Then, the traveling time difference between the upstream and the downstream wave propagations can be considerably increased due to multiple reflections inside the fluid. For the arrangement of the transducers in Fig. 10.21, the traveling path S of an ultrasonic wave becomes S = 2S1 + 2S2 + 6S3 .

340

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Fig. 10.21 Pair of flowmeter transducers installed on the pipe wall. A multi-reflection V-path was used to distinguish t B-A and t A-B better (see Table 10.1 for the geometric data) (Piao et al. 2020)

Table. 10.1 Geometric data for the flowmeter configuration in Fig. 10.21 Wedge

h(mm)

D(mm)

S1 (mm)

S2 (mm)

S3 (mm)

α0 (deg)

β1 (deg)

α2 (deg)

PEEK

3

60

31.9

2.9

19.8

35

43.4

18.3

39.8

2.2

16.9

35

36.2

15.7

Metaslab

Figure 10.22 compares the measured signals obtained using (a) a pair of the conventional PEEK wedge transducers and (b) the metaslab wedge transducers, respectively. The experiments were performed using 20 cycles of a sine wave at the frequency of f M = 420 kHz. Note that the metaslab was designed for f T = 440 kHz based on a 2D model. When the met-slab wedge was fabricated as a three-dimensional object, however, its frequency characteristics were slightly affected by some dimensional effect. The transmission efficiency with the fabricated metaslab dropped only a few percentages at f T = 440 kHz, but the experiments were performed at f M = 420 kHz for which the efficiency was found to be maximized. Figure 10.22 shows that the measured signal with the metaslab wedge is nearly 7 times larger in magnitude than that with the conventional PEEK wedge.23 This super-enhanced capability

23

The magnitudes of the recorded ultrasonic waves are given in mV, and it should be noted that the magnitude range in Fig. 10.22b is 20 times that in Fig. 10.22a.

10.5 Application of Perfect L-to-T Transmission in Ultrasonic Flowmeter

341

of the metaslab wedge can be extremely useful in pipes of large diameters and in making more compact flowmeters. In Fig. 10.22, t denotes the arrival-time difference between the two measured signals for the downstream and upstream wave propagations. Referring to Eq. (10.87), t corresponds to TB−A − T A−B . Table 10.2 compares the experimentally estimated values of TB−A and T A−B with the theoretical calculations. Certainly, the traveling time for the downstream propagation should be shorter than that for the upstream propagation as clearly shown in Table 10.2. In the calculation of TB−A and T A−B , only the traveling times in the fluid were used. The table also lists the calculated flow velocity V f using Eq. (10.87). The estimated flow velocity (1.29 m/s) by the metaslab wedge transducer is nearly equal to that (1.31 m/s) by the conventional wedge transducer. These two results favorably agree with the flow velocity

META

META

PEEK

PEEK

pipe

pipe

fluid

fluid

10 0 -10 -20 310

315

320

325

330

Time (μs)

20

Down stream Up stream

=1.26 m/s

35.6 mV

10 0 -10 -20 310

315

320

325

330

Time (μs)

20

Down stream Up stream

=5 m/s

37.1 mV

10 0 -10 -20 310

315

320

Time (μs)

(a)

325

330

Output voltage (mV)

=0 m/s

Output voltage (mV)

20

pipe

Output voltage (mV)

Output voltage (mV)

Output voltage (mV)

Output voltage (mV)

pipe

=0 m/s 400 200 0 -200 -400 300

305

310

315

320

Time (μs) Down stream Up stream

=1.26 m/s 400 200 0 -200

428 mV

-400 300

305

310

315

320

Time (μs)

400

Down stream Up stream

=5 m/s

200 0 -200

428 mV

-400 300

305

310

315

320

Time (μs)

(b)

Fig. 10.22 Signals obtained for different flow velocities (vf ) using a the conventional PEEK-base wedge transducer, and b the metaslab wedge transducer (Piao et al. 2020)

342

10 Perfect Transmission Across 2D Different Media Using Metamaterials

Table. 10.2 Traveling duration and estimated flow velocities by various approaches Wedge type

Traveling time through multireflection V-path (μs) T A−B

V f (m/s)

TB−A

Analytical

Experiment

Analytical

Experiment

Laser* method

Ultrasonic transducer

PEEK

256.05

261.85

256.38

262.00

1.26

1.31

Metaslab

252.43

265.32

252.71

265.50

1.29

* As depicted in Fig. 10.20, a laser-based method was used to measure flow velocity for comparison

(1.26 m/s) measured by the laser-based method. The experimental results presented in Fig. 10.22 and Table 10.2 confirm that the proposed metaslab wedge transducer performs equally as well as the conventional wedge transducer in estimating flow velocity. More importantly, the signal output can be substantially increased if the metaslab wedge transducers are employed as the ultrasonic transducers of a noninvasive ultrasonic flowmeter. This flowmeter application shows that the full modeconverting transmission realized by an anisotropic metamaterial slab can be critically advantageous in practical engineering applications.

10.6 Problem Set In Problems 10.1 to 10.5, consider a situation in which an anisotropic solid layer is inserted between different isotropic solids A and B. The wave speeds of longitudinal (L) and transverse (T) waves in isotropic solids A and B are denoted by c LA , cTA and c LB , cTB , respectively.

10.6 Problem Set

343

Problem 10.1 For an obliquely incident longitudinal wave in solid A, explain why it is impossible to fully transmit it into a longitudinal wave in solid B using the quarter-wave impedance-matching layer established in Sect. 7.6. Problem 10.2 Assume that a general wave vector of an incident longitudinal wave from solid A is given by k = k LA n = k LA (n x , n y ). (a) What are the wave modes in each of isotropic solid A, isotropic solid B, and the anisotropic solid layer? Indicate the mode type and propagation direction. (b) What are the wave vectors of all wave modes considered in (a) for isotropic solids A and B? (c) Calculate all possible critical angles for (i) c LA > c LB , (ii) c LB > c LA > cTB , and cTB > c LA . Problem 10.3 Assume that a general wave vector of an incident transverse wave from solid A is given by k = k TA n = k TA (n x , n y ). (a) What are the wave modes in each of isotropic solid A, isotropic solid B, and the anisotropic solid layer? Indicate the mode type and propagation direction. (b) What are the wave vectors of all wave modes considered in (a) for isotropic solids A and B?

344

10 Perfect Transmission Across 2D Different Media Using Metamaterials

(c) Calculate all possible critical angles for (i) cTA > c LB , (ii) c LB > cTA > cTB , and cTB > cTA . Problem 10.4 A transverse wave is obliquely incident from the isotropic solid A. Find the expressions for the reflection and transmission coefficients of longitudinal and transverse waves using Eq. (10.26). Express your results in terms of Si j , k LB,x , B and k T,x . Problem 10.5 Equation (10.30) represents the generalized phase-matching condition for the case of longitudinal wave incidence. Show that the generalized phasematching condition for the case of transverse wave incidence is also exactly the same as Eq. (10.30).

References COMSOL Multiphysics®, www.comsol.com. COMSOL AB, Stockholm, Sweden Dell’Isola M, Cannizzo M, Diritti M (1997) Measurement of high-pressure natural gas flow using ultrasonic flowmeters. Measurement 20:75–89 Guedes J, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83:143–198 Kim YY, Kwon YE (2015) Review of magnetostrictive patch transducers and applications in ultrasonic nondestructive testing of waveguides. Ultrasonics 62:3-19 Kweun JM, Lee HJ, Oh JH, Seung HM, Kim YY (2017) Transmodal Fabry-Pérot resonance: theory and realization with elastic metamaterials. Phys Rev Lett 118:205901 Lee J, Kweun M, Lee W, Park CI, Kim YY (2022a) Perfect transmission of elastic waves obliquely incident at solid–solid interfaces. Extreme Mech Lett 51:101606 Lee J, Park J, Park CW, Cho SH, Kim YY (2022b) Uni-modal retroreflection in multi-modal elastic wave fields. Int J Mech Sci 232:107655 Lee HJ, Lee HS, Ma PS, Kim YY (2016) Effective material parameter retrieval of anisotropic elastic metamaterials with inherent nonlocality. J Appl Phys 120:104902 Piao C, Yang X, Kweun JM, Kim H, Park H, Cho SH, Kim YY (2020) Ultrasonic flow measurement using a high-efficiency longitudinal-to-shear wave mode-converting meta-slab wedge. Sens Actuators, A 310:112080 Yang X, Kweun JM, Kim YY (2018) Theory for perfect transmodal Fabry-Perot interferometer. Sci Rep 8:69 Yang X, Kim YY (2018) Asymptotic theory of bimodal quarter-wave impedance matching for full mode-converting transmission. Phys Rev B 98:144110 Yang X, Kweun M, Kim YY (2019) Monolayer metamaterial for full mode-converting transmission of elastic waves. Appl Phys Lett 115:071901 Zheng M, Liu XN, Chen H, Miao H, Zhu R, Hu GK (2019) Theory and realization of nonresonant anisotropic singly polarized solids carrying only shear waves. Phys Rev Appl 12:014027 Zhu R, Chen YY, Wang YS, Hu GK, Huang GL (2016) A single-phase elastic hyperbolic metamaterial with anisotropic mass density. J Acoustical Soc Am 139:3303–3310 Zhu R, Liu XN, Huang GH, Huang HH, Sun CT (2012) Microstructural design and experimental validation of elastic metamaterial plates with anisotropic mass density. Phys Rev B 86:144307

Appendix

Solutions to Problems in Chap. 2 Problem 2.1 (a) Since the springs are connected in parallel, the equivalent stiffness of the given system is given by seq = s + s + · · · + s = N · s. p

Therefore, the natural frequency of this system ω0 becomes  ∴

p ω0

=

√ N ·s = N ω0 . m

Problem 2.1 (b) For the springs connected in series, the equivalent stiffness of the given system is calculated as seq =

1 s 1 = = . 1/s + 1/s + · · · + 1/s N /s N

Therefore, the natural frequency of this system ω0s becomes  ∴

ω0s

=

1 s/N = √ ω0 . m N

Problem 2.2 From the two initial conditions, we have © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim, Elastic Waves and Metamaterials: The Fundamentals, https://doi.org/10.1007/978-981-99-0205-7

345

346

Appendix

u(0) = A cos φ −

F sin  = 0, ωZ

(1)

u(0) ˙ = u(t)| ˙ t=0   F = −Aβe−βt cos(ωd t + φ) − Ae−βt ωd sin(ωd t + φ) + cos(ωt − ) Z t=0 F = −Aβ cos φ − Aωd sin φ + cos  = 0. (2) Z Equation (1) can be rewritten as A=

F sin  . ωZ cos φ

(3)

If Eq. (3) is substituted into (2), Eq. (2) becomes F F sin  cos  = (β cos φ + ωd sin φ). Z ωZ cos φ

(4)

The simplification of Eq. (4) becomes 1=

1 (β + ωd tan φ) tan . ω

(5)

One can find φ from Eq. (5). If β 1) and l (= ωl /ω0 < 1), we can find

348

Appendix

β u = + ω0



β + l = − ω0

β ω0



2

β ω0

+ 1,

(4)

2 + 1.

(5)

Using Eqs. (4) and (5), we can write the quality factor Q as Q=

ω0 1 = ωu − ωl u − l

= =



(β/ω0 ) +

1    (β/ω0 )2 + 1 − −(β/ω0 ) + (β/ω0 )2 + 1 

ω0 m ω0 = 2β c

(6)

(b) If we write β as  √ 2ζ ms c s = =ζ = ω0 ζ. β 2m 2m m

(8)

Q can be expressed as Q=

ω0 1 ω0 = = . 2β 2ω0 ζ 2ζ

(9)

Problem 2.5 Recall from Eqs. (2.69) and (2.70) and that U1 and U2 are given by 1 − 2 U1 , = 4 static  − (2 + rm ) 2 + 1 U1

(1)

U2 1 . = 4 static  − (2 + rm )2 + 1 U1

(2)

To find  satisfying U1 /U1static = 1, we set that the right-hand side of Eq. (1) be equal to 1. Then, the following equation is found: 4 − (1 + rm ) 2 = 0. Nontrivial solutions to Eq. (3) are

(3)

Appendix

349

=



1 + rm =

√

1 + 0.3 =

√

1.3 ≈ 1.140.

(3)

At this frequency, we can find U2 using Eq. (2) as U2 = −3.333. U1static

(9)

(The amplitude of mass 2 is nearly three times larger than that of mass 1). (a) To find  satisfying U1 /U1static = −1, we use Eq. (1) to find

4 − (3 + rm ) 2 + 2 = 4 − 3.3 2 + 2 = 0.

(10)

The nontrivial solutions to Eq. (10) are  = 0.894 or 1.581.

(11)

U1 U2 1 U2 = static · = −1 · ≈ −5 or 0.667. static U1 1 − 2 U1 U1

(12)

At these frequencies,

(In these cases, the amplitude of mass 2 can be larger or smaller than that of mass 1 depending on the selected frequency.) Problem 2.6 (a) False. (The mechanical impedance is the relation between the force and the velocity.) (b) True. (c) True. Problem 2.7 (a) The equations of motion can be written as

m 1 u¨ 1 = −s1 u 1 − s2 (u 1 − u 2 ) − c(u˙ 1 − u˙ 2 ) + f,

(1)

m 2 u¨ 2 = −s2 (u 2 − u 1 ) − c(u˙ 2 − u˙ 1 ).

(2)

Assuming that u 1 = U1 eiωt , u 2 = U2 eiωt , and f = Feiωt , Eqs. (1) and (2) become

350

Appendix

(s1 + s2 + iωc − m 1 ω2 )U1 = (s2 + iωc)U2 + F.

(3)

(s2 − m 2 ω2 + iωc) U2 = (s2 + iωc)U1 .

(4)

Solving Eq. (4) for U2 , U2 =

s2 + iωc U1 . s2 − m 2 ω2 + iωc

(5)

Substitute Eq. (5) into Eq. (3), U1 =

F (s1 − m 1 ω2 ) − (s2 + iωc) s2 −mm22ωω2 +iωc 2

.

(6)

Substitute Eq. (6) into Eq. (5) yields U2 =

F (s1 −m 1 ω2 )(s2 −m 2 ω2 +iωc) s2 +iωc

− m 2 ω2

.

(7)

(b) We write Eq. (6) as  −1   

U1 m 2 ω2 s2 iωc 2 = 1− − + , F/s1 s1 s1 s2 − m 2 ω2 + iωc

(8)

where  = ω/ω2 . Using the following parameters, s2 m2 c s2 2m 2 c 2rm c 2m 2 c 2 = = rm , = · = 2 β, = · = 2 · β, s1 m1 s1 s1 s2 2m 2 s2 s2 2m 2 ω2 ω2 we can write Eq. (8) as   −1 

 U1 β 2 2 = 1 −  − rm 1 + i  · 2 · . F/s1 ω2 1 − 2 + i  · 2β/ω2

(9)

Using Eq. (9), |U1 /(F/s1 )| is plotted as a function of  for varying β values in the figure below:

Appendix

351

β

β = 0.1

2

β

(c) Using U1 in Eq. (6),

U1 |β→0 =

F s1 − m 1

U1 |β→∞ =

ω2

2ω − s2 s2 m−m 2 2ω 2

(→ Curve 1).

F (→ Curve 2). s1 − m 1 ω 2 − m 2 ω 2

(11)

(12)

Because U1 |β→0 U1 |β→∞ have different signs at the intersecting points (otherwise, trivial solutions), we set U1 |β→0 = − U1 |β→∞ .

(13)

Substituting Eqs. (11) and (12) into Eq. (13) yields m 2 (2m 1 + m 2 )ω4 − 2ω2 (m 1 s2 + m 2 s1 + m 2 s2 ) + 2s1 s2 = 0.

(14)

If two solutions satisfying Eq. (14) are defined as ω A and ω B , we can find the following relation from Eq. (14): ω2A + ω2B =

2(m 1 s2 + m 2 s1 + m 2 s2 ) . m 2 (2m 1 + m 2 )

(15)

Now setting U1 (ω A ) = −U1 (ω B ) using Eq. (12), one can find 2s1 = ω2A + ω2B . m1 + m2 Because Eqs. (15) and (16) represent same equations, we can find 2(m 1 s2 + m 2 s1 + m 2 s2 ) 2s1 = , m 2 (2m 1 + m 2 ) m1 + m2 which can be simplified as

(16)

352

Appendix

s2 m1m2 = s1 (m 1 + m 2 )2

(17)

For the given values of m 1 , m 2 , and s1 , s2 = s1 ·

m1m2 = 1.78 × 103 N/m. (m 1 + m 2 )2

The plots for [U1 (ω)/(F/s1 )]β=0 and [U1 (ω)/(F/s1 )]β=∞ are given in the figure below:

β=0 (Curve 1) β=∞ (Curve 2)

ΩA

ΩB

Solutions to Problems in Chap. 3

Problem 3.1 (a) Considering the following free-body diagram for nth mass,

m1

d 2un = s2 (u n+2 − u n ) + s1 (u n+1 − u n ) − s2 (u n − u n−2 ) − s1 (u n − u n−1 ) dt 2

Appendix

353

∴ m 1 u¨ n = s1 (u n+1 + u n−1 − 2u n ) + s2 (u n+2 + u n−2 − 2u n )

(1)

(b) Assuming u n = U0 e jωt e− jβn , one can write (1) as



 −ω2 m 1 = s1 e jβ + e− jβ − 2 + s2 e2 jβ + e−2 jβ − 2 . Therefore, the following ω − β relation (dispersion relation) can be obtained: 

s2 β s1 sin2 + sin2 β ω =4 m 2 m



2

(2)

(c) Using Eq. (1) with m = 1 kg, the following curves can be plotted: (x-axis: β; y-axis: ω.)

Problem 3.2 (a) Group velocity positivity implies vg = dω > 0 ⇔ dω > 0. Differentiating dk dβ Eq. (2) in the solution to Problem 3.1 with respect to β yields, 2mω

s  dω 1 =4 sin β + 2s2 sin β cos β = 2 sin β(s1 + 4s2 cos β). dβ 2

Since sin β ≥ 0 for β ∈ [0, π ), s1 + 4s2 cos β > 0 or 4ss12 > − cos β if dω > 0. dβ Because cos β ∈ [1−, 1] the following condition needs to be satisfied: s1 ≥ 4s2 for β ∈ [0, π ). (b) For case (i) s1 = 1 and s2 = 0.5,

(3)

354

Appendix

Note that there is range where group velocities are negative because condition (3) is not satisfied. For case (ii) s1 = 1 and s2 = 0.2

Because condition (3) is satisfied, all group velocities are positive. Problem 3.3 Consider the following the symmetric unit-cell model,

From the definition of impedance, Fapp = − f n − ,n + = Z u˙ n = Z ( jωu n ) The equation of motion for the left m/2 mass can be written as  m u¨ n = Fn = f n − ,n + − s(u n − u n−1 ) − s(u n − u n−2 ), 2 or

Appendix

355

−

m 2 ω u n = −Z ( jωu n ) − s(2u n − u n−1 − u n−2 ). 2

(1)

With u n = U e j(ωt−βn) , Eq. (1) becomes −



m 2 ω = −Z ( jω) − s 2 − e jβ − e2 jβ . 2

(2)

From Eq. (2), one can find Z=

 1 m 2 ω + s(cos β − 1 + j sin β + cos 2β − 1 + j sin 2β) . jω 2

Using the dispersion relation, which is Eq. (3) can be reduced to

m 2 ω 2

 = 2s sin2

β 2

(3)

 + sin2 β ,

√ ms sin β + sin 2β Z= . 2 sin2 β2 + sin2 β

Problem 3.4 (a) u 1 = u inc + u ref , u 2 = u trans at u = 0(n = 0) Setting u 1 = u 2 at the interface yields A1 + A3 = A2 . (b) The time-averaged power flow is =

1 1 Z i |u˙ i |2 = Z i ω2 Ai2 . 2 2

Energy conservation implies that 1 = 2 + 3 ⇔ inc = ref + trans ∴ Z 1 A21 = Z 2 A22 + Z 1 A23 . (c) From part (a),

A1 + A3 = A2 ⇒ 1 + r = t .

(1)

Z 1 = Z 2 t 2 + Z 1r 2 ⇒ Z 1 (1 − r ) = Z 2 t.

(2)

From part (b),

Substituting Eq. (1) into Eq. (2) yields

356

Appendix

t=

Z1 (1 − r ). Z2

If Eq. (3) is substituted into Eq. (1), one can find 1 + r = r is determined as r=

(3) Z1 Z 2 (1

− r ) from which

Z1 − Z2 . Z1 + Z2

(4)

Substituting Eq. (4) into Eq. (3) yields t=

2Z 1 . Z1 + Z2

(5)

From Eq. (5), t = 1 (r = 0), is satisfied only if Z1 = Z2.

Solutions to Problems in Chap. 4 Problem 4.1 Considering equations of o motion for masses m 1 and m 2 : m 1 u¨ 2n = s(u 2n−2 + u 2n−1 + u 2n+1 + u 2n+2 − 4u 2n )

(1)

m 2 u¨ 2n+1 = s(u 2n−1 + u 2n + u 2n+2 + u 2n+3 − 4u 2n+1 )

(2)

Assuming the following form of u 2n and u 2n+1 . u 2n+1 = Ae j (ωt−(2n+1)β1 ) , u 2n = Be j (ωt−2nβ1 ) (β1 = kd/2), and substituting the expressions above in Eqs. (1) and (2) yield − m 1 ω2 B = s(Be j2β1 + Ae jβ1 + Ae− jβ1 + Be− j2β1 − 4B), − m 1 ω2 A = s(Ae j2β1 + Be jβ1 + Be− jβ1 + Ae− j2β1 − 4A). Rearranging the equations above, s(e jβ1 + e− jβ1 )A + [m 1 ω2 + (e j2β1 + e− j2β1 − 4)s]B = 0,

(3)

Appendix

357

[m 2 ω2 + (e j2β1 + e− j2β1 − 4)s]A + s(e jβ1 + Ae− jβ1 )B = 0.

(4)

For Eqs. (3) and (4) to have a nontrivial solution, it is required that [m 1 ω2 + (2 cos 2β1 − 4)s][m 2 ω2 + (2 cos 2β1 − 4)s] − 4s 2 cos2 β1 = 0. Rearranging the equation above, we can derive the following dispersion relation: m 1 m 2 ω4 + 2s(2 cos2 β1 − 3)(m 1 + m 2 )ω2 − 4s 2 (4 cos2 β1 − 9)(cos2 β1 − 1) = 0.

Problem 4.2 (a) We write the equations of motion as

m 1 u¨ 3n = s(u 3n−1 + u 3n+1 − 2u 3n ) m 2 u¨ 3n+1 = s(u 3n + u 3n+2 − 2u 3n+1 ) m 3 u¨ 3n+2 = s(u 3n+1 + u 3n+3 − 2u 3n+2 ) Assuming u 3n+2 = Ae j (ωt−(3n+2)β1 ) , u 3n+1 = Be j (ωt−(3n+1)β1 ) , u 3n = Ce j (ωt−3nβ1 ) , (β1 = kd/3) one can convert the equations of motion to −m 1 ω2 C = s(Ae jβ1 + Be− jβ1 − 2C) → (m 1 ω2 − 2s)C + se jβ1 A + se− jβ1 B = 0, −m 2 ω2 B = s(Ce jβ1 + Ae− jβ1 − 2B) → (m 2 ω2 − 2s)B + se jβ1 C + se− jβ1 A = 0, −m 1 ω2 A = s(Be jβ1 + Ce− jβ1 − 2A) → (m 3 ω2 − 2s)A + se jβ1 B + se− jβ1 C = 0. One can put the above equations in a compact form as ⎤⎡ ⎤ ⎡ ⎤ se− jβ1 m 1 ω2 − 2s A se jβ1 0 ⎦⎣ B ⎦ = ⎣ 0 ⎦. ⎣ se− jβ1 m 2 ω2 − 2s se jβ1 m 3 ω2 − 2s se jβ1 se− jβ1 C 0 ⎡

For a nontrivial solution,

358

Appendix

⎛⎡

⎤⎞ se− jβ1 m 1 ω2 − 2s se jβ1 ⎦⎠ = 0. det ⎝⎣ se− jβ1 m 2 ω2 − 2s se jβ1 2 jβ1 − jβ1 m 3 ω − 2s se se To express out the above equation yields (m 2 ω2 − 2s)s 2 + (m 3 ω2 − 2s)s 2 + (m 1 ω2 − 2s)s 2 −(m 1 ω2 − 2s)(m 2 ω2 − 2s)(m 3 ω2 − 2s) −s 3 e− j (3β1 ) − s 3 e j (3β1 ) = 0. Rearranging the above equation in the descending order of ω2 yields the following dispersion relation: m 1 m 2 m 3 ω6 − 2s(m 1 m 2 + m 2 m 3 + m 3 m 1 )ω4 + 3s 2 (m 1 + m 2 + m 3 )ω2 − 2s 3 (1 − cos 3β1 ) = 0.

(1)

(b) Substituting k = 0 (i.e., β1 = 0) into Eq. (1) yields

ω2 [(m 1 m 2 m 3 ω4 − 2s(m 1 m 2 + m 2 m 3 + m 3 m 1 )ω2 + 3s 2 (m 1 + m 2 + m 3 )] = 0. (2) Substituting m 1 = 1 kg, m 2 = 2 kg, m 3 = 2 kg, and s = 500 N/m in Eq. (2) yields ω = 0,

√

√ 750, and 1250.

Substituting kd = π (β1 = π/3) in Eq. (2) yields ω = 15.81, 18.96 and 27.29. The dispersion curve with stop and pass bands marked is given by

Appendix

359

(c) The dispersion curves are

As long as all of the three masses become identical, there occurs no bandingfolding phenomenon as in case (i). On the other hand, all of the three masses become identical, the banding-folding phenomenon occurs.

Solutions to Problems in Chap. 5

Problem 5.1 (a) The equations of motion for m 1 and m 2 can be written as

for m 1 : m 1 u¨ n = s1 (vn−1 + vn − 2u n )

(1)

360

Appendix

for m 2 : m 2 v¨n = s1 (u n + u n+1 − 2vn )

(2)

Assuming harmonic motion (at an angular frequency of ω) in Eq. (2), vn can be written as vn =

s1 (u n + u n+1 ) . 2s1 − m 2 ω2

(3)

Substituting vn in Eq. (3) Eq. (1) yields s1 (u n + u n+1 + 2u n ) − 2u n ) 2s1 − m 2 ω2 s1 (u n + u n+1 − 2u n ) 4s1 u n = s1 ( + − 2u n ) 2 2s1 − m 2 ω 2s1 − m 2 ω2

m 1 u¨ n = s1 (

(4)

If we introduce ω1 and ω2 such that m 1 ω12 = 2s1 , m 2 ω22 = 2s1 , Equation (4) can be written as   ω12 s1 ω22 un (u n + u n+1 − 2u n ) + 2s1 −m 1 u n ω = 2 ω22 − ω2 ω22 − ω2 2

(5a)

Rearranging Eq. (5a) yields,  −ω m 1 1 + 2

 ω12 s1 ω22 u = (u n + u n+1 − 2u n ) n 2 ω22 − ω2 ω22 − ω2

(5b)

Because u n ∼ ei(ωt−βn) , Eq. (5b) becomes  −ω2 m 1 1 +

 ω12 β s1 ω22 un = (4 sin2 )u n 2 2 2 ω22 − ω2 2 ω2 − ω

(6)

Because the whole system is to be treated as a monatomic system, Eq. (6) can be written as ef f

ef f

m 1 (ω)ω2 = 4s1 (ω) sin2 ef f

Comparing Eq. (6) and (7), one can identify m 1 ef f m 1 (ω)

 = m1 1 +

 ω12 , ω22 − ω2

β 2

(7) ef f

and s1

ef f s1 (ω)

as

  ω22 s1 = 2 ω22 − ω2

(8)

Appendix

361

(b) Plots:

1) Effective mass

2) Effective stiffness

3) Dispersion curve

Problem 5.2 As in Problem 5.1, the whole system is to be treated as a monatomic system with the ef f ef f effective properties m 1 and s1 . ef f

(a) First we find the effective mass m˜ 1 (ω) for the system consisting of m 1 , m 3 , and √ ef f s3 . To find m˜√1 (ω), we use Eq. (5.16) with ω2 and ω12 replaced by ω3 = s3 /m 3 and ω13 = s3 /m 3 + s3 /m 1 :  ef f

m˜ 1 (ω) = m 1 ef f

 2 ω13 − ω2 . ω32 − ω2

(1)

If m˜ 1 (ω) in Eq. (1) is substituted into Eq. (8a) in the solution of Problem 5.1(a), one can write   ω2 ef f ef f (2) m 1 (ω) = m˜ 1 (ω) 1 + 2 1 2 , ω2 − ω where

362

Appendix ef f

m˜ 1 (ω)ω12 = 2s1 and m 2 ω22 = 2s1 ef f

ef f

The effective stiffness s1 (ω) for this problem is exactly the same s1 (ω) given by Eq. (8b). (b) Plots: 1) Effective mass

2) Dispersion curve

Problem 5.3 ef f

(a) According to Eq. (5.30) in the main text, the effective stiffness s˜1 (ω) of the vertically resonating system consisting of m y , s1 and s is calculated as

ef f

s˜1 (ω) =

  s1 ω 2 − ω 2 , 2 ω21 − ω2

(1)

where 

 0.5s1 + s s ω1 = 2 tan α , ω = 2 tan α . my my ef f

If s˜1 (ω) in Eq. (1) is substituted into Eq. (8b) in the solution of Problem 5.1(a), one can write ef f s1 (ω)

  ef f ω22 s˜1 (ω) , = 2 ω22 − ω2

where ef f

ef f

m 1 ω12 = 2˜s1 (ω) and m 2 ω22 = 2˜s1 (ω). ef f

The effective mass m 1 (ω) is the same as that given in Eq. (8a) in the solution of Problem 5.1(a).

Appendix

363

(b) Plots:

1) Effective mass

2) Effective stiffness

3) Dispersion curve

Problem 5.4 According to Eq. (5.34), the effective mass is calculated as  ef f

m 1 (ω) = m 1 s

s

2 ω12 − ω2 2 ω2 − ω2

 (1)

s

2 with ω12 = m22 + m21 and ω22 = m22 . Using Eq. (1) and the given data, one can find that the effective mass becomes negative if

54.78 < ω < 67.08 [rad/s].

(2)

According to Eq. (5.34), the effective stiffness is calculated as ef f s1 (ω)

with ω13 = 2 tan α



0.5s1 +s3 my

  s1 ω2 − ω23 = 2 ω2 − ω213

and ω3 = 2 tan α



s3 . my

(3)

364

Appendix

Using Eq. (3) and the given data, one can find that the effective mass becomes negative if 36.52 < ω < 63.24 [rad/s].

(4)

From the results given by (2) and (4), the effective mass and stiffness become simultaneously negative if 54.78 < ω < 63.24 [rad/s]. Problem 5.5 ef f

(a) The effective stiffness s2 (ω) serves to connect m 1 and m 2 . One can express ef f s2 (ω) in terms of s2 , s3 , and m y by using Eq. (34),1

ef f s2 (ω)

with ω23 = 2 tan α



0.5s2 +s3 my

  s2 ω23 − ω2 , = 2 ω223 − ω2

and ω3 = 2 tan α



(1)

s3 . my ef f

Utilizing Eq. (1) in the Solution of Problem 5.2, the effective mass m 1 (ω) can be written as  2  ω12 (ω) − ω2 ef f , (2) m 1 (ω) = m 1 ω22 (ω) − ω2 where ef f

2 ω12 (ω) =

1

ef f

ef f

s2 (ω) s2 (ω) s (ω) + and ω22 = 2 . m2 m1 m2

We put ½ here because s2 is defined differently from s2 in Fig. 5.8.

Appendix

365

Plot: Effective mass

ef f

(b) As α → π/2, the effective stiffness s2 (ω) becomes frequency-independent as   s3 s2 ef f = 75N/m. s2 (ω) = 2 0.5s2 + s3 ef f

Therefore, the pattern of the frequency dependency of the effective mass m 1 will be different from that calculated for α = π/2. Plot (for α =π/2): Effective mass

In part (a), there are two frequency ranges in which the effective mass is negative. In part (b), however, there is only one frequency range in which the effective mass ef f is negative because the effective stiffness s2 (ω) is no longer frequency-dependent.

366

Appendix

Solutions to Problems in Chap. 7

Problem 7.1 Fy = −T sin θ ≈ −T tan θ (assuming small θ ) ∂u y = −T , ∂x vy =

∂u y . ∂t

(1) (2)

Substituting u y = U0 e j(ωt−kx) into Eqs. (1) and (2) yields Fy = −T U0 (− jk)e j (ωt−kx) = j T ku y ,

(3)

v y = jωU0 e j (ωt−kx) = jωu y .

(4)

From (3) and (4), Fy = T Using the relations c =

√

k uy. ω

(5)

T /ρ (or T = ρc2 ) and c = k/ω, Eq. (5) becomes

1 Fy = ρc2 v y = (ρc)v y = Zv y . c Problem 7.2. In the following power balance equation given by Eq. (7.63), 1 2 1 1 ω Z 1 |A1 |2 = ω2 Z 1 |B1 |2 + ω2 Z 2 |A2 |2 , 2 2 2 we can put A2 = 2 A1 and Z 2 = 0 to obtain 1 2 1 ω Z 1 |A1 |2 = ω2 Z 1 |B1 |2 . 2 2 Therefore, the result A2 = 2 A1 does not violate the physical law of energy balance. Problem 7.3

Appendix

367

   1

Pout x=0 = − Re vx (σx )∗ S  2 x=0 

 1 = − Re [(iω)An sin kn x][An kn cos kn x]∗  2 x=0 =0

   1

Pout x=L = − Re vx (σx )∗ S  2 x=L 

 1 ∗  = − Re [(iω)An sin kn x][An kn cos kn x]  2 x=L =0 ∴ Pout  = Pout x=0 + Pout x=L = 0. Therefore, ∂UT /∂t = 0 from Eq. (7.53). Problem 7.4 2π/ω

 1

[−Re( f )Re(vx )]x dt = − Re F(x)V(x)∗ . 2

1 < P >x = 2π/ω 0

< P >x+d 2π/ω

 1

[−Re( f )Re(vx )]x+d dt = − Re F(x + d)V(x + d)∗ 2

1 = 2π/ω 0

1 = − Re [T11 V(x) + T12 F(x)][T21 V(x) + T22 F(x)]∗ 2

1 ∗ ∗ ∗ ∗ |V(x)|2 + T11 T22 |F(x)|2 = − Re T11 T21 V(x)F∗ (x) + T12 T21 F(x)V∗ (x) + T12 T22 2  

∗

" 1! ∗ ∗ ∗ = − |V(x)|2 Re T11 T21 + |F(x)|2 Re(T12 T22 F(x)V∗ (x) . ) + Re T11 T22 + T12 T21 2 To satisfy < P >x =< P >x+d , the following relation must hold: 

∗ ∗ ∗ ∗ = 0, Re(T12 T22 ) = 0, T11 T22 + T12 T21 = 1. Re T11 T21

(1)

Because T11 and T22 are real-valued, T12 and T21 are purely imaginary. For a ∗ = −T21 . One can variable Q that is purely imaginary, Q∗ = −Q. Therefore, T21 now write the last expression in Eq. (1) as ∗ ∗ T22 + T12 T21 = 1 → T11 T22 − T12 T21 = 1 → det T = 1. T11

368

Appendix

Problem 7.5 From Eq. (7.40), (A1 − B1 ) =

Z2 ∗ Z2 A → (A1 − B1 )∗ = A Z1 2 Z1 2

(1)

Multiplying Eq. (7.36) by Eq. (1) yields (A1 + B1 )(A1 − B1 )∗ =

Z2 A2 A∗2 . Z1

(2)

Expanding and simplifying Eq. (2) yields (A1 + B1 )(A1 − B1 )∗ =

Z2 A2 A∗2 Z1

Z2 |A2 |2 → |A1 |2 − |B1 |2 − A1 B∗1 + A∗1 B1 = Z1

 → Z 1 |A1 |2 − |B1 |2 = Z 2 |A2 |2   Z1 − Z2 used A1 B∗1 = A∗1 B1 since B1 = A1 Z1 + Z2

(3)

Multiplying ω2 /2 to both sides of Eq. (3) and rearranging the resulting terms in the equation yields, 1 2 1 1 ω Z 1 |A1 |2 = ω2 Z 2 |A2 |2 + ω2 Z 2 |B1 |2 2 2 2 Problem 7.6 (Here, “r” and “t” are denoted by “R” and “T,” respectively.) Using Eqs. (7.41) and (7.42), R0 = T0 = R = T =

# 1 − Z2 Z1 Z1 − Z2 1−ξ # = = Z1 + Z2 1+ξ 1 + Z2 Z1 2Z 1 2 2 # = = Z1 + Z2 1+ξ 1 + Z2 Z1 # Z2 Z1 − 1 Z2 − Z1 ξ −1 # = = Z1 + Z2 1+ξ 1 + Z2 Z1 # 2Z 2 Z 1 2Z 2 2ξ # . = = Z1 + Z2 1+ξ 1 + Z2 Z1

Transmission from medium 1 to medium 2: T A = T0 , Propagation from x = 0 to x  = 0: TB = e−ik2 d ,

Appendix

369

Transmission from medium 2 to medium 1: TC = T  . Therefore T1 = T A TB TC = T0 T  e−ik2 d . Reflection from medium 2 to medium 1: TD = R  , Propagation from x  = 0 to x = 0: TE = e− jk2 d , Reflection from medium 2 to medium 1: TF = R  , Propagation from x = 0 to x  = 0:: TG = e−ik2 d ,  Transmission from medium 2 to medium  1: TH = T . 2 Therefore, T2 = T1 TD TE TF TG TH = T1 R  e−2ik2 d . Referring to Figure (a), the difference between T j and T j−1 is due to path from D → E → F → G. Therefore, Rˆ becomes

2 Rˆ = TD→E→F→G = TD TE TF TG = R  e−2ik2 d . The total transmission T can be obtained as T = T1 + T2 + T3 + · · ·   = T1 1 + Rˆ + Rˆ 2 + Rˆ 3 + · · · T1

=

1 − Rˆ T0 T  e−ik2 d = . 1 − (R  )2 e−2ik2 d

Imposing |T | = 1,    

2  → T0 T   = 1 − R  e−2ik2 d , and using T0 =

2 , 1+ξ

T =

2ξ , 1+ξ

R =

ξ −1 , 1+ξ

  → 4ξ = (1 + ξ )2 − (ξ − 1)2 e−2ik2 d  ∴ e−2ik2 d = 1 → k2 d = nπ (n = 1, 2, 3, · · ·). Problem 7.7 Referring to Eq. (7.20), the general solution can be written as u = Aei(ωt−kx) + Bei(ωt+kx) Applying the boundary conditions at x = 0 and x = l yields u(x = 0) = 0 → (A + B)eiωt = 0

370

Appendix

∴ B = −A, u(x = l) = 0

 → Aeiωt e−ikl − eikl = 0 → −2iAeiωt sin kl ∴ sin kl = 0. Therefore, kl = nπ (n = 1, 2, · · ·), , ∴ u n (x) = sin ∴

nπ x l

ωl nπ c = nπ → ωn = c l

Problem 7.8 (a) To plot the dispersion curve, we use β = cos−1 (a(ω)) from the first equation of (7.180), where a (ω) is given by Eq. (7.179) and T(21) = T(b) · T(a) . Therefore, the dispersion relation can be plotted as

Dispersion relation of periodic structure in Problem 7.8.

(b, c) For N = 1, we use t = For N = 3,

1 S22

from Eq. (7.215b) and T = |t|2 .

−1   1 1 1 1  T2 T2 T2 S −Z 1 Z 1 −Z 1 Z 1 −1    1 1 1 3 1 , = (T2 ) −Z 1 Z 1 −Z 1 Z 1 $ % $ %  $ % N =3 N =3 t 1 1 S S 11 12 = S N =3 = N =3 N =3 S S 0 r r 21 22 N =3



Appendix

371

   1 2  → T = |t| =  N =3  . S 2

22

Likewise, for N = 10, S N =10 



1 1 −Z 1 Z 1

−1

 (T2 )10

 1 1 , −Z 1 Z 1

   1 2  T = |t| =  N =10  . S 2

22

Comment: As N increases, the transmittance T approaches zero in the stopband. However, there is oscillatory behavior observed in the passband—this phenomenon remains unless N → ∞. Problem 7.9 (a) Displacement continuity at x = 0 yields  

−ikx + r eikx x=0 = te−ikx x=0 1e → 1 + r = t.

(1)

372

Appendix

Force balance at x = 0 yields ∂u 0− ∂ 2 u 0+ ∂u 0+ = −ω2 Mt = −E A + EA 2 ∂t ∂x ∂x → −ω2 Mt = E Aik(1 − r ) − E Aikt

M

Mω2 t = 1−r −t ik E A Mω Mω2 t = 1−r −t →− E t =− iZ iω c A   Mω → 1−r = 1− t. iZ

→−

(2)

Using the Eqs. (1) and (2), Mω t iZ Mω t. →r = 2i Z 2r =

Substituting the above equation into Eq. (1), Mω t =t 2i Z   1 −iq Mω ∴t = , r= where, q = . 1 + qi 1 + iq 2Z

1+

If M → 0, then q → 0, therefore, r → 0, t → 1. This result shows that the limiting case simulates a 1D continuum bar. (b) The force balance at x = d − gives Aσd − = Fs (where, Fs = S(u d + − u d − ))



 ∂u d − → EA = E A −ike−ikd + ikr eikd = S te−ikd − e−ikd − r eikd ∂ x



 → −ik E A e−ikd − r eikd = S te−ikd − e−ikd − r eikd  S

t − 1 − r e2ikd . → 1 − r e2ikd = − ik E A If we define S =

S , ik E A

1 + Se2ikd r − St = 1 − S. The force continuity in the spring yields

(3)

Appendix

373

Aσd − = Aσd + ∂u d + ∂u d − =E →E ∂x ∂x → −ike−ikd + ikr eikd = −ikte−ikd → 1 − r e2ikd = t → e2ikd r + t = 1.

(4)

Using Eqs. (3) and (4), we can obtain the following equation %  $ % $ r 1−S 1 + S e2ikd −S = 1 e2ikd t 1 $ % %  $ r 1−S 1 S 1

 → =

 t 1 1 + 2S eikd −e2ikd 1 + S e2ikd



∴r = If we define p =

1 1 + 2S 2S , ωZ

e−2ikd , t =

2S 1 + 2S

.

then, r=

1 − pi e−2ikd , t = . 1 − pi 1 − pi

If S → 0, p goes 0. Therefore, r → e−2ikd , t → 0. This limiting case simulates a bar with a free-end boundary at x = d. If S → ∞, p goes ∞, therefore, r → 0, t → 1. This limiting case s case simulates a 1D continuum bar. (c) The displacement continuity at x = 0 yields  

−ikx + r eikx x=0 = ae−ikx + beikx x=0 e ∴ 1 + r = a + b.

(1)

The force balance at x = 0 yields  



−ω2 M e−ikx + r eikx x=0 = E A −ikae−ikx + ikbeikx x=0 

− E A −ike−ikx + ikr eikx x=0 → −ω2 M(1 + r ) = ik E A(1 − r − a + b). Let M =

Mω2 , ik E A

(5)

then Eq. (5) becomes − M(1 + r ) = 1 − r − a + b



 ∴a−b = M −1 r + M +1 .

(2)

374

Appendix

The force balance at x = d yields Fd − = S(u d + − u d − )



 ∂u d − → EA = E A −ikae−ikd + ikbeikd = S te−ikd − ae−ikd − beikd ∂x 

−ikd − beikd = −S te−ikd − ae−ikd − beikd → ae 

→ a − be2ikd = −S t − a − be2ikd



 ∴ a 1 − S − 1 + S e2ikd b = −St. (3) The force continuity at x = d yields Fd − = Fd + → Aσd − = Aσd + ∂u d + ∂u d − = EA → EA ∂x ∂x → −ikae−ikd + ikbeikd = −ikte−ikd → ae−ikd − beikd = te−ikd ∴ a − be2ikd = t.

(4)

Therefore, the four expressions are now obtained: (1) 1 + r = a + b, 



(2) a − b = M − 1 r + M + 1 ,



 (3) a 1 − S − 1 + S e2ikd b = −St, (4) a − be2ikd = t.

Solutions to Problems in Chap. 8

Problem 8.1 By applying the boundary conditions with Ci = 0 and Di = 0, ⎡ ⎤ Ci  ⎢ ⎥ $ % 3 3 3 3 0 −i E I k + K t E I k + K t i E I k + K t −E I k + K t ⎢ Di ⎥ , ⎢ ⎥= −E I k 2 E I k2 −E I k 2 E I k2 ⎣ Cr ⎦ 0 Dr

Appendix

375

we can find Cr and Dr as k 3 (i + 1) − 2ξ Cr = 3 Ci , k (i − 1) + 2ξ

2ik 3 Ci Dr = 3 k (i − 1) + 2ξ

  Kt . where, ξ = EI

As K t goes 0, ξ → 0 then, Cr → −iCi , Dr → (1 − i)Ci : free-end condition. As K t goes ∞, ξ → ∞ then, Cr → −Ci , Dr → 0: Hard-wall (or fixed) condition. Problem 8.2 By applying boundary conditions with Ci = 0 and Di = 0, ⎡ ⎤ Ci  ⎢ ⎥ $ % 2 2 2 2 0 −E I k − ikα E I k − kα −E I k + ikα E I k + kα ⎢ Di ⎥ . ⎢ ⎥= 3 3 3 3 EIk iEIk −E I k −i E I k ⎣ Cr ⎦ 0 Dr we can find Cr and Dr as Cr =

k 2 (i + 1) + 2ikξ Ci , k 2 (i − 1) + 2ikξ

Dr =

 α  2k 2 . where, ξ = C i k 2 (i + 1) + 2kξ EI

As α goes 0, ξ → 0 then, Cr → −iCi , Dr → (1 − i)Ci : Free-end condition. As α goes ∞, ξ → ∞ then, Cr → Ci , Dr → 0: Hard-wall condition. Problem 8.3 (a) Let vm = Vm eiωt , vr = (Cr eikx + Dr ekx )eiωt , v(x, t) = vi (x, t) + vr (x, t)   3 2 Boundary conditions at x = 0 remark:V = −E I ∂∂ xv3 , M = E I ∂∂ xv2 , (i) Force: V + s(v − vm ) = 0. (ii) Moment: M = 0. (iii) Equation of motion for mass m: m v¨m + s(vm − v) = 0. We have three equations (i), (ii), and (iii) for three unknown variables (Vm , Cr , Dr ). Solving the equations yields   s (−2α + k 3 + ik 3 )Ci 2ik 3 Ci s 1− , Dr = whereα = Cr = 2α − k 3 + ik 3 2α − k 3 + ik 3 EI s − mω2 1 ((−2α + k 3 + ik 3 )eikx + 2ik 3 ekx )Ci eiωt . → vr = 2α − k 3 + ik 3

376

Appendix

(b) (1) If m → ∞, it represents an elastic support at x = 0. α → the boundary condition reduces to: V + sv = 0, and M = 0.

s EI

 . Therefore,

(2) if m → 0, it represents the free-end boundary conditions. (α → 0). Therefore, the boundary condition b reduces to: V = 0 and M = 0. (c) If m, s → ∞, it represents a simply supported support. Therefore, the boundary condition b reduces to: v = 0 and M = 0. Problem 8.4 ρ = 2713 kg/m3 , E = 70 GPa,m  = 1 kg/m, s  = 50 N/m2 , A = (a × a) m2 , a = 0.03 m, Izz = (a 4 /12) m4

(a)

for the beam segment (x). • equation of motion in the y-direction:ρ Ax ∂∂t v2 = V + S  x(w − v). 2

→ ρA

∂ 2v ∂V + S  (w − v). = ∂t 2 ∂x

• moment equilibrium : M +

(1)

x x V ≈ 0 (∵ EB beam). (V + V ) + 2 2 (2) ∂ 2w = S  x(v − w). ∂t 2

(3)

   ∂M ∂ 2v ∂ ∂ 2v →V =− used: M = E I 2 =− EI 2 ∂x ∂x ∂x ∂x

(4)

• equation of motion for m  m  x Using Eq. (2), M + V x = 0

Using Eq. (3),

Appendix

377

m

∂ 2w = S  (v − w). ∂t 2

(5)

Assuming harmonic motion with frequency ω yields, ∂ 2w = −ω2 w. ∂t 2

(6.1)

∂ 2v ∂ 4v = −ω2 v and 4 = k 4 v. 2 ∂t ∂x

(6.2)

w(x, t) = W ei(ωt−kx) → v(x, t) = V ei(ωt−kx) → Equation (6.1) → Eq. (5):

 S  − ω2 m  w = S  v → w =

S

Sv . − m  ω2

(7)

Now, we express S  (w − v) in terms of v as S  (w − v) = S 



 Sv S  m  ω2 Sm  ∂ 2v − v = v = − S  − m  ω2 S  − m  ω2 S  − m  ω2 ∂t 2

(8)

To obtain Eq. (8), we used Eqs. (7) and (6-2). Substituting Eqs. (4) and (8) in Eq. (1) yields ρA

∂ 4v Sm  ∂ 2v ∂ 2v = −E I − ∂t 2 ∂x4 S  − m  ω2 ∂t 2 ( )  m  ρSA ∂ 2v ∂ 4v → ρA 1 +  + EI 4 = 0  2 2 S − m ω ∂t ∂x

(9)

The quantity in the parenthesis in Eq. (9) can be written as 

1+

m  ρSA S  − m  ω2



=

S  − m  ω2 + m  ρSA S  − m  ω2

 S  1/m  + 1/ρ A − ω2 = . S  /m  − ω2

(10)

If we introduce the following symbols ω2p 

S , m

(11)

and ωq2 Equation (9) can be written as

S





1 1 +  ρA m

 (12)

378

Appendix

ρeff A

∂ 2v ∂ 4v + E I = 0, ∂t 2 ∂x4

(13)

where the effective density ρeff is defined as ρeff = ρ

ωq2 − ω2 ω2p − ω2

.

(14)

The range of negative value is ω p < ω < ωq , where ω p = 7.07, ωq = 8.40. 

(b) −ω2 ρeff A + k 4 E I v = 0 →∴ k 4 = ρEeffIA ω2 for nontrivial solution. If ρeff > 0,  k 2 = ±ω

ρeff A EI

 41  41 √  √  → k = ± ω ρEeffIA or ±i ω ρEeffIA .  41 √  For right propagation wave, k = + ω ρEeffIA , and  41 √  for right decaying wave, k = −i ω ρEeffIA . else if ρeff < 0,  k = ±iω 2

|ρeff |A EI

Appendix

379

1 1   1 + i √ −ρeff A 4 1 − i √ −ρeff A 4 →k= √ ω , √ ω , EI EI 2 2 1 1   −1 + i √ −ρeff A 4 1 + i √ −ρeff A 4 ω ,− √ ω . √ EI EI 2 2 √  −ρeff A  41 √ , − 1+i ω EI , 2 1 1     √ −ρeff A 4 −1+i √ −ρeff A 4 √ ω EI , √2 ω E I . For left decaying wave, k = 1+i 2 See Fig. 8.10 for the dispersion curve.

For right decaying wave, k =



√ −ρeff A 1−i √ ω EI 2

 41

Problem 8.5 From the result in Problem 8.4, we can consider the additional rotational support. In this case, Eq. (4) derived in Problem 8.4 should be changed as M +

∂v x x (V + V ) + V − R  x = 0, 2 2 ∂x

Thus, ∂v ∂M + V = R ∂x ∂x ∂2 M ∂ 2v ∂ 2v ∂ 3v ∂V = − 2 + R  2 = −E I 3 + R  2 ⇒ ∂x ∂x ∂x ∂x ∂x

(1)

Substituting Eq. (1) into Eq. (1) of Problem 8.4 and following the derivation procedure given in Problem 8.4 yields ρeff A

∂ 2v ∂ 2v ∂ 4v = R 2 − E I 4 2 ∂x ∂x ∂x

Thus, we can obtain (Using, e.g., (6.1) in Problem 8.4): k R 2 ρeff A 2 k − ω = 0. EI EI    R  2 ρeff (ω)A 2 R 2 ± ω + k =− 2E I 2E I EI ∴ k4 +

(b) From (a), we choose the 2nd branch for a pass band can exist:

R k2 = − + 2E I



R 2E I

2 +

ρeff A 2 ω EI

380

Appendix

The 2nd branch can have a pass band condition if k 2 > 0 (k should be a purely real value) R + k2 = − 2E I



R 2E I

2 +

ρeff A 2 ω >0 EI

  2   2 Therefore 2ER I + ρEeffIA ω2 > 2ER I , i.e., ρEeffIA ω2 > 0 (i.e., ρeff > 0). Otherwise, the branch represents    a stop band.  Because ρeff = ρ 1 +

S ρA S  −m  ω2

m

,

the parameters affecting ρeff are S  and m, not R  . (c) To determine the frequency range ω1 ≤ ω ≤ ω2 of the stopband, we examine ⎛ ρeff = ρ ⎝1 + S ρA 2 ω0 −ω2

< −1, where ω02 =

m



S ρA

 ⎞

S  − m  ω2

⎠ −ω02 + ω2 ρA S → ω2 < ω02 + ρA  ∴ ω0 < ω
ω2 S < −ω02 + ω2 ρA S . → ω2 > ω02 + ρA In this case, no solution exists. ∴ ω0 < ω < ω02 + ρSA → Stop band frequency range.

Appendix

381

(d) dispersion curves:

Problem 8.6 Using Eqs. (8.56) and (8.57), group velocities for the first and second branches are plotted as

The limiting group velocity of the first branch is slightly different from the exact velocity of the theory of 3D elasticity, c R . And the limiting group velocity of the second branch is far different from the exact velocity, cs . If one wish to make the limiting group velocity of the first branch equal to c R , the following equation should be satisfied: c˜s = c R   GK (0.862 + 1.14ν) G → = ρ ρ (1 + ν)

382

Appendix

∴ K = 0.8662.

Problem 8.7 From Eq. (8.89), m I ω4 + A(k)ω2 + B(k) = 0. (a) To derive A(k) and B(k), we calculate  D1 + mω2 D2 =0 det D2∗ D3 + I ω 2



 → D1 + mω2 D3 + I ω2 − D2 D2∗ = 0

(1)

 D1 = α eika + e−ika − 2 = 2α(cos ka − 1)

(2a)



 1 ∗ ika a α e − e−ika = (iaα) sin ka 2  

ika   1 ∗ 2 ika −ika D3 = β e + e a −2 − α e + e−ika + 2 2 1 2 = 2β(cos ka − 1) − a α(cos ka + 1) 2 D2 =

(2b)

(2c)

If Eqs. (2a–2c) are substituted into Eq. (1), we can find 1 A(k) = 2(α I + βm)[cos(ka) − 1] − αma 2 [cos(ka) + 1], 2 B(k) = 4αβ[cos(ka) − 1]2 . (b) by using quadratic formula, the given dispersion relation can be solved with respect to ω2 as, 

A(k)2 − 4I m B(k) (acoustic branch) 2I m  A(k)2 − 4I m B(k) −A(k) + ω2 = (optical branch) 2I m

ω = 2

−A(k) −

(c) By differentiating the given dispersion relation with respect to wavenumber k, we can find

Appendix

383

∂ω ∂ω + A (k)ω2 + 2 A(k)ω + B  (k) = 0 ∂k ! ∂k " − A (k)ω2 + B  (k) ∂ω ! " → = ∂k 2ω 2I mω2 + A(k)

4 I mω3

(1)

where   1 ∂ A(k) = − 2(α I + βm) − αma 2 sin(ka)a, ∂k 2 ∂ B(k) B  (k) = = 8αβ[cos(ka) − 1][− sin(ka)]a ∂k A (k) =

(d) Find the condition that the optical branch has negative group velocity regardless  of ω. (Hint: A2 (k) − 4ImB(k) is always real, positive in the optical branch. Hint 2: Consider the condition only for 0 < kaπ .) For the optical branch, the following relation holds: 2I mω2 = −A(k) +

A(k)2 − 4I m B(k)

(2)

Substituting Eq. (2) into Eq. (1) yields ! " " ! − A (k)ω2 + B  (k) − A (k)ω2 + B  (k) ∂ω ! " =   =  ∂k 2ω 2I mω2 + A(k) 2ω −A(k) + A(k)2 − 4ImB(k) + A(k) " ! − A (k)ω2 + B  (k)   = (3) 2ω A(k)2 − 4ImB(k) In Eq. (3), the denominator is always positive for the optical branch. Therefore, the is governed by the sign of the numerator. The numerator can be expressed sign of ∂ω ∂k as. ! " numerator = − A (k)ω2 + B  (k) = sin(ka)

   1 2(α I + βm) − αma 2 ω2 + 8αβ[cos(ka) − 1] . 2

For 0 < ka < π , sin(ka) is always positive. Thus, the sign of 

∂ω sign ∂k



 = sign

∂ω ∂k

(4)

is determined by

  1 2 2 2(α I + βm) − αma ω + 8αβ[cos(ka) − 1] . 2

384

Appendix

Furthermore, cos(ka) − 1 has always negative for 0 < ka < π . Therefore, if the is always negative regardless of ω. following condition is satisfied, ∂ω ∂k 1 2(α I + βm) − αma 2 < 0. 2

Solutions to Problems in Chap. 9

Problem 9.1 For isotropic media, we have two independent coefficients C11 and C66 because C12 = C11 − 2C66 , C16 = C26 = 0, C22 = C11 . Thus, we can express i j as 11 = C11 n 2x + C66 n 2y , 22 = C66 n 2x + C11 n 2y , 12 = 21 = (C12 + C66 )n x n y = (C11 − C66 )n x n y .

(1)

Then, , which is the determinant of [i j ], reduces to 



2 4 k .  = 11 k 2 − ρω2 22 k 2 − ρω2 − 12

(2)

Substituting Eq. (1) into Eq. (2),

2 C11 C66 n 2x + n 2y k 4 − ρω2 (C11 + C66 )k 2 + ρ 2 ω4 = 0.

(3)

Because n 2x + n 2y = 1, Eq. (3) becomes 



2 ρω − C11 k 2 ρω2 − C66 k 2 = 0.

(4)

Solutions to Eq. (4) are ρω2 − C11 k 2 = 0 or ρω2 − C66 k 2 = 0 Using c L =

√ √ C11 /ρ and cT = C66 /ρ, we obtain k = ±ω/c L , k = ±ω/cT .

Problem 9.2 (1) For the case of k = ±ω/c L Substituting ρω2 = C11 k 2 in Eq. (9.24) yields 

−n 2y n x n y (C11 − C66 ) n x n y −n 2x



Ux Uy



  0 = . 0

(1)

Appendix

385

Simplifying (1), we have 

−n y n x n y −n x





Ux Uy

  0 = . 0

(2)

Equation (2) is satisfied if 

Ux Uy



 =A

nx ny

 ≡ AP L .

(2) For the case of k = ±ω/cT As in Case (1), we obtain the following result, if ρω2 = C66 k 2 is substituted into Eq. (9.23), 

n y nx n y nx



Ux Uy



  0 = . 0

Equation (3) is satisfied if 

Problem 9.3 (a)

Ux Uy





−n y =A nx

 ≡ APT

(3)

386

(b)

Appendix

Appendix

387

Problem 9.4 (a) For longitudinal wave incidence,

As shown in the above figure, there is no critical incidence angle in which only a transverse wave is transmitted to the steel plate. Also, there is no critical incidence angle in which no longitudinal or transverse wave is transmitted to the steel plate. The reason for this result is that the speed of the longitudinal wave in aluminum, 5,393 m/s, is greater than the speed of the longitudinal and transverse waves in steel, 5,291 m/s and 3,130 m/s, respectively. (b) For transverse wave incidence,

As shown in the above figure, the critical angle above which no longitudinal wave is transmitted to the steel plate is 36.2°. The critical incidence angle above which no longitudinal or transverse wave is transmitted to the steel plate is 85.8°.

388

Appendix

Problem 9.5 In the case of transverse wave incidence, the denominator appearing in Eqs. (9.106) and (9.107) should be  +  1

A ∗   A A ∗  T ,A  A · M33 + M23 · M43 P  = M13 . 2 Therefore, the reflectance and transmittance can be defined as  −   −   +   +   P L ,A   P T ,A   P L ,B   P T ,B          R L =  T + ,A , RT =  T + ,A , TL =  T + ,A , TT =  T + ,A      P P P P the coefficients  To   

calculate  M A · M A ∗ + M A · M A ∗  as 13 33 23 43

appearing

in

Eq.

(1),

we

(1) express



A ∗ 

  A A ∗ A · M43  M13 · M33 + M23  = ω2 sin θTA · Z TA sin 2θTA + cos θTA · Z TA cos 2θTA

 = ω2 2Z TA sin2 θTA cos θTA + Z TA cos θTA cos 2θTA

 = ω2 Z TA cos θTA 2 sin2 θTA + cos 2θTA = ω2 Z TA cos θTA

(2)

Substituting Eq. (2) into Eq. (1) (and using the results given in Eq. (9.104)) yields the results in Eqs. (9.106) and (9.107) in the main text. Problem 9.6 For longitudinal wave incidence,

For transverse wave incidence,

Appendix

389

Solutions to Problems in Chap. 10

Problem 10.1 The matching layer considered in Sect. 7.6 is only valid when there is only a single wave mode, either longitudinal or transverse. To match the interface conditions at the interfaces between dissimilar media, an obliquely incident longitudinal wave generates not only the longitudinal wave but also transverse wave in the reflected and transmitted wave fields. Therefore, a more general matching theory which consider multimodal wave fields is needed. Problem 10.2 (a) Isotropic solid A: Incident longitudinal wave (L + ), reflected longitudinal wave (L-), reflected transverse wave (T-). Isotropic solid B: Transmitted longitudinal wave (L + ), transmitted transverse wave (T + ). Anisotropic solid layer: Quasi-longitudinal wave (QL + , QL-), quasitransverse wave (QT + , QT-). (b) Isotropic solid A: – L+: (k LA n x , k LA n y ) – L-: ( (−k LA n x , k LA n y ) )   A 2 c LA k LA cT n y – T-: − c A 1 − c A , k LA n y T

L

Isotropic solid B: (  – L+ : ( – T+ :

c LA k LA c LB c LA k LA cTB

1−  1−

 

c LB n y c LA cTB n y c LA

2 2

) , k LA n y ) , k LA n y

390

Appendix

(c) For c LA > c LB : no critical angle.

 A c For c LB > c LA > cTB : θcr,L = sin−1 c LB .  A L  A c c For cTB > c LA : θcr,L = sin−1 c LB , θcr,T = sin−1 c LB . L

T

Problem 10.3 (a) Isotropic solid A: Incident transverse wave (T + ), reflected longitudinal wave (L-), reflected transverse wave (T-). Isotropic solid B: Transmitted longitudinal wave (L + ), transmitted transverse wave (T + ). Anisotropic solid layer: Quasi-longitudinal wave (QL + , QL-), quasitransverse wave (QT + , QT-). (b) Isotropic solid A: – T +(: (k TA n x , k TA n y ) )   A 2 cTA k TA cL n y A – L-: − c A 1 − c A , k T n y L

T

– T-: (−k TA n x , k TA n y ) Isotropic solid B: (  – L+: ( – T+:

cTA k TA c LB cTA k TA cTB

1−  1−

 

c LB n y cTA cTB n y cTA

2 2

) , k TA n y ) , k TA n y

(c) For cTA > c LB : no critical angle.

 A c For c LB > cTA > cTB : θcr,L = sin−1 cTB .  A L  A c c For cTB > cTA : θcr,L = sin−1 cTB , θcr,T = sin−1 cTB . L

T

Problem 10.4 For transverse wave incidence, the reflection and transmission coefficients have the following relation with the scattering matrix: ⎧ jk B d ⎫ ⎡ S11 e L ,x t L ⎪ ⎪ ⎪ ⎪ ⎬ ⎢ ⎨ 0 S 21 =⎢ B ⎣ S31 ⎪ e jkT,x d tT ⎪ ⎪ ⎪ ⎭ ⎩ S41 0

S12 S22 S32 S42

S13 S23 S33 S43

⎤⎧ ⎫ 0 ⎪ S14 ⎪ ⎪ ⎨ ⎪ ⎬ r S24 ⎥ ⎥ L S34 ⎦⎪ 1 ⎪ ⎪ ⎩ ⎪ ⎭ S44 rT

By solving the equations in the second and fourth rows, the reflection coefficients are obtained as follows:

Appendix

391

rL =

S43 S24 − S23 S44 S42 S23 − S22 S43 , rT = S22 S44 − S42 S24 S22 S44 − S42 S24

Substituting these into the equations in the first and third rows, the transmission coefficients are obtained as follows: B

B

t L = eik L ,x d (S13 + S12 r L + S14 r T ), tT = eikT,x d (S33 + S32 r L + S34 r T )

Problem 10.5 For transverse wave incidence, scattering matrix components should satisfy the following equations for the perfect transmission: S13 = 0, S23 = 0, S43 = 0 (for T-to-T perfect transmission) S23 = 0, S33 = 0, S43 = 0 (for T-to-L perfect transmission) In Eq. (10.14), scattering matrix components can be expressed as S pq = A pq e−iτα d + B pq e−iτβ d + C pq e−iτγ d + D pq e−iτδ d Then, the conditions for the scattering matrix components becomes ⎡ −i(τβ −τα )d ⎤ ⎡ ⎤−1 ⎡ ⎤ e B13 C13 D13 −A13 ⎣ e−i(τγ −τα )d ⎦ = ⎣ B23 C23 D23 ⎦ ⎣ −A23 ⎦ (for T-to-T perfect transmission) e−i(τδ −τα )d B43 C43 D43 −A43 ⎤−1 ⎡ ⎤ ⎡ −i(τβ −τα )d ⎤ ⎡ B23 C23 D23 −A23 e ⎣ e−i(τγ −τα )d ⎦ = ⎣ B33 C33 D33 ⎦ ⎣ −A33 ⎦ (for T-to-L perfect transmission) e−i(τδ −τα )d B43 C43 D43 −A43 Here, all components on the left-hand side must be real-valued because all compo



nents on the right-hand side are real-valued. As a result, τβ − τα d, τγ − τα d and (τδ − τα )d must be integer multiples of π. Therefore, the following conditions must be satisfied for perfect transmission: ⎧ ⎨ φ Q L + ,Q L − ≡ (τβ − τα )d = lπ, φ Q L + ,QT + ≡ (τγ − τα )d = mπ, ⎩ φ Q L + ,QT − ≡ (τδ − τα )d = nπ, (l, m, n : integers) This is exactly the same result as for longitudinal wave incidence.