Ocean Acoustics (Springer Textbooks in Earth Sciences, Geography and Environment) 3030358836, 9783030358839

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Anatoly Kistovich Konstantin Pokazeev Tatiana Chaplina

Ocean Acoustics

Springer Textbooks in Earth Sciences, Geography and Environment

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Anatoly Kistovich • Konstantin Pokazeev Tatiana Chaplina

Ocean Acoustics

123

•

Anatoly Kistovich All-Russian Scientific Research Institute for Physical-Engineering and Radiotechnical Metrology Moscow, Russia

Konstantin Pokazeev Faculty of Physics Lomonosov Moscow State University Moscow, Russia

Tatiana Chaplina Institute for Problems in Mechanics RAS Moscow, Russia

ISSN 2510-1307 ISSN 2510-1315 (electronic) Springer Textbooks in Earth Sciences, Geography and Environment ISBN 978-3-030-35883-9 ISBN 978-3-030-35884-6 (eBook) https://doi.org/10.1007/978-3-030-35884-6 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The World Ocean is the last reserve of human development, the World Ocean resources, which are not fully been using yet, are able to provide the implementation of the program for the survival of humanity—the concept of sustainable development. The oceans are also the most important element in the biosphere, ensuring its stability, the oceans play a huge role in the Earth’s climate system and global warming. That is why the study of the oceans is the most important scientific task of mankind. One of the powerful tools for studying the ocean is acoustics (hydro-acoustics), so familiarity with the basics of hydro-acoustics is necessary for everyone who studies the ocean. The methods of studying acoustic processes described in the book can be successfully applied in the study of other types of wave processes in the ocean. The book includes the main results of theoretical and experimental studies on hydro-acoustics, conducted in recent years by leading experts in this field. The book indicates further directions and prospects for the study of the ocean by acoustic methods. The book pays great attention to simplified mathematical models of acoustic processes in the ocean and, at the same time, a physically rigorous description of the main acoustic phenomena in the ocean. The theoretical and experimental material presented in the monograph includes the most recent results of scientific research and corresponds to the modern world level. Each chapter of the book includes examples of using the studied mathematical models of acoustic phenomena in solving practical problems. A presentation of theoretical results is provided with the detailed derivation, which includes, if possible, all intermediate mathematical calculations. This allows the reader to follow all the important steps of the methods used and to perceive the final formulas as a logical conclusion of the realized idea. Moscow, Russia

Anatoly Kistovich Konstantin Pokazeev Tatiana Chaplina

v

Introduction

The world’s oceans, in their continuous dynamics, are permeated by a variety of wave movements, among which the acoustic waves that exist in seawater due to its compressibility are particularly prominent. The science of underwater sound, its emission, propagation, absorption, dispersion, reflection and reception is called ocean acoustics or hydro-acoustics. The role of this science is very great, because of all the energy types discovered so far, the sound energy spreads in the water at the greatest distance [1–4]. The invention and use of sonar have revolutionized hydrography. Bathymetric maps of the World Ocean have been obtained, and deep water depressions and ridges have been discovered. Hydroseismic methods have enabled the study of the geological structure of the ocean floor. Hydro-acoustic methods are used to study currents, surface and internal waves, sea ice, water mass structure. Hydro-acoustics is widely used to solve applied problems, such as fishing, mineral prospecting and navigation. The huge role of hydro-acoustics in scuba diving. The history of the submarine fleet is inextricably linked to the development of hydro-acoustics. The conditions of acoustic wave propagation in the ocean have a number of specific features due, on the one hand, to the properties of the aquatic environment of the ocean itself, and, on the other hand, to the properties of the boundary media, i.e., the atmosphere and bottom. Sound is the longitudinal elastic waves, and the speed of its propagation is determined by the ratio: .pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ~=q ¼ 1 Kr q c¼ v ð1Þ ~ is the adiabatic modulus of volumetric elasticity where q is the seawater density, v and Kr is the adiabatic coefficient of compressibility of seawater, at the same time ~ ¼ Kr1 . v Values Kr and density q depend on temperature T, salinity S and pressure p. Therefore, the speed of sound in the water will be a function of these characteristics and can be recorded in a form: ð2Þ c ¼ c0 þ D cT þ D cS þ D cp þ D cSTp

vii

viii

Introduction

It is Wilson’s formula where c0 ¼ 1449:14 m/s is the sound velocity at T ¼ 0  C, S ¼ 35‰, p ¼ 1 atm and D cT ; D cS ; D cp ; D cSTp are corrections due to variations of T, S and p from above announced values. Wilson’s formula is valid for the following intervals of change in values T, S and p: 40  C\T\300  C, 0\S\37‰, 1\p\103 kG/sm2. Along with Wilson’s formula, there are a number of other empirical formulas in which corrections are described by polynomials of different degrees of pressure, temperature and salinity. One of these formulas looks like c ¼1449:2 þ 4:6T  0:055T 2 þ 0:000297T 3 : þ ð1:34  0:010TÞðS  35Þ þ 0:016z A change in water temperature per 10  C leads to a change in sound velocity of 3 m/s, a change in salinity S per 1‰ leads to a change of c by 1.2 m/s and a pressure increase or decrease of 1 atm changes by 0.2 m/s. The greatest change in the speed of sound in the sea causes a change in the temperature of the seawater. The increase in the sound velocity when the temperature changes by one degree depends on the temperature value (see Table 1). In real-world seawater conditions, the sound velocity varies within c  1440 . . . 1540 m/s. The first scientific mention of sound propagation in the water is already in the works of Leonardo da Vinci, who wrote: “If you, being at sea, put into the water a hole in the pipe, and the other end of it is attached to the ear, you will hear the ships going far away.” With this remark, Leonardo anticipated the modern practice of underwater eavesdropping of ships and one can only be surprised by his foresight, as the ships of his era are rowing and sailing ships, which were much more silent than the modern fleet. After Leonardo da Vinci, it will be more than three centuries before Arago and then Colladen will once again notice that the water conducts sound perfectly and that this circumstance can be used for practical purposes. The vertical sound velocity gradient in the ocean is usually three orders of magnitude greater than the horizontal gradient. An exception is the convergence of warm and cold currents, where the horizontal gradient can be commensurate with the vertical sound velocity gradient. Therefore, for many practical applications, the ocean can be seen as a flat stratified environment in which properties change only vertically. However, in some areas of the World Ocean currents, frontal zones, vortex systems and internal waves cause significant disturbance of the horizontally layered nature of sound propagation. In these areas, the structure of the sound field is very complex, with significant spatial and temporal fluctuations of sound signals.

Table 1 Temperature dependence of sound velocity DT, °C DcT , m/s

5 4, 1

10 3, 6

15 3, 1

20 2, 7

25 2, 4

30 2, 1

Introduction

ix

This dependence of acoustic fields on environmental properties is used to investigate the characteristics of hydrophysical fields. The thin vertical structure of ocean waters, which is widespread almost everywhere, also has a significant impact on the structure of acoustic fields, especially at high frequencies. In the process of sound propagation in the ocean, the sound energy is absorbed and dissipated. Quantitative process of sound absorption in the ocean can be described by the formula: ~ pðRÞ ¼ pð0ÞebR ð3Þ ~ is the absorption coefficient Here, R is the distance from the sound source, b ~ ¼ f ðTÞ. (db/km) and it depends on the temperature of the medium, i.e., b Absorption is caused by air bubbles, marine organisms with gas inclusions, bottom and viscous dissipation. As is known, there is a first (shear) kinematic viscosity and a second (volumetric) kinematic viscosity. Since the volumetric kinematic viscosity is approximately three times greater than the first one for the marine environment, the effects of viscous volumetric attenuation play a role in the high-frequency range. At the same time, the effects of sound wave attenuation in the process of its propagation in a real environment are primarily determined by the scattering on inhomogeneities and spatial divergence of the wave (except for the flat one). Compared to these mechanisms, the effect of the viscous attenuation mechanism is negligibly small and can therefore be neglected. Usually, to describe the effect of gas inclusions on the attenuation of sound, an example is given with two glasses of sparkling wine filled to the edges—one of the wine has stood still, the other—freshly poured. The bell at the first glass is stronger than at the second glass, as the gas bubbles strongly dampen the sound. The attenuation of sound as it spreads in the ocean is due not only to the process of its absorption, but also to the process of dispersion. The law of sound dissipation in the sea looks pðRÞ ¼ pð0ÞevR ð4Þ where v is the scattering coefficient (dB/km). Reasons for scattering include air bubbles, which are a product of plankton life and during waves and storms; small-scale temperature irregularities; sea surface roughness caused by excitement; accumulations of small marine animals forming sound dispersive layers in the depths of the World Ocean; and fluctuations in the refractive index of sound rays. The greatest scattering and absorption of sound is caused by “resonant” bubbles, where the frequency of their own radial vibrations coincides with the frequency of sound. The resonance frequency of the bubble f0 with fixed radius r0 (in centimeters) at a depth z (in meters) below the sea surface can be determined by the approximate formula 327 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f0 ¼ 1 þ 0:1 z; Hz: r0 Therefore, the study of the distribution of air bubbles by radius depending on the hydrometeorological conditions, depth is of considerable interest.

x

Introduction

Sound dispersion layers formed by the accumulation of small marine animals have a considerable length. They are characterized by daily migration with depth. At dawn, they descend to a depth of 300–600 m, with sunset they rise to a depth of 100–150 m. It is now considered that the sound dispersion in such layers is due to the resonance vibrations of small fish bubbles of several cm in size. The resonance frequency of swimbladders depends on their size and depth. In underwater acoustics, it is important to have sound scattered in the opposite direction, i.e., in the direction of the sound source. The backscatter characteristic is based on a backscatter coefficient equal to the ratio of the acoustic power dissipated by a single surface per unit of solid angle to the intensity of the incident wave. The combined effect of sound absorption and dissipation leads to attenuation of sound pressure or sound attenuation: ~ pðRÞ ¼ pð0ÞeKR ð5Þ ~ is attenuation coefficient (dB/km). K According to experimental data and theoretical calculations, sound attenuation depends on the frequency of sound oscillations: The sound of low frequencies attenuates less, i.e., spreads over long distances. Decrease in 10 times at the frequency 100 Hz takes place at a distance of 8333 km! No other type of radiation has such a range. The attenuation of acoustic waves in seawater is stronger than in fresh water. Usually only the total effect of sound absorption and dissipation—attenuation—is measured. Separation of absorption and dispersion is only possible in individual cases. The effect of various factors on the attenuation of sound does not lend itself to a strict theoretical description, therefore, experimental methods of investigation of the attenuation of acoustic waves are widely used and therefore, there are many empirical formulas for determining the attenuation coefficient in different frequency ranges of acoustic waves. Detailed questions of hydrosphere physics are covered in [5, 6]. Since acoustics is a scientific discipline, the description of the diversity of these processes and phenomena is based on the general scientific approach, which has its roots in the equations that determine the laws of sound propagation and emission in the marine environment. The ocean environment is the most complex thermodynamic system, and the study of its acoustic properties is a very difficult task. Two approaches are used to address acoustic issues. On the one hand, since acoustics is a subsection of hydro-aerodynamics, the equations underlying its mathematical model are a particular case of the fundamental equation system of hydrodynamics. From another point of view, successively in space and time of medium compression and rarefaction can be considered as manifestations of some elastic properties of the object in which the sound wave propagates. And then the acoustic equations can be obtained as a result of limiting transition from the general equations of elastic oscillations for the case of liquid medium. Both of these approaches eventually produce the same result, but their application shows a close relationship between the adjacent sections of physics.

Introduction

xi

References 1. Egorov NI (1974) Physical oceanography. L.: Gidrometeoizdat, p 455 2. Egorov NI (1978) In: Doronin YuP (ed) Ocean physics, p 296 3. Brekhovskikh LM, Goncharov VV (1982) Introduction to continuum mechanics. (Appendix to the theory of waves). M.: Science, p 336 4. Kistovich AV, Pokazeev KV (2006) Introduction to the acoustics of the ocean. Tutorial. M.: LLC MAX Press, p 136 5. Anisimova EP, Pozkadeev KV (2002) Introduction to the physics of the hydrosphere. M.: Physical Faculty of Moscow State University, p 171 6. Trukhin VI, Pokazeev KV, Kunitsyn VE et al (2002) Fundamentals of environmental geophysics. Lan publishing house, p 384

Contents

1 The Mathematical Model of Acoustic Processes . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 12

2 Acoustic Phenomena in the Language of Elasticity Theory . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 22

3 General Properties and Character Types of Sound Waves . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 41

4 Plane Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 57

5 Geometric Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 70

6 Ray Description of the Sound Field in Inhomogeneous Media . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 85

7 Wave Description of the Sound Field in Inhomogeneous Media . . . . 87 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8 Sound Wave Ref lection from the Ocean Floor . . . . . . . . . . . . . . . . . 105 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9 Scattering of the Sound by Surface and Ocean Floor Irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Methodical Instructions and Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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The Mathematical Model of Acoustic Processes

Acoustics as a science studies the movements of the compressible liquid, namely oscillatory movements, which are called sound (acoustic) waves. An acoustic wave is an alternating compression and rarefaction of the medium through which it propagates. The main local characteristics of acoustic waves are determined by such properties of the medium as temperature, pressure and concentration of impurities. The global properties of the propagation of these waves depend on the spatial and temporal distributions of the above factors, as well as on the shape of the medium boundaries, if any. Before embarking on the study of the ocean as an acoustic object, it is necessary to consistently study the properties of the simplest acoustic phenomena using examples of simplified marine models. First, we will outline the area of acoustic processes, the description of which will be the main focus. Let the initially homogeneous and isotropic medium is characterized by density q0 and pressure p0 . Acoustic wave propagation in such an environment is characterized by spatial and temporal alternations of the increased and decreased (relative p0 ) pressure areas. It is clear that such local pressure deviations entail local changes in the density of the medium. Let’s consider that in some small area of space, the pressure has changed by a value Dp, and the corresponding change in density was a value Dq. The properties of the studied acoustic processes in the environment are determined by the nature of the relationship between the values Dp and Dq. Let’s assume that in the most general way, this connection looks like Dq ¼ f ðDp; p0 ; q0 Þ

ð1:1Þ

If the values of pressure Dp and density Dq deviations are arbitrary, the ratio (1.1) is not informative and it is difficult to obtain any constructive results. In order to take the next step in our study, we will impose the first restriction: We will consider such acoustic phenomena, in which the values of deviations Dp and Dq are © Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_1

1

2

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The Mathematical Model of Acoustic Processes

significantly lower than the initial values p0 and q0 , that is, there are weak disturbances of the equilibrium state of the environment. In the formal mathematical language, it means fairness of relations jDp=p0 j  1, jDq=q0 j  1. Using the proposed restriction, we will place the right side (1.1) in the Taylor series of deviations Dp: Dq ¼ f0 ðp0 ; q0 Þ þ aðp0 ; q0 ÞDp þ bðp0 ; q0 ÞðDpÞ2 þ cðp0 ; q0 ÞðDpÞ3 þ   

ð1:2Þ

Guided by the principle of invariability of the state of the thermodynamic equilibrium system in the absence of effects on it, we assume that the deviations of density Dq are equal to zero at Dp ¼ 0. Substitution of values in (1.2) Dq ¼ 0, Dp ¼ 0 leads to consequence f0 ðp0 ; q0 Þ ¼ 0. As a result, the ratio (1.2) takes the form Dq ¼ aðp0 ; q0 ÞDp þ bðp0 ; q0 ÞðDpÞ2 þ cðp0 ; q0 ÞðDpÞ3 þ   

ð1:3Þ

The constants a, b, c, etc., included in (1.3) are the acoustic parameters of the environment. If the acoustic properties of the medium are such that for the possible variations of pressure Dp the ratio is valid

jaðp0 ; q0 Þj 
bðp0 ; q0 ÞDp þ cðp0 ; q0 ÞðDpÞ2 þ   


ð1:4Þ

then Eq. (1.3) can be further simplified, and its approximate record can be used Dq ¼ aðp0 ; q0 ÞDp

ð1:5Þ

Ratio (1.4) is the second constraint that we used in the transition to Eq. (1.5). The strict mathematical conclusion of the ratio (1.5) requires the establishment of a relationship between dimensionless values of relative deviations Dp=p0 and D q=q0 , those resulting from the acoustic influence on the environment. However, in a completely similar way, the result will remain the same. In general, there may be multiple sources of acoustic signals. In order to stay within the framework of the proposed environmental model, a third restriction is imposed, the essence of which is that if for each individual source the ratio (1.5) is executed, the total impact is also subject to this law, i.e., Dq ¼ Dq1 þ Dq2 þ Dq3 þ    ¼ aðp0 ; q0 ÞDp1 þ aðp0 ; q0 ÞDp2 þ aðp0 ; q0 ÞDp3 þ    ¼ aðp0 ; q0 ÞðDp1 þ Dp2 þ Dp3 þ   Þ ¼ aðp0 ; q0 ÞDp

ð1:6Þ

1

The Mathematical Model of Acoustic Processes

3

In this case, of
P as a consequence

Pthe adopted
model, it is necessary to fulfill the conditions
i Dpi =p0
 1 and
i Dqi =q0
 1; otherwise, the approximation of weak perturbations of the equilibrium state of the system is disturbed. The meaning of these limitations is easily understood by a simple example. Let each of the acoustic sources separately leads to the fact that at some point in space at a given time, the density of the environment increases by one percent. In other words, Dqi =q0 ¼ 0:01 for every source. If these sources, for example, are present 50, their total impact will result in an environment density that is one and a half times higher, so DqR =q0 ¼ 0:5. So we are going to go beyond the first restriction adopted. Equation (1.5) also shows that if the pressure deviation Dp changes by k times, the density deviation Dq changes by the same number of times. The formal mathematical record of this property, as well as the properties given by the expression (1.6), is Dq ¼ f ðDpÞ;

f ðkDpÞ ¼ k f ðDpÞ

f ðDp1 þ Dp2 Þ ¼ f ðDp1 Þ þ f ðDp2 Þ

ð1:7Þ

The ratios (1.7) determine the linear dependence of density perturbations on pressure perturbations. Thus, the subject of further study will be a subsection of general acoustics called linear acoustics. Before we start to get the basic linear acoustic ratios, let’s look again at Eq. (1.5). The analysis of dimensions of this equation shows that the parameter a is the inverse square of some velocity c constant in the whole space occupied by the medium. Let’s try to figure out what the speed is and what it describes. In other words, which physical object or phenomenon should be attributed this speed c? As it is known, the motion of a liquid (or gaseous) medium is described by a set of parameters, which include the velocity field vðx; y; z; tÞ (in the Euler description). And this field is the only kinematic value influencing the character of motion. Is it possible to associate an undefined speed c with the hydrodynamic field of speed v? Turns out it is not! Suppose it is a hydrodynamic speed. Then, since c is constant in all space, it would follow from (1.5) that acoustic phenomena are such processes in which all particles of the medium move either in circles with a single center (which requires the presence of infinitely large force near such a center), or along a single chosen direction, that is, they are a constant current (or wind). The first variant is physically unrealizable, and in the second variant, there is no spatio-temporal change of areas of the raised and lowered pressure as in the system of coordinates moving together with a stream of particles, all points of the medium are thermodynamically equivalent. Since, according to the above, the velocity c is not hydrodynamic, it is reasonable to assume that it is the velocity of the actual acoustic phenomena. Simplified, this is the rate of propagation of the picture of compression and discharge of the environment. However, such a statement still needs to be proved. It is impossible to do this with one Eq. (1.5). Here, it is necessary to use equations describing the movement of the liquid medium.

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The Mathematical Model of Acoustic Processes

Since we are interested not in the most general type of equations of motion, but in their form, which refers to the problem of sound propagation, let’s start with the fact that we derive equations describing the acoustic processes in the environment. As mentioned in the introduction, two approaches are acceptable: a hydrodynamic approach based on common concepts of aeromechanics and an approach based on an elastic model of the marine environment. Let’s start with the hydrodynamic approach. The basic laws of mechanics of a moving continuous medium are described by the equations of change of its mass, diffusion of impurities, impulse and energy transfer. The physical properties of the medium itself, and not the hydrodynamic currents developing in it, are described by the equation of state, which connects the thermodynamic parameters of liquid or gas. Often, the equation of state is represented in the form of the dependence of the density of the medium on other thermodynamic variables, for example, in the form q ¼ qðp; fSn g; rÞ

ð1:8Þ

where fSn g—mass concentrations of impurities and r—specific entropy of the environment. Sometimes the ratio (1.8) of specific entropy is replaced by temperature. If we now add to the equations of motion and equation of state (1.8) all sorts of conditions, which must be satisfied by the entered physical fields (speed, pressure, gradients of impurities, etc.) on the solid and free boundaries of the medium, there will be a set of relations describing virtually all possible phenomena in solid media —convection, internal and surface waves, diffusion of impurities, etc. Thus, a set of equations will be created that is extremely inconvenient for studying the basics of acoustic phenomena in the marine environment. Since any scientific approach requires progress from simple to complex, we will simplify the system of equations of motion, rejecting those factors on which the existence of sound waves does not depend, as well as satisfy the condition of linearity of the ongoing acoustic processes. First of all, let’s ignore the Earth’s rotation and Archimedean force. Secondly, the medium will be considered clean, without any impurities. This makes it possible not to take into account the processes of impurities transfer, but also to exclude from the equation of state (1.8) the dependence of density on their mass concentrations fSn g. Thirdly, since at this initial stage we are interested in the problems of sound propagation rather than its radiation, we will consider that there are no sources of forces, mass, impurities and energy in the environment. The assumption that there are no heat fluxes in the medium makes it possible to completely exclude from consideration the energy evolution equation, which within the framework of the accepted approximations becomes a combination of equations of motion and mass preservation.

1

The Mathematical Model of Acoustic Processes

5

Finally, limiting ourselves to the framework of linear acoustics, we will consider only small sound oscillations in intensity, that is, refusing to study shock waves, linearize the equations of motion. In this case, the mathematical model of sound waves in the medium takes the form @~ q @v rp þ q0 r  v ¼ 0; ¼ þ mDv þ ðl þ m=3Þrr  v @t @t q0 q ¼ qðp; rÞ

ð1:9Þ

where q0 is the density of the medium in the absence of sound waves, m ¼ g=q0 and l ¼ f=q0 are the first and second kinematic viscosity; v is the hydrodynamic field of velocity. In accordance with the traditional acoustic terminology, we will call the velocity v the oscillatory velocity in the future. The physical nature of the oscillatory velocity at any point in the space ðx; y; zÞ is the velocity of a liquid particle that is currently at a specified point. Further simplification of the model (1.9) is related to the study of the equation of state. Let’s write down the differential of this equation  dq ¼

   @q @q dp þ dr: @p r @r p

Since the heat fluxes are considered negligibly small in the adopted model, due to their deterministic nature, acoustic processes are characterized by almost constant ~ disturbances due to specific entropy ðdr ¼ 0Þ, and small pressure ~p and density q the sound wave allow the use of decomposition ~ p ¼ p0 þ ~p; q ¼ q0 þ q; so that the differential analog of the equation of state is simplified and reduced to d~ q¼

  @q 1 d~p  d~ q ¼ 2 d~p; @p r c

c2 ¼

 1 @q @p r

ð1:10Þ

where c is the estimated speed of sound in the environment. Comparison of Eqs. (1.5) and (1.10) shows that the parameter a introduced earlier is nothing but the degree of condensation of the medium in a small volume of fixed particles with the change in pressure applied to it, taking into account the additional condition: the absence of heat exchange between the allocated volume and the surrounding liquid. Since the density q in the allocated volume V is determined by the ratio q ¼ mN=V (where m is mass of a single particle and N is number of particles), so dq ¼ mN dV=V 2 ¼ q d ln V and a ¼ qðd ln V=dpÞr . The value ðd ln V=dpÞr determines the adiabatic relative compressibility of the

6

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The Mathematical Model of Acoustic Processes

medium when applied overpressure. Thus, the parameter a is an elastic characteristic of the medium. The use of the ratio (1.10) makes it possible to link changes in the time of perturbations of density and pressure @~ q 1 @~p ¼ @t c2 @t

ð1:11Þ

Substitution of (1.11) in the first equation of the system (1.9) leads to the equation @~p þ q0 c 2 r  v ¼ 0 @t

ð1:12Þ

which explicitly defines the relationship between pressure changes in the medium and its compressibility. Replacement of the equation of state in the system (1.9) by (1.12) gives the final form to the equations of motion of the viscous compressible medium without impurities @v r~p ¼ þ mDv þ ðl þ v=3Þrr  v @t q0 @~p @~ q þ q0 c2 r  v ¼ 0; þ q0 r  v ¼ 0 @t @t

ð1:13Þ

The structure of Eqs. (1.13) in determining the hydrophysical fields of acoustic processes requires the following sequence of actions. First, the first two equations determine the pressure ~p and oscillatory velocity fields of v the sound wave, and ~. then the third equation determines the behavior of the density perturbation q In the previous presentation, the term “sound wave” was constantly used without explaining the reasons for attributing the wave character to the type of motion of the medium under study. The rationale for using this term is based on the consideration of the first two equations of the system. Application to the first equation of the system (1.13) of the operation r (divergence operations) using the known property of vector algebra r  Da  r  ðrr  a  r  r  aÞ ¼ ðr  rÞr  a  ðr  rÞr  a ¼ Dðr  aÞ and to the second equation the operation of time differentiation @=@t reduces them to the equations of the form

1

The Mathematical Model of Acoustic Processes

7

@A D~p ¼ þ ðl þ 4m=3ÞDA @t q0 @ 2 ~p @A ¼ 0; A ¼ r  v þ q 0 c2 2 @t @t

ð1:14Þ

In the absence of dissipation in the medium, the exclusion of the first two equations of magnitude @A=@t leads to the equation for pressure perturbation @ 2 ~p  c2 D~p ¼ 0; @t2

ð1:15Þ

which is nothing but a wave equation. In the one-dimensional case, when D ¼ @ 2 =@x2 , the solution (1.15) looks кoтopoe являeтcя ничeм иным, кaк вoлнoвым ypaвнeниeм. ~p ¼ f1 ðx  ctÞ þ f2 ðx þ ctÞ where f1 and f2 are arbitrary functions. The first component of the solution, the function f1 , describes the pressure perturbations running in the positive direction of the axis x with the speed c, and the second component, the function f2 , describes the perturbations spreading in the negative direction with the same speed. It is this fact—the propagation of perturbations with the speed c—that allows us to attribute a certain value c ¼ ð@q=@pÞ1=2 to r the physical meaning of the speed of sound. Let’s consider a particular case of a sound wave, a harmonic wave, the temporary changes of characteristics of which occur according to the law expði x tÞ. Substitution of the form function ~p ¼ QðxÞ expði x tÞ into one-dimensional analog of Eq. (1.15) reduces it to the equation k2 QðxÞ þ

d2 QðxÞ ¼ 0; dx2

k¼

x c

ð1:16Þ

The solution of this equation does not cause any difficulties and is presented in the form QðxÞ ¼ Q þ expðikxÞ þ Q expðikxÞ which allows the solution (1.15) to be shaped ~p ¼ Q þ expðiðkx  x tÞÞ þ Q expðiðkx  x tÞÞ describing two sinusoidal waves running in opposite directions of the axis x. The second expression in (1.16) ðk ¼ x=cÞ is the dispersion relation of one-dimensional sound waves. The values of phase cph ¼ x=k and group cgr ¼ @x=@k velocities of such waves coincide and are equal to the previously entered sound velocity c.

8

1

The Mathematical Model of Acoustic Processes

In the three-dimensional case, the solution (1.15) in the form of a plane harmonic wave is recorded in the form ~p ¼ ~p0 expðiðk  r  x tÞÞ

ð1:17Þ

where k is the wave vector, r is the radius-vector of the observation point and ~ p0 is the amplitude of the wave. Substitution of (1.17) in (1.15) determines the relationship between frequency and wave vector k2 ¼

x2 ; c2

k2 ¼ kx2 þ ky2 þ kz2

ð1:18Þ

which are the components of the wave vector k in the Cartesian coordinate system kx ðx; y; zÞ. Let’s define the vector of group wave velocity (1.17). According to the definition, the i component of this vector is defined by the ratio cgr i ¼

@x @ki

ð1:19Þ

To obtain the right side (1.19), we differentiate the dispersion ratio (1.18) by ki : 2ki ¼

2x @x c2 @ki

ð1:20Þ

Since, as follows from (1.18), x ¼ cjkj, it follows from (1.19, 1.20) cgr i ¼

@x cki ¼ @ki jkj

As a result, the vector of group velocity will be determined by the ratio    c  k kx ; ky ; kz ¼ c cgr ¼ cgr x ; cgr y ; cgr z ¼ jkj jkj

)
cgr
¼ c

ð1:21Þ

which indicates that the direction of the group velocity of the sound wave coincides with the direction of the wave vector, and its value is equal to the velocity of the sound. In order to finally find out the nature of sound waves, it is necessary to define the expression for the velocity field for the given form of sound pressure (1.17). For this purpose, we can use the first equation of the system (1.13) in a non-dissipative approximation ðl ¼ m ¼ 0Þ, specifying the representation for the velocity v as

1

The Mathematical Model of Acoustic Processes

9

  v ¼ vx ; vy ; vz expðiðk  r  x tÞÞ

ð1:22Þ

Substitution of (1.17) and (1.22) in Eq. (1.13) leads to the vector formula v¼

~p0 k expðiðk  r  x tÞÞ q0 x

ð1:23Þ

Thus, sound waves are longitudinal ðvjjkÞ and isotropic (i.e., their properties do not depend on the direction) perturbations of the medium density. So far, the sound waves have been considered in a non-dissipative environment. Let’s study how dissipation affects on the properties of acoustic waves. First, let’s derive the dispersion equation of waves in the medium with dissipation, for which we will return to the system (1.13) and substitute in it the solution in the form of a plane wave, that is, let’s set the representation of physical fields of the form ~¼q ~0 expðiuÞ; vi ¼ vi0 expðiuÞ; u ¼ k  r  x t ~p ¼ ~p0 expðiuÞ; q

ð1:24Þ

Substitution of (1.24) in (1.13) and elimination of the exponential factor expðiðk  r  x tÞÞ reduce the system (1.13) to a system of linear algebraic equa~0 , vi0 tions with respect to amplitudes ~p0 , q  ixvi0 ¼ iki q1 p0  m k2 vi0  ðl þ m=3Þki k  v0 0 ~  x~p0 þ q0 c2 k  v0 ¼ 0;

x~ q 0 þ q0 k  v 0 ¼ 0

the condition of non-trivial solvability of which (equal to zero of the main determinant) leads to a dispersion relation for sound waves in the medium with dissipation    2 k 1  ix~m c2  x2 =c2 ðx þ i m k2 Þ2 ¼ 0;

~ m ¼ l þ 4m=3

ð1:25Þ

It can be seen that the dispersion equation (1.25) is divided into two independent equations ðx þ i m k2 Þ2 ¼ 0    k2 1  i x~m c2  x2 =c2 ¼ 0

ð1:26Þ

pffiffiffiffiffiffiffiffiffiffi The solution of the first equation (1.26) k ¼ i x=m describes a twofold degraded boundary layer (the periodic boundary layer of Stokes) that exists in the environment only when there are solid boundaries oscillating with frequency x. The second equation is a dispersion equation of sound waves in a dissipative medium. In the absence of dissipation ð~m ¼ 0Þ, it passes into Eq. (1.18) for sound waves in a non-contiguous medium.

10

1

The Mathematical Model of Acoustic Processes

Since, as follows from the second equation (1.26), 2

@x ki ðc2  i x~mÞ ¼ ¼ ki f ðxÞ @ki x ðc2  i x~m=2Þ2 the group velocity is directed to the wave vector k. For the same reason, the field of oscillatory velocity v also has this property. Thus, as in the non-dissipative case, sound waves are longitudinal and isotropic. Now let’s find out how badly the sound waves fade out due to the viscous effects. Let a plane harmonic wave propagate along a direction, e.g., x. Well, then kx ¼ k, ky ¼ kz ¼ 0. We need to determine the actual and imaginary parts kx . The solution of the second equation (1.26) leads to the expression   x 1 x~ m=c2 kx ¼ k ¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ i a c 2 1 þ ðx ~m=c2 Þ2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a¼

1þ

1 þ ðx ~m=2c2 Þ2

Let’s make a numerical assessment of the values included in the obtained ratio. Using the value of the first (shear) kinematic viscosity for the marine environment ðm 106 m2 =sÞ and the fact that in the wide diapasones of temperature and pressure l 2:81m (the one of twenty abnormal features of water), the value of ~m ¼ l þ 4m=3 can regard which equals to 4  106 m2 =s. The speed of sound has m=c2 an order of magnitude c 1:5  103 m/s. Then, the dimensionless complex x~ 12 1 is comparable with the unit only for frequencies x higher than 10 s . Since such frequencies of sound oscillations are not observed in nature, we get, with good approximation, the expression for the wave number kx ¼ k

 x x~m=c2 1þi : c 2

ð1:27Þ

It can be seen from (1.27) that the coefficient of linear attenuation of the sound wave due to viscous shear stresses is determined by d¼

 x2 ~m  16 10 . . .106 m1 ; 3 2c

  x ¼ 1. . .105 s1

The obtained result means that the attenuation of the sound wave in the presented frequency range due to the viscosity can be safely ignored. At the same time, the effects of sound wave attenuation in the process of its propagation in a real environment are primarily determined by the scattering on inhomogeneities and spatial divergence of the wave (except for the plane one). Compared to these mechanisms, the viscous attenuation mechanism is negligibly

1

The Mathematical Model of Acoustic Processes

11

small and therefore will not be considered in the future. In the future, all environments under study will be considered non-dissipative. The system of linear acoustic equations (1.13) is simplified and takes the form of @v r~p ¼ ; @t q0

@~p þ q0 c2 r  v ¼ 0; @t

@~ q þ q0 r  v ¼ 0 @t

ð1:28Þ

Applying the operation r to the first equation (1.28) leads to the result that, r  v ¼ 0, it is possible to enter the u velocity potential according to v ¼ ru

ð1:29Þ

Substitution of the ratio (1.29) in the system (1.28) makes it look like ~p ¼ q0

@u ; @t

@~p þ q0 c2 Du ¼ 0; @t

@~ q þ q0 Du ¼ 0 @t

ð1:30Þ

Excluding the pressure from the first two equations of the obtained system leads to the fact that the velocity potential u must satisfy the wave equation @2u  c2 Du ¼ 0; @t2

ð1:31Þ

which is exactly the same as the wave equation for the pressure ~ p of the sound wave. Comparison of the second and third equations of the system (1.30) leads to a correlation between pressure and density perturbations in the sound wave: ~¼ q

~p c2

ð1:32Þ

The expressions (1.28–1.32) are valid in non-viscous liquids for all types of waves—plane, spherical, cylindrical, etc—and have a comprehensive character. Thus, the system of equations underlying the theory of acoustic phenomena connects acoustic pressure ~p and velocity potential u ~p ¼ 

@u ; @t

@~p þ q0 c2 Du ¼ 0 @t

ð1:33Þ

after the solution of which the density perturbations caused by sound vibrations and the field of vibrational velocity v are determined by ~¼ q

~p ; c2

v ¼ ru

ð1:34Þ

12

1

The Mathematical Model of Acoustic Processes

Ratios (1.33–1.34) are the basis of mathematical models used to describe the radiation and propagation of sound in the following sections. A more detailed description of the acoustic phenomena based on the hydrodynamic approach is presented, for example, in the manuals [1, 2].

References 1. Landau LD, Livshits EM (1986) Theoretical physics, vol VI. Hydrodynamics. Science, 736 p 2. Brekhovskikh LM, Lysanov YP (1982) Theoretical basics of ocean acoustics. L: “Gidromemoizdat”, 264 p

2

Acoustic Phenomena in the Language of Elasticity Theory

Let’s now turn to the second approach, in the framework of which the physical essence of acoustic phenomena, consisting in the spatio-temporal change of compressions and depressions, is directly related to the manifestation of elastic properties of the medium. Since any compression or rarefaction is accompanied by deformation, internal stresses arise in the medium, which are related to the value of deformation by ratios including various coefficients so-called elastic characteristics of the medium. The change of internal deformations from negative to positive ones, as well as the spatial and temporal dynamics of the internal stress field, are directly related to the propagation of compression and depression in liquid or gas. In this regard, it is useful to consider sound waves as the ultimate case of elastic waves arising in a solid medium. Since the deduction of the basic equations of elastic wave processes is based on the use of concepts that differ from the basic accepted concepts of acoustics, it is necessary to give the necessary definitions of the values that make up the terminology of the elasticity theory [1]. The deformation vector n is the displacement of a marked point in the body from the position r it occupied before the deformation began to a point r0 : n ¼ r0  r

ð2:1Þ

The component record of the ratio (2.1) in the orthogonal coordinate system (e.g., Cartesian) is ni ¼ x0i  xi

ð2:2Þ

At small deformations of the elastic body, the relative changes in the distances between the two close selected points will also be small. A set of relative elongation in the direction of xk at low deformation ni determines the strain tensor, the explicit form of which is given by the expression © Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_2

13

14

2 Acoustic Phenomena in the Language of Elasticity Theory

nik ¼


 1 @ni @n þ k ; 2 @xk @xi

nik ¼ nki

ð2:3Þ

Any deformations of an elastic body that was previously in a state of equilibrium lead to changes in the acting forces of intermolecular interaction. As a consequence, changes in the state of the body lead to changes in its basic thermodynamic characteristics, including its free energy F. The change in free energy as a result of the relative elongation of the body caused by small deformations determines the amount of work done on the body. Since the relative elongation is not a scalar but a tensor (2.3), the change in free energy when the corresponding element of the strain tensor changes determines the element of the stress tensor
rik ¼

@F @nik

 T

¼ rki

ð2:4Þ

where the lower index T in the ratio (2.4) indicates that the derivative is calculated at a constant medium temperature. Thus, the value of the element rik is the work performed on the body against the forces in the direction of xk arising under deformation in the direction of xi . The force itself is defined by the ratio known in mechanics (indirect origin) as the speed of change of work with distance, i.e., =i ¼

@rik @2F ¼ @nik @xk @xk

ð2:5Þ

The ratio (2.5) determines the force that acts on the infinitesimal element of the medium located at the point of r. Since under the action of this force the body at this point is subject to deformation n (i.e., the medium element is subject to displacement n), Newton’s second law for the elastic body takes the form q

@ 2 ni @rik ¼ @t2 @xk

ð2:6Þ

where q is the density of the environment at the point of r. Until free energy F is undetermined, the force (2.5) and therefore the right side of the equation of motion (2.6) cannot be determined. In order to write this equation in explicit form, it is necessary to set the model of free energy of the weakly deformed body. Since the free energy value cannot depend on the sign of relative elongation, and the resulting force, within the framework of the model of small deformations, must be linearly dependent on elongation, there is only one possibility to compose a free energy function F ¼ F0 þ

k 2 n þ ln2ik 2 ii

ð2:7Þ

2

Acoustic Phenomena in the Language of Elasticity Theory

15

where F0 is some constant, the values k and l are called Lamé coefficients, k is the comprehensive compression coefficient and l is the shift elastic coefficient. Substitution (2.7) in the ratio (2.4) determines explicitly the stress tensor
rik ¼

@F @nik

 T

¼ k npp

@nqq @npq þ 2l npq @nik @nik

ð2:8Þ

¼ k npp dqi dqk þ 2l npq dpi dqk ¼ k npp dik þ 2l nik where the repetitive index p is supposed to be summed up. Now let’s determine the force by substituting (2.8) in (2.5): @npp @rik @n ¼ kdik þ 2l ik @xk @xk @xk
2  2 @ np @ ni @ 2 nk ¼ kdik þl þ @xk @xp @xk @xi @x2k
 @ @ ¼k r  n þ l D ni þ rn @xi @xi @ ¼ lDni þ ðk þ lÞ rn @xi

=i ¼

The vector record of the obtained ratio looks like = ¼ lDn þ ðk þ lÞrr  n

ð2:9Þ

After the force (2.9), i.e., the right side (2.6), is determined, the equation of motion becomes final q

@2n ¼ lDn þ ðk þ lÞrr  n @t2

ð2:10Þ

Since the right part of the equation contains both Lamé coefficients describing the elastic properties of both compressible (k) and incompressible (l) bodies, it makes sense to present the deformation n in the form of a sum of two deformations n1 and n2 of the simple compression and the simple shear, respectively, such as, and r  n1 6¼ 0, r  n1 ¼ 0 and r  n2 ¼ 0, r  n2 6¼ 0: This property r  n1 ¼ 0 indicates that a scalar potential u can be entered u to describe pure compression n1 , such that Du 6¼ 0 (satisfaction with the property r  n1 6¼ 0). Similarly, the property r  n2 ¼ 0 allows to enter the vector potential w for which the ratio r  r  w 6¼ 0 (condition r  n2 6¼ 0) is met. Thus, it is permissible to imagine n ¼ ru þ r  w

ð2:11Þ

16

2 Acoustic Phenomena in the Language of Elasticity Theory

of the most general kind for which r  n 6¼ 0, r  n 6¼ 0. Substitution of (2.11) in equation of motion (2.10) brings the latter to mind q

@2 @2 ru  ðk þ 2lÞDr u ¼  q r  w þ lDr  w @t2 @t2

ð2:12Þ

Let’s apply the operation r (divergence) to the obtained equation. Since for an arbitrary vector a the relation r  ðr  aÞ ¼ 0 is valid, the right part of the Eq. (2.12) will turn to zero, as a result of which the ratio describing the change in scalar potential will be formed u q

@2 Du  ðk þ 2lÞDDu ¼ 0 @t2

Since, as noted above, D u 6¼ 0, the last equation can be reduced to q

@2u  ðk þ 2lÞDu ¼ 0 @t2

ð2:13Þ

which is by its type and properties a wave equation describing the propagation of the rarefaction and compression regions in space–time. Since for this type n ¼ ru; the waves are longitudinal and their propagation velocity is determined by the c2k ¼

k þ 2l q

ð2:14Þ

Applying the operation r (rotor) to Eq. (2.12) resets the left side of the Eq. (2.12), because for any scalar, a there is one r  ra ¼ 0. As a result, an equation of the kind is formed q

@2 r  r  w  lDr  r  w ¼ 0 @t2

Using the property r  r  w 6¼ 0 allows you to simplify the resulting equation by converting it to the form q

@2w  lDw ¼ 0 @t2

ð2:15Þ

It is also a wave equation that describes the transverse shear waves that propagate in space at a rate of c2? ¼

l q

ð2:16Þ

2

Acoustic Phenomena in the Language of Elasticity Theory

17

It should be specifically noted that the making of wave Eqs. (2.13, 2.15) neglects the result of the Hamilton operator’s action r on the density of the medium, that is, 2 2 the members rq  @t@ 2 ru and rq  @t@ 2 r  w of the species and which are members of the second order of magnitude in the framework of the approximation of small deformations were discarded. In this regard, a strict approach requires replacing the density q with an average value in Eqs. (2.13, 2.15). As always, the equations of motion need to be supplemented by boundary conditions. In general, at the boundary of the elastic media interface, they should be equal in size and opposite in direction of force, with which the media act on each other. If n is a unit vector of normal to the surface of the section, the condition formulated above is presented in the form of ð1Þ

ð2Þ

ni rij ¼ ni rij

ð2:17Þ

where the indexes (1) and (2) indicate the contacting elastic media. Besides, if there is no slippage between the media, the displacement vectors will be equal at the partition boundary nð1Þ ¼ nð2Þ

ð2:18Þ

In the case of slippage, a ratio describing the equality of normal to the boundary component displacement vectors in the mediums should be performed n  nð1Þ ¼ n  nð2Þ

ð2:19Þ

The connection of the presented description of elastic media dynamics with acoustic processes in water is carried out by means of the liquid non-contiguous medium model, in which the shear elastic coefficient l is equal to zero. Since the all-round adiabatic compression ratio is set by the ratio [1]


 1 1 @V 1 @1=q 1 @q 1 ¼ ¼ ¼ ¼ 2 k þ 2l=3 V @p r 1=q @p r q @p r qc in a liquid medium model k ¼ qc2 , where q is the density of the liquid, c is the speed of sound. This ratio is analogous to the differential variant (1.10) of the equation of state. The parameter entered in the first chapter a is related to the comprehensive fluid compression ratio by an obvious ratio a ¼ q=k. Absence of shear effects in the non-contiguous liquid allows to set the vector of particle displacement by means of a single scalar potential nf ¼ ru where the lower index f specifically indicates the case of a liquid medium.

ð2:20Þ

18

2 Acoustic Phenomena in the Language of Elasticity Theory

The wave Eq. (2.15) for the vector potential disappears, because in this case there is no potential itself w; and Eq. (2.13) is simplified and takes the form @2u  c2 Du ¼ 0 @t2

ð2:21Þ

Since in an inviscid case slippage of one medium relative to another is acceptable, the Eqs. (2.20, 2.21) are supplemented by boundary conditions (2.17, 2.19). However, in acoustics it is not customary to describe the process in terms of deformations and the values derived from them. The most common is the formulation of basic patterns by means of velocity, pressure and density fields. In order to pass these thermodynamic variables, it is necessary to notice only that the velocity field v is connected with the displacement field by nf a simple ratio v¼

@nf @t

ð2:22Þ

The time differentiation of expressions (2.20, 2.21) with the use of renaming @u=@t ! u defines the velocity field by means of scalar potential v ¼ ru

ð2:23Þ

which satisfies the wave equation exactly the same as Eq. (2.21). The time differentiation of the dynamic boundary condition (2.17), taking into account the definition (2.8), transforms it into a form qð1Þ cð1Þ2 r  vð1Þ ¼ qð2Þ cð2Þ2 r  vð2Þ  qð1Þ cð1Þ2 D u ð1Þ ¼ qð2Þ cð2Þ2 D u ð2Þ

ð2:24Þ

where qðiÞ and cðiÞ are the densities and sound speeds in the contact liquid media. Since, as follows from (1.23), qc2 Du ¼  @p @t , the relations (2.24) can be shaped in @pð1Þ =@t ¼ @pð2Þ =@t, which after integration in time sets the final formulation of the dynamic boundary condition pð1Þ ¼ pð2Þ

ð2:25Þ

The kinematic condition obtained by time differentiation of the ratio (2.19) describes the equality of normal speed components on the different sides of the interface n  vð1Þ ¼ n  vð2Þ

ð2:26Þ

In the absence of slippage between the media, differentiation of equality (2.18) leads to equality of velocities on both sides of the border

2

Acoustic Phenomena in the Language of Elasticity Theory

vð1Þ ¼ vð2Þ

19

ð2:27Þ

In the liquid limit l ¼ 0, the expression for the force resulting from the deformation of the medium takes the form = ¼ krr  n

ð2:28Þ

The form representation (2.28) represents the gradient of the symmetrical tensor trace, which allows us to associate this result as a value coinciding with the term  r~p on the right side of the Euler equation. Thus, the representation of pressure perturbations in the form ~p ¼ qc2jj r  n

ð2:29Þ

reduces the Eq. (2.10) of the elasticity theory to the first equation (Euler) of the system (1.28), obtained on the basis of the general hydrodynamic relations. The time differentiation of the ratio (2.29) generates the second equation of the system (1.28). Since the sum of the diagonal components of the deformation tensor gives a relative change in body volume [1], the rn¼

~ q dV 0  dV dV 0 =m  dV=m 1=q0  1=q q  q0 ¼ ¼ ¼  q0 dV dV=m 1=q q

~ is the density deviation from its average value. where q From this equality, the obvious relationship is ~ þ q0 r  n ¼ 0 q

ð2:30Þ

whose differentiation by time gives the third system Eq. (1.28) of linear acoustics equations. As a result of the performed operations, we managed to reformulate the description of waves in elastic media in terms of velocity–pressure–density in the liquid (gaseous) medium limit, and the obtained equations and boundary conditions coincided exactly with the corresponding relations derived on the basis of fundamental equations of hydrodynamics. In addition to this expected result, it was also possible to show that when describing sound waves in liquid non-dissipative media, the acoustic potential of the velocity field [ratio (2.23)] appears in the theory quite organically. Here it is necessary to make a comment concerning the introduction of the velocity field potential. In general, its appearance is not connected at all with the acoustic nature of the phenomena under consideration. The true reason is the use of Euler’s linearized equation

20

2 Acoustic Phenomena in the Language of Elasticity Theory

@v ¼ r~p @t

ð2:31Þ

The often introduced representation of the velocity field as a solution to the Eq. (2.31) is v ¼ rU þ r  W

ð2:32Þ

which is inadequate to the task at hand. The matter is that the application of the operator r (rotor) to the Eq. (2.31) leads to the ratio @r  v ¼0 @t

ð2:33Þ

from where it comes r  v ¼ FðrÞ. As shown, for example, in [2], the velocity field is determined by its vector of swirliness by the ratio 1 v¼ r 4p

Z

Fðr0 Þ 3 0 dr jr  r0 j

which means that the vortex part r  W of the speed (2.32) is time independent. Thus, the time-frozen distribution of the vorticity in the whole space does not affect the dynamics of the current in any way, because it does not show its effect in the linearized Euler equation. This part of the velocity also does not contribute to the linearized continuity q equation @~ @t þ q0 r  v ¼ 0 because r  ðr  WÞ  0. For the same reason, the vortex component does not appear in the equation @~p þ q0 c2 r  v ¼ 0. @t Therefore, without detracting from the generality, describing the acoustic phenomena can be put r  v ¼ 0. Substitution of this representation ratio (2.32) leads to the expression rrW¼0

ð2:34Þ

The last equality is always executed when r  W  rU2 . But then, the representation (2.32) is transformed into a form v ¼ rU

ð2:35Þ

i.e., the velocity field is fully described through its potential. At the same time, decomposition U ¼ U1 þ U2 is acceptable, so that r  v ¼ DðU1 þ U2 Þ; when DU1 6¼ 0 and the potential U1 describes acoustic oscillations, and DU2 ¼ 0 so the

2

Acoustic Phenomena in the Language of Elasticity Theory

21

potential U2 refers to non-acoustic phenomena, such as the potential swirl-free flow of the body. Thus, both the hydrodynamic approach and the limit transition from the equations of the elasticity theory lead to the general main result: The concept of the acoustic potential u satisfying the wave equation is the basis for the linear description of the dynamics of sound waves @2u  c2 Du ¼ 0 @t2

ð2:36Þ

the solution of which defines the physical fields with the expressions v ¼ ru;

~p ¼ q0

@u ; @t

~¼ q

~ p c2

ð2:37Þ

The initial and boundary conditions for the potential are formulated from the corresponding conditions for speed, pressure and density with the help of (2.37). In conclusion of the first two chapters devoted to the conclusion of the basic equations of linear acoustics, it is necessary to mention some limitations of the constructed mathematical model of the elementary sound phenomena. In the whole previous statement, it was implicitly assumed that the speed of sound is a characteristic constant medium and does not depend on the parameters of the wave propagating in the medium. Actually, it is a little different. At the end of the twentieth century, a phenomenon was discovered that the speed of sound wave propagation in water depends on frequency, namely it increases with its increase. This effect is evident only in the hypersonic region ðf  109 . . .1012 ; HzÞ and represents a smooth transition from the usual sound velocity ðc  1500 textm=sÞ at the lower boundary of the specified frequency range to about double the value at the upper one. Two models of the mechanism of hypersonic wave propagation were used to explain this phenomenon. In the first model, according to which water at ultrahigh frequencies was considered as viscoelastic, the environment with increasing frequency was considered as more and more elastic, in the sense that liquid particles do not have time to move essentially from the positions for the period of wave oscillations ðT  109 . . .1012 ; sÞ Water in relation to such fluctuations becomes a kind of solid body, and in solids, the speed of sound increases, e.g., the speed of sound in ice is approximately equal c  3000 m=s. The second model considers water as a two-component medium consisting of hydrogen and oxygen ions (between which, of course, there are chemical bonds). Since hydrogen ions are significantly lighter than oxygen ions, high-speed, high-frequency sound will be transmitted exclusively through hydrogen atoms. The experiment carried out by a group of Italian physicists in the f  109 . . .1011 Hz frequency region showed the fairness of the viscoelastic model of water and complete inconsistency with the conclusions of the two-component theory [3].

22

2 Acoustic Phenomena in the Language of Elasticity Theory

The graph below shows the results of the comparison of experimental data on the measurement of the dynamic structural water factor in the frequency range from 100 MHz to 100 GHz for different values of the free path of molecules Q at temperature T ¼ 0  C [3].

Triangles, circles and rhombuses belong to experimental points, and lines represent dependences calculated according to the viscoelastic theory. A more detailed description of the problem with the description of some of the features of the experiments is carried out, as well as additional references to this problem can be found in [4]. Thus, the approximated linear acoustics model is not comprehensive and has limitations on the frequency of waves propagating in the environment. At the same time, the vast majority of sound phenomena recorded in the environment lie far from the lower boundary of the hypersonic frequency range and can be described within the limits of the accepted approximations.

References 1. Landau LD, Livshits EM (1987) Theoretical physics. VII. Theory of elasticity. Science, 248 p 2. Bachelor J (1973) Introduction to fluid dynamics. Mir, Moscow, 758 p 3. Santucci S, Fioretto D, Gomez L, Gessini A, Masciovecchio C (2006) Is there any fast sound in water? Phys Rev Lett 97:225701 4. http://elementry.ru/news/430411

3

General Properties and Character Types of Sound Waves

In the previous sections, it was shown that sound is alternating in the space and time of compression and rarefaction of the environment in which it is distributed. Hence, the sound is a wave. Let’s ask the question: What is the equal of energy and sound wave impulse? According to the definition, the energy per unit of liquid volume is equal ~; e ¼ e0 þ ~e; where e0 , ~e are the specific internal qðe þ v2 =2Þ. Since q ¼ q0 þ q energies of a stationary liquid and a sound wave; accordingly, the energy of a volume unit will be determined by the ratio ~Þv2 =2 E ¼ qe þ ðq0 þ q

~2 @ 2 ðqeÞ

q @ðqeÞ

~ ~Þv2 =2 ¼ q 0 e0 þ q þ þ    þ ðq0 þ q @q
q¼q0 2 @q2
q¼q0

ð3:1Þ

The derivatives in (3.1) are taken at constant entropy because the sound wave in neglect of thermal transfer effects is an adiabatic process. Using the thermodynamic ratio de ¼ Tdr  pdV ¼ Tdr þ

p dq q2

allows to determine the first derivative @ðqeÞ=@q: 

@ðqeÞ @q



  @e p ¼ eþ q ¼ eþ 2 ¼ w @q r q r

ð3:2Þ

where w is the specific enthalpy.

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_3

23

24

3

General Properties and Character Types of Sound Waves

The second derivative is defined by the expression        2  @ ðqeÞ @w @w @p c2 ¼ ¼ ¼ @q2 r @q r @p r @q r q

ð3:3Þ

Let’s substitute (3.2) and (3.3) in expression (3.1) for energy of volume unit, keeping terms up to the second order of smallness on perturbations of physical fields. ~þ E ¼ q0 e0 þ w0 q

~2 c2 q v2 þ q0 2q0 2

ð3:4Þ

Since the first term in this expression is the energy of the unit volume of a stationary liquid, it has nothing to do with the sound wave and should be eliminated. The total energy of the sound wave is determined by the integration of the ratio (3.4) across the entire volume of the liquid  Z  ~2 c2 q v2 ~þ E ¼ E dV ¼ w0 q þ q0 dV 2q0 2  Z  2 2 ~ c q v2 ¼ þ q0 dV 2q0 2 Z

ð3:5Þ

The last equality is conditioned by the fact that the law of liquid mass preservation is fulfilled; therefore, Z

Z ~ dV ¼ w0 w0 q

~ dV ¼ 0 q

The integrative expression in (3.5) is the density of the sound energy E¼

~2 c2 q v2 þ q0 2q0 2

ð3:6Þ

Now let’s consider the question of the impulse of the unit volume of the liquid in which the sound wave propagates. By definition, the unit volume impulse is the density of the mass flow j ¼ qv

ð3:7Þ

~; and q ~ ¼ ~p=c2 , the substitution of these relations in the Since q ¼ q0 þ q expression (3.7) defines the representation of the required value in the form of j ¼ q0 v þ ~pv=c2

ð3:8Þ

3

General Properties and Character Types of Sound Waves

25

The value ~pv makes sense of the density of the sound energy flux, as it satisfies the equation @E þ r  ð~pvÞ ¼ 0 @t

ð3:9Þ

which is the law of sound wave energy conservation. The total wave impulse is equal to the integral over the whole volume occupied by it: Z

Z Z 1 jdV ¼ q0 v dV þ 2 ~ pv dV c Z Z 1 ¼ q0 ru dV þ 2 ~pv dV c I Z Z 1 1 pv dV ¼ q0 u dS þ 2 ~pv dV  2 ~ c c

J¼

ð3:10Þ

The last equality in (3.10) is conditioned by the fact that at the boundaries of the region occupied by the wave, the sound potential is equal to zero: u ¼ 0. Since the latter integral does not turn to zero, the result obtained means that the sound wave is accompanied by a transfer of substance. It should be noted that since ~; ~p and v are the values of the first order of smallness, the transfer of matter is the q value of the second order of smallness, and when considering certain problems, it can be neglected. An acoustic wave in a medium is considered to be fully defined if all its spatial ~ and ~ and temporal characteristics are known: scalar fields q p; vector field v and velocity c. The field ~p describes the time-varying spatial distribution of pressure ~; which differs from the field ~ disturbances generated by sound waves. The field q p only by the numerical coefficient ð~ q ¼ ~p=c2 Þ; determines the actual picture of compression–rarefaction in the sound wave. As has been said many times before, the value c determines the velocity of propagation in the space of this picture. What then determines the oscillating speed v? In order to understand this issue, let’s turn to the simplest example of the propagation of a plane harmonic sound wave. As follows from the expression (1.23) obtained in Chapter 1 for such a wave, the pressure perturbation and the oscillatory velocity in it are related by the ratio ~¼~ ~p ¼ q0 x v=k ¼ q0 c v; where v ¼ jvj. As for any type of sound waves q p=c2 , there is a connection between the perturbation of density and the amplitude of the ~ ¼ q0 v=c for a plane harmonic wave. Substitution of this oscillatory velocity q result in expression (3.6) for the density of sound energy leads to an obvious sequence of relations ~2 =2q0 þ q0 v2 =2 ¼ q0 v2 =2 þ q0 v2 =2 ¼ q0 v2 E ¼ c2 q

26

3

General Properties and Character Types of Sound Waves

Thus, the oscillatory velocity v (more precisely, its amplitude) determines the spatial and temporal distribution of the intensity of the propagating sound wave. Since the energy density vector ~pv at the selected point in space coincides with the direction of oscillatory velocity, it is the oscillatory velocity that shows where the sound energy concentrated in the elementary volume surrounding a given point is transferred to. When a sound wave hits the membrane of the human ear, for example, some of its energy is released to the reflected wave and the other part is transferred to the energy stored in the membrane. The higher the energy of the incident wave, the greater the amplitude of the membrane deflection from the equilibrium position and the louder the sound perceived by the ear. Thus, in everyday language, the oscillatory velocity determines the volume of the sound. The waves emitted by sound sources can be of a wide variety. The most famous are plane, spherical and cylindrical waves. Let’s proceed to the consistent study of their main characteristic properties. ~ Since the sound wave manifests itself through the pressure ~ p and density q deviations from their equilibrium values, as well as through the velocity field v; which are determined by the acoustic potential u by means of relations 2.37, it is sufficient to study the properties of the potential of any type of the wave of interest to answer all the questions. So, let’s start with a plane wave. In order to emit a plane wave, it is necessary to have a source of infinite dimensions, since one of the most important properties of a ~; v; and consequently uÞ when plane wave is its invariable characteristics (~p; q moving in any direction perpendicular to the vector of its group velocity. Without detracting from the commonalities, let’s place the source of the plane wave in the plane z ¼ 0. Since the properties of the wave are unchanged along the axes x and y, @u=@x ¼ @u=@y ¼ 0 and Eq. (1.31) for the velocity potential takes the form @2u @2u  c2 2 ¼ 0 2 @t @z

ð3:11Þ

Then, the potential of the running in the positive direction of the z plane wave axis is determined by the expression up ¼ f ðz  ctÞ

ð3:12Þ

Because v ¼ @up =@z ¼ f 0 ðz  ctÞ; ~p ¼ q0 @up =@t ¼ q0 cf 0 ðz  ctÞ; the pressure and velocity in the plane wave are related by the ratio v¼

~p q0 c

ð3:13Þ

3

General Properties and Character Types of Sound Waves

27

The obtained results refer to the plane wave of the most general form (3.12), and therefore are also valid for the monochromatic plane wave, the potential of which is usually recorded in the form up ¼ u0 expðiðk  r  x tÞÞ

ð3:14Þ

where k and x are the wave vector and the frequency of the sound wave, respectively, and u0 is its constant amplitude. Further, considering plane monochromatic sound waves, we will assume that the potential of their velocity is described by the expression (3.14). Thus, the energy density of the plane monochromatic wave is set by the ratio ~2 =2q0 þ q0 v2 =2 ¼ q0 v2 . Considering the value c2 q ~2 =2q0 ¼ k q ~2 =2 q20 (k is E ¼ c2 q the comprehensive compression coefficient) as a density of potential energy of elastic interactions in the wave, and, accordingly, the value q0 v2 =2 as a density of kinetic energy of liquid elements involved in acoustic oscillations, we can draw the following conclusion: in a plane monochromatic wave at each point of space at any given time moment the energy of sound oscillations is divided equally into potential and kinetic components. Now let’s consider another type of sound waves, namely those whose characteristics—speed, pressure and density perturbation—depend only on the distance to a certain point in space. As follows from the assumption made, such waves have spherical symmetry and are therefore called spherical waves. Based on the general wave equation @2u  c2 Du ¼ 0 @t2 let’s simplify its appearance by using the mentioned spherical symmetry of the sound propagation process. In this case, the Laplace operator D is presented in the form:   1 @ 2 @ r D¼ 2 r @r @r and that’s why the wave equation is taking shape   @2u 2 1 @ 2 @u r c 2 ¼0 @t2 r @r @r

ð3:15Þ

Substitution in (3.15) of the representation u ¼ f ðr; tÞ=r leads to the equation for the function f 2 @2f 2@ f  c ¼0 @t2 @r 2

28

3

General Properties and Character Types of Sound Waves

whose solution is already known. f ¼ f1 ðr  ctÞ þ f2 ðr þ ctÞ: Thus, the solution for the spherical wave potential is written in the form of u¼

f1 ðr  ctÞ f2 ðr þ ctÞ þ r r

ð3:16Þ

which is the sum of the divergent and convergent waves. Since the pressure and density perturbations in the sound wave are determined through the potential by ~p ¼ q0

@u ; @t

~¼ q

q0 @u c2 @t

their distribution is given by formulas of the same kind as (3.16). The radial component of the velocity (and only it is different from zero in a spherical wave), defined as a potential gradient, has the form v¼

  @ f1 ðr  ctÞ f2 ðr þ ctÞ þ @r r r

ð3:17Þ

If there is no sound source in the space, the potential must also be limited at the r ¼ 0. But then, it follows from (3.16) f1 ðctÞ þ f2 ð þ ctÞ ¼ 0; therefore, the potential should look like u¼

f ðr  ctÞ  f ðr þ ctÞ r

ð3:18Þ

which corresponds to the potential of a standing spherical wave. If in space at the point r ¼ 0 there is sound source the only a spherical wave will spread. For a similar monochromatic wave, its potential is set by the expression u¼A

expðiðkr  x tÞÞ ; r

k ¼ x=c

ð3:19Þ

It is important to underline the fact that such potential is a solution to the equation @2u  c2 Du ¼ 4p AdðrÞ expði xtÞ @t2

ð3:20Þ

in the right part of which there is a sound source function; dðrÞ is the Dirac delta function; moreover, dðrÞ ¼ dðxÞdðyÞdðzÞ.

3

General Properties and Character Types of Sound Waves

29

In contrast to a plane wave, the amplitude of spherical wave decreases at distance from the source in proportion to the distance from it. When a sound source is switched on briefly, the pressure field at a fixed point in the space has the property that Zþ 1

Zþ 1 ~p dt ¼ q0

1

1

@u dt ¼ q0 ðuð1Þ  uð þ 1ÞÞ ¼ 0 @t

ð3:21Þ

This means that when the wave passes through this point, both condensation ð~p [ 0Þ and rarefaction ð~p\0Þ will be observed, in contrast to the plane wave, which may consist of condensation or rarefaction alone. The same change of condensation and rarefaction is observed in the space at the fixed moment of time as Zþ 1

Zþ 1 r ~p dr ¼ q0

0

r 0

Zþ 1 ¼ cq0

@u dr @t ð3:22Þ f 0 dr ¼cq0 ðf ð þ 1Þ  f ð0ÞÞ ¼ 0

0

This property also distinguishes a spherical wave from a plane one. Another type of wave in which the distribution of all physical quantities is homogeneous along a certain direction, such as an axis z; and has axial symmetry with respect to this axis, is called a cylindrical wave. In the case of a linear source located along the axis of z a cylindrical coordinate system ðz; r; hÞ, the potential of a cylindrical wave is a solution to the equation @2u dðrÞ expði xtÞ  c2 Du ¼ 2p A @t2 r

ð3:23Þ

which has a sound source function on its right side. Laplace’s operator in the cylindrical coordinate system for the problem (3.23) considered looks like   1@ @ D¼ r r @r @r For the monochromatic wave in free space, Eq. (3.23) is reduced to the ordinary differential equation d2 u 1 du þ k2 u ¼ 0; þ dr 2 r dr

k¼

x c

ð3:24Þ

30

3

General Properties and Character Types of Sound Waves

whose solution u ¼ J0 ðkrÞ expði x tÞ represents a standing cylindrical wave. The traveling divergent wave satisfying Eq. (3.23) with the sound source is described by ð1Þ

u ¼ AH0 ðkrÞ expði x tÞ

ð3:25Þ

ð1Þ

where H0 is Hankel’s function is of the first kind. At large distances, at kr  1; the asymptotic formula for the potential of the divergent cylindrical wave is valid rffiffiffiffiffiffiffiffiffi 2 expðiðkr  x t  p=4ÞÞ uA pkr

ð3:26Þ

One can see that the amplitude of the cylindrical wave drops like a square root from the distance to the emitting point. A cylindrical wave has the property that may have a leading edge, but may not have a backward edge: Once the sound disturbance reaches the observation point, it slowly attenuate at the observation point when t ! 1. As in the case of a spherical wave, Zþ 1 ~p dt ¼ 0 1

which means there are both condensations and rarefactions in the cylindrical wave. Earlier it was shown (3.9) that the sound wave is characterized by a nonzero energy flow. To create such a flow, it is necessary that there are sound energy sources in the environment, because the sound wave cannot come out of nowhere for nothing. Certain types of sources (3.20, 3.23) have just been considered in the study of the properties of spherical and cylindrical waves. We are now exploring the problem of sound sources in the general approach. By their very nature, sound sources can be force or mass. An example of a force source is, for example, a palm slapping the water surface. The mass source is the end of the tube through which the injection or suction of a medium with constant or variable flow takes place. Description of sound radiation requires the introduction of force and mass sources into the acoustic Eq. (1.28) q0

@v ¼ r~p þ f; @t

@~ q þ q0 r  v ¼ m; @t

@~ q 1 @~ p ¼ 2 @t c @t

ð3:27Þ

where f and m are the densities of the force and mass sources consequently, which are, in general, the functions of coordinates and time.

3

General Properties and Character Types of Sound Waves

31

Let’s assume that force sources f have potential Uðr; tÞ; so f ¼ rUðr; tÞ. This approach ensures the potentiality of the generated velocity field v and makes it possible not to consider, along with acoustic waves, vortex uncompressible currents. The exclusion of pressure and density from Eq. (3.27) leads to the equation for the acoustic potential   @2u 1 @U 2 2  c Du ¼  c mþ @ t2 q0 @t

ð3:28Þ

with a source on the right side. The velocity, pressure and density are determined by the potential in terms of v ¼ ru;

~p ¼ q0

@u  U; @t

~¼ q

~ p c2

ð3:29Þ

Sometimes the source function in the right side (3.28) is inconvenient when integrating the wave equation. In this case, it makes sense to abandon the approach based on the acoustic potential and directly integrate the equation for pressure, which in this case looks like   @ 2 ~p 2 2 @m þ DU  c D ~p ¼ c @t @ t2

ð3:30Þ

After integrating Eq. (3.30), knowledge of acoustic pressure ~ p allows to determine the density perturbation by means of ~¼ q

~p c2

and based on any of the following equations @u 1 ¼  ð~p þ U Þ; @t q0

Du ¼

  1 1 @~ p m 2 q0 c @t

ð3:31Þ

determines the acoustic potential. Naturally, the approach based on the integration of the pressure equation is justified only if the function @m=@ t þ DU is simpler (in the sense of integration of the wave equation) than the function c2 m þ @U=@ t. One of the most important types of sound sources is the point source. If the problem of point source radiation is solved, then on the basis of the obtained solution it is possible, using the Green function method, to construct a solution for the radiation of the source system with a given space distribution. First of all, let’s consider the problem of sound emission by a point source with mass flow rate mðtÞ in one-dimensional case. Such a source can be represented in

32

3

General Properties and Character Types of Sound Waves

the form of two parallel planes, making mutually opposite movements along the axis x normal to them. Let’s consider that the planes experience infinitely small displacements from the position x ¼ 0. Let the coordinate of one of the planes be described by the function for the certainty of the one that shifts from zero in the positive part of the axis x. Then, the amount of substance per unit of time, which this plane pushes directly in front of it, is the value q0 S

d xðtÞ ¼ q0 S vðtÞ dt

ð3:32Þ

where v is the speed of the plane and S is its area. Since the liquid medium has the same velocity near the plane (it is assumed that the continuity of the liquid is not disturbed), it can be determined by the concept of acoustic potential by the ratio

@u

@u

vðtÞ ¼  @x
x¼xðtÞ þ 0 @x
x¼ þ 0

ð3:33Þ

The last approximation of the equation is justified by the assumption of infinitesimal plane displacements xðtÞ: Substitution of (3.33) in (3.32) results in act that the flow rate near the selected plane is determined
dxðtÞ @u

q0 S ¼ q0 S
dt @x x¼ þ 0

ð3:34Þ

Since the flow rate produced by the second plane is equal to the flow rate from the first plane, and in total, according to the problem definition, they are equal mðtÞ
@u

mðtÞ q0 S
¼ @x x¼ þ 0 2

ð3:35Þ

Since we consider the acoustic phenomena in which perturbation is transmitted with the speed of sound c, using the one-dimensionality of the problem, we can assert that at a distance x from the plane, the acoustic potential is associated with its value on the plane by the ratio uðt; xÞ ¼ uðt  x=cÞ: Because the function uðt  x=cÞ has the property @uðt  x=cÞ 1 @uðt  x=cÞ ¼ @x c @t

3

General Properties and Character Types of Sound Waves

33

then the ratio (3.35) is converted to a form

mðtÞ @u

q0 S @u

¼ q0 S
¼ 2 @x x¼ þ 0 c @t
x¼ þ 0 In turn, since pressure and potential are linked by a ratio ~ p ¼ q0 @u=@t; the latter equation gives the necessary connection between flow and pressure near the plane ~pjx¼ þ 0 ¼

c mðtÞ 2S

ð3:36Þ

The result received allows to record the solution for the pressure value to the right of the point source (at the x [ 0Þ in the form of ~p ¼

c mðt  x=cÞ 2S

ð3:37Þ

To the left of the source, the same pattern of movement should be observed as to the right, so that when x\0, there is ~p ¼

c mðt þ x=cÞ 2S

ð3:38Þ

and the sign + shows that the acoustic wave is running to the left. The combination of the results obtained expresses the pressure field from a point mass source in a one-dimensional case by the ratio ~p ¼

c ½mðt  x=cÞ#ðxÞ þ mðt þ x=cÞ#ðxÞ 2S

ð3:39Þ

where #ðxÞ is the Heaviside’s single function. This result was obtained on the basis of a non-strict approach. Now we will achieve the same, but on the basis of the solution of the wave Eq. (3.30) for the acoustic pressure. Since the source is concentrated at a point x ¼ 0; the pressure equations take the form @ 2 ~p @ 2 p~ c2 @m dðxÞ  c2 2 ¼ 2 @t @x S @t

ð3:40Þ

where dðxÞ is the Dirac’s delta function. In Eq. (3.40), the value m is no longer understood to be the mass flow density as in Eq. (3.30), but the mass flow rate itself as in (3.35). That is why the right part is divided into a square S.

34

3

General Properties and Character Types of Sound Waves

Let’s look for the solution of Eq. (3.40) in the form of the sum of the disturbances running to the right and to the left of the source, i.e., let’s present the pressure in the form ~p ¼ f ðt  x=cÞ#ðxÞ þ gðt þ x=cÞ#ðxÞ

ð3:41Þ

However, the functions f and g are still unknown. Let’s substitute the representation (3.41) in the wave Eq. (3.40). For this purpose, let’s calculate the necessary derivatives beforehand. So, the first derivative of the x: @ ~p @ ¼ ½f ðt  x=cÞ#ðxÞ þ gðt þ x=cÞ#ðxÞ @x @x @f ðt  x=cÞ @#ðxÞ #ðxÞ þ f ðt  x=cÞ ¼ @x @x @gðt þ x=cÞ @#ðxÞ #ðxÞ þ f ðt þ x=cÞ þ @x @x 1 @f ðt  x=cÞ ¼ #ðxÞ þ f ðt  x=cÞdðxÞ c @t 1 @gðt þ x=cÞ þ  gðt þ x=cÞdðxÞ @t c  1 @gðt þ x=cÞ @f ðt  x=cÞ #ðxÞ  #ðxÞ þ dðxÞðf ðtÞ  gðtÞÞ ¼ c @t @t Now let’s find the second derivative   @ 2 ~p 1 @ 2 f ðt  x=cÞ @ 2 gðt þ x=cÞ ¼ #ðxÞ þ #ðxÞ @t2 @t2 @x2 c2   dðxÞ @gðtÞ @f ðtÞ  þ þ d0 ðxÞðf ðtÞ  gðtÞÞ c @t @t It is clear that the second derivative in time is determined by the expression @ 2 ~p @ 2 f ðt  x=cÞ @ 2 gðt þ x=cÞ ¼ #ðxÞ þ #ðxÞ @t2 @t2 @t2 By substituting these expressions in Eq. (3.41), we obtain the following ratio  cdðxÞ

 @gðtÞ @f ðtÞ c2 @m þ dðxÞ  c2 d0 ðxÞðf ðtÞ  gðtÞÞ ¼ @t @t S @t

ð3:42Þ

3

General Properties and Character Types of Sound Waves

35

For the ratio (3.42) will be identity the conditions should be valid f ðtÞ ¼ gðtÞ;

@gðtÞ @f ðtÞ c2 @m þ ¼ @t @t S @t

ð3:43Þ

Asked for identity (3.43), the conditions must be met f ðtÞ ¼ gðtÞ ¼

c2 m 2S

ð3:44Þ

Now substitution of (3.44) in the representation (3.41) for acoustic pressure gives the result ~p ¼

c ½mðt  x=cÞ#ðxÞ þ mðt þ x=cÞ#ðxÞ 2S

exactly the same as (3.39). Thus, the problem of point source radiation in one-dimensional space is solved. Let’s now consider the problem of point source radiation in isotropic three-dimensional space. In this case, it is easier to solve the wave equation for the acoustic potential @2u c2 m dðrÞ 2  c Du ¼  @t2 q0 4pr 2

ð3:45Þ

Since in an isotropic case, in a spherical coordinate system whose origin coincides with  2the position of the source, the Laplace operator is reduced to a form @ @ ; the solution of Eq. (3.45) can be represented, by analogy with the D ¼ r12 @r r @r previous case, as u ¼ f ðrÞmðt  r=cÞ þ gðrÞmðt þ r=cÞ: In order for there to be a wave gðrÞmðt þ r=cÞ in space that converges to the beginning of the coordinates, it is necessary to have infinitely distant sound sources. As there are no such sources in the task, the potential should be sought in a simplified form u ¼ f ðrÞmðt  r=cÞ

ð3:46Þ

Substitution of this solution in Eq. (3.45) leads to the ratio     2 @f @2f 2 f @f m dðrÞ þ þ m0 ¼ m r @r @r 2 c r @r t q0 4pr 2

ð3:47Þ

This ratio is identical at any flow rate dependencies m on time, if the function f ðrÞ looks

36

3

General Properties and Character Types of Sound Waves

f ðrÞ ¼ 

1 4p rq0

Thus, the potential, and through it the pressure, is determined by the relations u¼

mðt  r=cÞ ; 4prq0

~p ¼ q0

@u m0 ðt  r=cÞ ¼ @t 4pr

ð3:48Þ

If we were to act, as in the previous case, through the pressure equation, we would experience great difficulty in solving it. The obtained ratio shows that, unlike the one-dimensional case, the amplitudes of the acoustic fields decrease proportionally to the distance. The intermediate case of a two-dimensional, cylindrically symmetrical flow is equally complex both for the integration of the equation for the potential and for the pressure. As it was rightly noted in [1], because in the one-dimensional case the pressure is proportional to the flow rate, and in the three-dimensional case, the pressure is a derivative of the flow rate in time, and in the two-dimensional case, there should be an intermediate result. That is, the pressure must be proportional to the derivative of order 1/2. Similarly, the acoustic potential in the two-dimensional case is expressed by the order of 1/2 integral. u¼

1 ð1=2Þ pffiffi I mðt  r=cÞ; q0 r

1 ~p ¼ pffiffi Dð1=2Þ mðt  r=cÞ r

ð3:49Þ

where fractional operators [2] (I ðaÞ is integral, DðaÞ is differential) are determined by I

ðaÞ

1 f ðtÞ ¼ CðaÞ

Zt

ðt  sÞa1 f ðsÞ ds

0

1 d DðaÞ f ðtÞ ¼ Cð1  aÞ d t

Zt

ð3:50Þ a

ðt  sÞ f ðsÞ ds 0

where CðxÞ is the gamma function and the order of integration (differentiation) a takes on values in the interval ð0; 1Þ: As one would expect, the potential and pressure in the two-dimensional case fall inversely proportional to the square root of the distance. No less important than the monopoly is the source of the dipole type. An acoustic dipole is a source–drain system in which the total mass flow through the surface covering both the source and the drain is zero. Despite this, this system emits sound waves, of course, less effective than a single monopoly, but emits. In order to reveal the peculiarities of acoustic dipole radiation, let’s consider a three-dimensional problem about the radiation of the system of two point sources,

3

General Properties and Character Types of Sound Waves

37

the flow rates of which are equal in absolute value and opposite in sign. The sources are located in the points r0 and r0 . The equation for the acoustic potential in this case is @2u c2 m dðr  r0 Þ  dðr þ r0 Þ 2  c Du ¼  @t2 q0 4pr 2

ð3:51Þ

In connection with the linearity of the problem, its solution is expressed in the form of a linear combination of solutions of the form (3.48), so that in this case the pressure is determined by the expression ~p ¼

  1 m0 ðt  jr þ r0 j=cÞ m0 ðt  jr  r0 j=cÞ  4p j r þ r0 j j r  r0 j

ð3:52Þ

This is the exact representation of the pressure from the acoustic dipole but in general case is not informative. Its asymptotics is of interest at distances jrj  jr0 j significantly exceeding the linear dimensions of the dipole. From these distances, the dipole seems to be a point and only in this case it is called a point dipole. Since j r  r0 j  j rj j r0 j j rj2 ¼ r 2 , the decompositions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ jr r0 j ¼ j rj2 2r  r0 þ j r0 j2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 ¼ j rj 1 2 r  r0 =j rj2 þ j r0 j2 =j rj2  r 1 r  r0 =jrj2  1 1

1  r  r 0 =j r j 2  r r r  r0 00 0 m ðt  r=cÞ m ðt  r =cÞ  m0 ðt  r=cÞ  rc

ð3:53Þ

are valid. Substitution of (3.53) in (3.52) determines the asymptotic expression for the dipole pressure field 

 1 h 0 r  r0 00 m ðt  r=cÞ  m ðt  r=cÞ 1  r  r0 =jrj2 4p r rc 

i r  r0 00 0 m ðt  r=cÞ 1 þ r  r0 =jrj2  m ðt  r=cÞ þ rc  rl 0 r ¼ m ðt  r=cÞ  m00 ðt  r=cÞ 3 4p r c

~p 

ð3:54Þ

where l ¼ 2r0 is the vector that connects the source to the drain. The value q ¼ m l is called dipole moment and is an internal characteristic of the dipole. In this terminology, the Formula (3.54) can be converted to

38

3

~p  

General Properties and Character Types of Sound Waves

 1 r 00 0 q r  q ð t  r=c Þ  ð t  r=c Þ 4p r 3 c

ð3:55Þ

which only shows the distance to the observation point and the dipole moment. In order to analyze the obtained ratio (3.55), we will introduce the value of time delay of the signal s ¼ r=c from the source (dipole) to the observation point. In terms of parameter s, expression for dipole pressure in the far zone takes the form of ~p  

c2 s  ðq0 ðt  sÞ  s q00 ðt  sÞÞ; 4p s3

s¼s

r r

ð3:56Þ

If the characteristic time of the dipole moment change is significantly longer than the signal delay time, the above formula is simplified ~p  

c2 s  q0 ðt  sÞ; 4p s3

s¼s

r r

ð3:57Þ

It can be seen at once that in this situation, the dipole field at large distances decreases much faster than the point-type monopole field. The structure of the dipole field is much more complex than the field structure of the monopole. This is explained by the fact that the ratio (3.55) includes the scalar product of the radius-vector r  q ¼ r jqj cos h of the observation point at the dipole moment, where h is the angle between the dipole moment direction and the direction to the observation point. The use of such a record for a scalar multiplication r  q allows to give the expression for pressure the form ~p  

 1 0 r jqj ðt  r=cÞ  jqj00 ðt  r=cÞ cos h 2 4p r c

ð3:58Þ

It follows from the analysis (3.58) that the highest radiation intensity of the dipole is along the dipole’s moment line and the lowest one is perpendicular to it. Build the surface in space according to the following rules: Choose the origin of coordinates at a point r ¼ 0 that coincides with the center of the dipole (a point between the source and the drain, equidistant from them), select a selected axis that coincides with the line, passing through the source and drain, along each line, passing through the center of the dipole and making a component with the selected axis angle h; postpone the value cos h: The resulting spatial figure, often referred to as a “dumbbell,” is called a dipole pattern. It characterizes the amplitude of the emitted sound along the direction of the observation point. The cross section of the pattern of any plane passing through the selected axis is a plane, closed curve that resembles a figure of the number eight. The results of the analysis given here refer to the so-called far field of the dipole, when the distance r to the observation point significantly exceeds the geometric dimensions of the dipole 2j r0 j: In the near field, r j r0 j; the analysis of the acoustic field should be carried out on the basis of the exact ratio (3.52). In this

3

General Properties and Character Types of Sound Waves

39

case, the dipole moment cannot be included in the analysis, as decomposition (3.53, 3.54, 3.55, 3.56, 3.57, 3.58) is no longer valid. It can be seen that the structure of the dipole sound field is much more complex than the structure of the monopole field. If for a monopole, the law of sound pressure drop is the same for any removal of the observation point; for a dipole, it is possible to identify the near and far zones of radiation, where the behavior of the sound differs significantly. At the end of this section, we will consider the problem of sound emission from harmonic sources, i.e., such sources, the flow functions of which are subject to the law of temporal dependence of the species mðtÞ ¼ m0 expði x tÞ; where x is the constant frequency of oscillations. Such a source can be represented as a “breathing” impenetrable sphere, the radius of which changes according to the harmonic law. According to the expression (3.48) for a point monopole field, such a harmonic source creates a pressure in the space determined by the ratio m0 ðt  r=cÞ i x m0 ¼ expðiðx r=c  x tÞÞ 4pr 4p r ix m0 expðiðk  r  x tÞÞ ¼ 4p r

~p ¼

ð3:59Þ

where k ¼ j kj ¼ x=c ¼ 2p=k is wave number and k is wavelength. The acoustic potential corresponding to the monopole is also determined from (3.48) by substituting the flow function mðtÞ: u¼

m0 expðiðk  r  x tÞÞ 4prq0

ð3:60Þ

Comparison of the obtained relations with the expression (3.19) shows that in this case, the point monopole emits a harmonic spherical wave. At the same time, for such a wave, the connection between potential and pressure is significantly simplified and reduced to the following ~p ¼ i x q0 u

ð3:61Þ

The obtained property is common for all harmonic sound waves and is inherent not only in the spherical wave. This can be seen directly for any type of acoustic source and any type of sound wave. Let’s consider now the dipole (3.52), the source and flow of which are made by the in-phase harmonic oscillations. Substitution of an expression in (3.52) mðtÞ ¼ m0 expði x tÞ results in

40

3

General Properties and Character Types of Sound Waves

 ix m0 expðiðk  ðr þ r0 Þ  x tÞÞ 4p j r þ r0 j  expðiðk  ðr  r0 Þ  x tÞÞ  jr  r0 j   ix m0 expðik  r0 Þ expðik  r0 Þ  ¼ expðiðk  r  x tÞÞ 4p j r þ r0 j jr  r0 j

~p ¼ 

ð3:62Þ

Considering that the observation of the dipole field occurs from a distance significantly exceeding its size, i.e., j rj  2j r0 j; we transform (3.62) to the form ~p 

x m0 sinðk  r0 Þ expðiðk  r  x tÞÞ 2pr

ð3:63Þ

which describes the far field of the point harmonic dipole. By its characteristics, the obtained expression is close to the field of the point harmonic monopole (3.59). However, there are also significant differences. Firstly, the dipole field is shifted in phase relative to the monopole field on p=2: Second, the dipole field has an additional amplitude factor 2 sinðk  r0 Þ: In the case when the wavelength of the emitted wave is much greater than the linear dimensions of the dipole, i.e., the condition 2k  r0 1 (condition of the acoustic compactness of the source system) is met, and the expression (3.63) allows the transformation to the form x m0 k  r0 expðiðk  r  x tÞÞ 2p r x xk k  q expðiðk  r  x tÞÞ ¼ ¼ jqj cos h expðiðk  r  x tÞÞ 4p r 4p r

~p 

ð3:64Þ

where h is the angle between the direction to the observation point and the dipole moment q: As follows from the obtained result, the far field of a point dipole is equivalent in its structure to the field of a point monopole multiplied by a directivity diagram cos h: The approach presented here allows to consider more complex sound source systems. In case of acoustic compactness of such system for case of radiation of harmonic oscillations, the full sound field can be represented in the form of a superposition of fields from separate monopolies and, at calculation of the far field, to make in the received relation limiting transitions at r  D and kD 1; where D is the maximum linear size of system that will allow to calculate necessary characteristics of acoustic fields in a far zone by means of rather convenient relations.

References

41

References 1. Lighthill J (1981) Waves in liquids. Moscow, Mir, p 598 2. Dzhrbashyan MM (1966) Integral transformations and representations of functions in the complex domain. Moscow, “Science”, p 672

4

Plane Sound Waves

Plane waves are both an independent type of sound wave and a tool used in the study of the characteristics of wave processes of the most general type. By its structure, the plane wave is probably the simplest wave object. The most general description of this wave can be given as follows. A plane sound wave is such a physical object, for which in three-dimensional space it is always possible to distinguish the direction along which the wave properties can be changed. In planes perpendicular to the direction given, all physical characteristics associated with a plane wave (oscillatory velocity, pressure and density) change only in time. The velocity field of plane wave velocity, its energy flow vector, pressure and density perturbation gradients are always in time and everywhere in space parallel to the allocated direction. Instantaneous picture of physical fields of a plane wave runs along the selected axis with the speed of sound. That is all the limitations of a plane wave. The law of change in time of characteristics of a plane wave in a fixed point is absolutely arbitrary. The object described above is still somewhat complex. The reason for this is the arbitrariness of the law of changing the properties of the wave in time. But bearing in mind that the system of acoustic equations is linear and recalling the harmonic analysis, any temporal dependence of a plane wave can be represented as a superposition (discrete or continuous) of monochromatic oscillations. Knowledge of the properties of each of the monochromats that make up the plane wave allows to describe the properties of the entire wave as a whole. For this reason, it is necessary to study in detail the properties of plane monochromatic waves, the peculiarities of their propagation, reflection from the boundaries of the medium, etc. The description of the plane monochromatic wave will be carried out on the basis of its potential, which, according to (3.14), is represented by the ratio of up ¼ u0 expðiðk  r  xtÞÞ

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_4

ð4:1Þ

43

44

4

Plane Sound Waves

where k and x are the wave vector and the frequency of the sound wave, respectively, and u0 is constant amplitude. The representation (4.1) should be supplemented with a dispersion relation of a homogeneous isotropic space x ¼ kc;

k ¼ j kj ¼

2p k

ð4:2Þ

where c is the speed of sound and k is the wavelength. Expressions (4.1, 4.2) fully describe sound in a homogeneous isotropic and boundless space. But the real environment is not always homogeneous and isotropic. Often, it has boundaries that reflect the sound. Studying the process of plane wave propagation in such situations requires, at the very least, solving the simplest problems that reveal its properties. First of all, let’s consider the problem of the plane wave falling on the plane interface boundary between two mediums. We believe that the interface boundary coincides with the plane ðy; zÞ , x ¼ 0. A plane wave drops to this boundary from an area x\0 that is characterized by the speed of sound c1 . The speed of sound equals to c2 in the area x [ 0. As a result of the interaction of the incident wave with the interface, the reflected wave will appear in the half-space x\0 and the transmitted wave will appear in the half-space at the x [ 0. Since the boundary conditions at the x ¼ 0 are independent of either time or coordinates y and z, the corresponding characteristics of all waves (incident, transmitted and reflected), namely the frequency ky and kz components of the wave vectors, must coincide. According to this, let us present the wave vectors of the incident, reflected and transmitted waves in the form of kin ¼ ðkxin ; ky ; kz Þ;

kref ¼ ðkxref ; ky ; kz Þ;

ktr ¼ ðkxtr ; ky ; kz Þ

ð4:3Þ

Let’s denote the angles of incidence, passage and reflection (i.e., angles between the directions of wave vectors and the axis) by symbols h1 , h2 and h3 accordingly (see Fig. 4.1). Since, as mentioned earlier, the components of wave vectors ky and kz all waves coincide, the value ky2 þ kz2 is an invariant in the whole space. Let’s write down the representation of this invariant for each of the waves: ky2 þ kz2 ¼ k12 sin2 h1 ;

ky2 þ kz2 ¼ k22 sin2 h2

ky2 þ kz2 ¼ k12 sin2 h3 ;

k12 ¼ x2 =c21 ; k22 ¼ x2 =c22

ð4:4Þ

By equating the right parts of the first and third equations, we obtain h1 ¼ h3

ð4:5Þ

4

Plane Sound Waves

45

Fig. 4.1 Illustration of a plane wave fall on the interface boundary

That is, the angle of incidence is equal to the angle of reflection, and the equation of the right parts of the first and second equations leads to sin h1 c1 ¼ sin h2 c2

ð4:6Þ

which is the law of Snellius. Let us now determine the reflection and propagation coefficients of the plane wave through the boundary of the two mediums. For this purpose, it is necessary to use the boundary conditions consisting in the fact that at x ¼ 0 pressure and normal to the boundary components of oscillatory velocity in the upper and lower half-spaces should coincide. In order to realize this idea, it is necessary to set the type of velocity potentials of all the waves under consideration. Let it be uin ¼ Ain expðiðkin  r  xtÞÞ uref ¼ Aref expðiðkref  r  xtÞÞ utr ¼ Atr expðiðktr  r  xtÞÞ

ð4:7Þ

where uin , uref , utr are potentials, Ain , Aref , Atr are complex amplitudes and kin , kref , ktr are wave vectors of the incident, reflected and transmitted waves, respectively.

46

4

Plane Sound Waves

In the lower half-space ðx\0Þ, the velocity potential is equal to the sum of the potential of the incident and reflected waves, i.e., u1 ¼ uin þ uref ; in the upper half-space ðx [ 0Þ, the velocity potential is equal to the potential of the transmitted wave: u2 ¼ utr . The condition of pressure ~p1 and ~p2 equality and on the different sides of the boundary is determined, taking into account the ratio (1.23) for a plane wave, by the ratio ~ p1 jx¼0 ¼ ~p2 jx¼0

,

q1



@u1

@u2

¼ q 2 @t
x¼0 @t
x¼0

ð4:8Þ

and the equality of normal velocity components leads to the expression v1n jx¼0 ¼ v2n jx¼0



@u1

@u2

¼ @x
x¼0 @x
x¼0

,

ð4:9Þ

The use of submissions (4.7) makes the boundary conditions (4.8, 4.9) in explicit form q1 ðAin þ Aref Þ ¼ q2 Atr ; k1x ¼ x cos h1 =c1 ;

k1x ðAin  Aref Þ ¼ k2x Atr

k2x ¼ x cos h2 =c2

ð4:10Þ

The reflection coefficient is defined as the ratio of the time average normal components to the interface of reflected and incident energy flux densities, and the propagation coefficient is the ratio of similar values for transmitted and incident waves. As already shown in (3.9), the density of the flow of acoustic energy is determined by S ¼ ~pv. Since the physical characteristics of natural processes are real values, when using the complex representation (4.7), it is necessary to correctly calculate the effective quantity S and average it over the period T ¼ 2p=x (the characteristic time of the harmonic process). According to the general rules, when specifying pressure ~ p and field of oscillatory velocity v by their complex representations, the actual value S is determined by the expression S¼

~p þ ~p v þ v 1 ¼ ð~pv þ ~p v þ ~ pv þ ~ p vÞ: 4 2 2

ð4:11Þ

In the case of harmonic processes, the averaging over the period vanishes the first two terms, since tZ 0 þT

tZ 0 þT

~pvdt ¼ t0

t0

~p v dt ¼ 0;

4

Plane Sound Waves

47

where t0 is an arbitrary time moment, and the expression (4.11) takes the form 1 S ¼ ð~pv þ ~p vÞ; 4

ð4:12Þ

where the bar above denotes averaging over time. If the acoustic potential of a plane wave in a medium with a density q is represented as u ¼ Aeikx x eiðky y þ kz zxtÞ ; then the x component of the vector (4.12) according to (1.33, 1.34) has the form Sx ¼

xq 2 j Aj k x : 2

ð4:13Þ

Thus, based on (4.13), the reflection and transmission coefficients of a plane wave are determined by the relations R¼

jAref j2 jAin j2

;

T¼

q2 k2x jAtr j2 : q1 k1x jAin j2

ð4:14Þ

Let’s represent, for example, the reflection coefficient in an explicit form. Excluding from (4.10), the amplitude Atr of the transmitted wave leads to the q1 k2x ðAin þ Aref Þ ¼ q2 k1x ðAin  Aref Þ from which it follows Aref q2 k1x  q1 k2x ¼ Ain q2 k1x þ q1 k2x

ð4:15Þ

Substitution of (4.15) and expressions for k1x , k2x , in (4.14) using Snellius law (4.6) results in consistent relations   q2 k1x  q1 k2x 2 R¼ q2 k1x þ q1 k2x     q2 cos h1 =c1  q1 cos h2 =c2 2 q2 tg h2  q1 tg h1 2 ¼ ¼ q2 cos h1 =c1 þ q1 cos h2 =c2 q2 tg h2 þ q1 tg h1

ð4:16Þ

which, in the end, determine the desired reflection coefficient. It is evident from (4.16) that the reflection coefficient is zero at q2 tg h2 ¼ q1 tg h1 . Since the initial conditions for the considered process of the plane wave transient through the interface are the parameters of the mediums c1 , q1 , c2 , q2 and the angle of incidence h1 , it is necessary to write down the condition of

48

4

Plane Sound Waves

complete reflection in such a way as to express the maximum angle of incidence h1 only through the characteristics of the medium. Let’s write down the condition q2 tg h2 ¼ q1 tg h1 first in the form q22 tg2 h2 ¼ q21 tg2 h1 Using the properties of trigonometric functions, the last equality can be given the form q22 sin2 h2 cos2 h2 ¼ q21 sin2 h1 cos2 h1 The right part of this expression by means of trigonometric ratios, and also by means of already written out equality q22 tg2 h2 ¼ q21 tg2 h1 , is transformed to cos2 h2 1 þ tg2 h1 1 þ tg2 h1 ¼ ¼ cos2 h1 1 þ tg2 h2 1 þ q21 tg2 h1 =q22 and the right side, taking into account the Snellius law, is limited to the expression q22 sin2 h2 q22 c22 ¼ q21 sin2 h1 q21 c21 Equating the obtained results, we obtain the equation q22 c22 1 þ tg2 h1 ¼ 2 2 q1 c1 1 þ q21 tg2 h1 =q22 whose decision relatively tg2 h1 has the form tg2 h1 ¼

q22 c22  q21 c21 q21 ðc21  c22 Þ

ð4:17Þ

The obtained solution determining the tangent of the angle of full internal reflection takes place only in cases when c1 [ c2 and at the same time q2 c2 [ q1 c1 , as well as when the sign of inequality in both relations is reversed. The ocean, as a real environment, is bounded in space by the free surface on which it comes into contact with the atmosphere and the bottom. Since the acoustic properties of the ocean itself and the environments bordering it differ significantly, the real limiting surfaces, from the point of view of sound physics, are sometimes called the acoustic boundaries of the ocean. At these boundaries, there are specific processes related to the reflection, propagation and transformation of sound waves. The study of the acoustic properties of the ocean requires a separate consideration of physical processes at different types of boundaries. Water and air—the two

4

Plane Sound Waves

49

environments where acoustic processes are of most interest to us—differ significantly in density and speeds of sound. This situation leads to some unexpected results when reflecting the wave from the boundary of the interface of these environments. In order to identify these unusual properties in the whole volume, it is necessary, in addition to the reflection coefficient (4.16), to determine the coefficient of transmission of the wave in general. Substitution of the ratio jAtr j2 =jAin j2 in the formula (4.14), which is calculated from (4.10) by exclusion Aref , leads to an expression for T T¼4

q1 c1  q2 c2

cos h1 cos h2 cos h1 þ

q 1 c1 q 2 c2

cos h2

ð4:18Þ

2

For convenience of the further presentation, we will result, the formula (4.16) received earlier, in defining the value of the reflection coefficient  R¼

q2 cos h1 =c1  q1 cos h2 =c2 q2 cos h1 =c1 þ q1 cos h2 =c2

2

¼

cos h1  qq1 cc12 cos h2 cos h1 þ

2

q 1 c1 q 2 c2

cos h2

!2 ð4:19Þ

Let the sound wave first fall to the water–air interface from the water; i.e., the values with the index “1” correspond to the water and with the index “2” to the air. Because q1  q2 , c1  c2 , then q1 c1 =q2 c2  1. As a result, at cos h1 6¼ 0 (the wave drop on the partition boundary is not sliding one), reflection and transmission coefficients are set with approximate values R  1;

T 4

q2 c2  3:0  105 q1 c1

ð4:20Þ

Thus, when a sound wave falls from the water into the air, it is almost completely reflected and does not escape into the atmosphere. When a wave falls to the airborne interface and when the values with the index “1” correspond to the air and with the index “2” to the water, there are ratios: q1  q2 , c1  c2 , q1 c1 =q2 c2  1. As a result at the cos h1 6¼ 0, the reflection and propagation coefficients are set with approximate values R  1;

T 4

q1 c1  3:0  105 q2 c2

ð4:21Þ

It follows from (4.21) that there is almost complete reflection of the sound wave ðR  1Þ, but in contrast to the previous case, the pressure in the water near the surface is twice greater than the pressure in the incident wave. This does not contradict the law of energy conservation in any way, since always the ratio

50

4

Plane Sound Waves

RþT 1

ð4:22Þ

is valid. Since R and T also characterize the energy flux density of the reflected and transmitted waves in relation to the incident wave, there is no violation of physical laws. Another form of ratio recording (4.13–4.20), which uses the concept of acoustic impedance of a continuous environment, is given in [1]. Quite specific effects are manifested in the propagation of sound waves in moving media. Let in the homogeneous flow of fluid moving at speed U the source of monochromatic frequency sound x and the receiver are placed at a fixed point in the space. In this case, the linear acoustic system of equations (1.28) takes the form   q0 ru0t þ ðUrÞru ¼ rp ~0t þ q0 Du þ U  r~ q ¼ 0; q

~¼ q

p : c2

ð4:23Þ

The elimination of density and pressure reduces system (4.23) to the wave equation for potential u00t þ 2U  ru0t þ U  rðU  ruÞ  c2 Du ¼ 0

ð4:24Þ

In the case of a plane wave of the form u ¼ u0 expðiðk  r  xtÞÞ, Eq. (4.24) is converted to the form ðx  u  kÞ2 c2 k2 ¼ 0, from which follows 

Uk x ¼ ck þ U  k ¼ k c þ k

 ¼ kðc þ U  ek Þ

ð4:25Þ

the dispersion equation in a moving medium, where ek is a unit vector in the direction of the wave vector of the emitted wave. A distinctive feature of the dispersion equation (4.25) is the constancy of the frequency, in the sense that the same frequency x as that excited by the radiator is recorded at the receiver. This fact is explained by the fact that since the flow of the medium is stationary, the time interval T between the minima (maxima) of radiation remains unchanged throughout the propagation time from the emitter to the receiver, which means that the frequency of the wave remains constant. At the same time, as it follows from the dispersion equation (4.25), the wavelength in the flow differs from the wavelength in a stationary medium. If in a fixed medium the wave number is determined by the expression k ¼ xc , then it takes place in the flow k ¼ c þxUek . This phenomenon is called the Doppler effect in a moving medium. Now, the Doppler effect is considered, when the medium is at rest, and either the source or the receiver moves. Let x be the frequency emitted by the source. First, we consider the case when the source is standing, and the receiver moves with

4

Plane Sound Waves

51

speed U. The value U  k=k determines the component of the receiver speed U along the direction of the phase change of the wave coming from the radiator. Let at some moment of time some maximum (minimum) of the radiated wave (we shall call this maximum first) reach the receiver. At this moment, the next maximum (minimum) (we call it second) is at a distance from the receiver L ¼ cT, where c is the speed of sound in the resting medium. As the receiver moves, this second maximum will reach the receiver at a time T 0 after the arrival of the first maximum. This time is determined by the ratio cT 0 ¼ L þ

Uk 0 Uk 0 T cT þ T k k

ð4:26Þ

0 where L þ Uk the second maximum before meeting k T is the distance traveled by   0 2p 0 2p with the receiver. From (4.26) follows: 1  Uk ck T ¼ T. Since T ¼ x , T ¼ x0 , 0 then the frequency of the signal x received by the receiver is determined by the expression

x0 ¼

  Uk 1 x ck

ð4:27Þ

Let the emitter now move, and let the receiver be still. At the initial moment of time t ¼ 0, the first maximum of the sound wave is emitted and the distance L between the emitter and receiver is equal at this moment. Then, the first maximum will reach the receiver at the moment of time t1 ¼ L=c. The second maximum is emitted at the moment of time t ¼ T, while the emitter is at a distance from the receiver L þ Uk k T. Consequently, the second maximum will reach the receiver at the time t2 ¼ T þ ðL þ U  kT=kÞ=c. Since the difference between the moments of arrival of signals at the receiver is equal tothe desired  period of the received signal T 0 , there is a relationship T 0 ¼ t2  t1 ¼ T 1 þ Uk ck , from which it follows that the frequency of the signal x0 received by the receiver is determined by the expression x0 ¼

x 1 þ Uk ck

ð4:28Þ

From the ratio obtained, it follows that if the emitter approaches the receiver ðU  k\0Þ, then the frequency of the received signal is higher than the frequency of the emitted signal, and if it is removed ðU  k [ 0Þ, then vice versa. At the beginning of this section, the problem of reflection of a plane acoustic wave from the boundaries of the medium interface was studied. Since a plane wave is a specific object for whose radiation it is necessary to have a source of infinite length, the question arises on how to determine the wave field of other, more real sources, in the presence of reflective boundaries. For spherical or cylindrical waves, for example, the Snellius law is not directly applicable. At the same time, there is a natural desire to use the results obtained for plane waves in relation to waves of other types. But how can this desire be put into practice?

52

4

Plane Sound Waves

At the initial stage of wave disturbance studies, the Huygens principle was widely used to describe their propagation. Its essence is that each point of the wave front can be considered as a source of secondary waves propagating in the form of elementary spherical waves in an isotropic medium. The surface that envelops all these elementary waves forms a new wave front. The assumption (which is the most important part of the principle) is that the elementary waves have the maximum intensity in the direction of propagation of the front, which gradually drops to zero in the opposite direction. In Fig. 4.2, the Huygens principle is illustrated by the example of plane (a) and spherical (b) wave propagation. The directivity coefficient of elementary waves for these cases is described by different ratios. Since the Huygens principle, in fact, suggests that any wave should be considered as a spherical wave decomposition from a distributed set of spherical sources, the question arises: Can we choose not spherical but plane waves as elementary ones? Thus, the idea of determining the characteristics of the process of reflection of a wave of any type from a plane surface is reduced to the following process. First, the wave is divided into a set of plane waves, for each of which, according to Snellius law, the reflected (transmitted) wave is determined, and then the results for all reflected (transmitted) plane waves are summed up (integrated), forming the reflected (transmitted) wave field of the sound wave.

Fig. 4.2 Huygens principle

4

Plane Sound Waves

53

So, at the first stage of solving the problem of the reflection of a wave of an arbitrary type from a plane surface, the problem arises of its decomposition into plane waves. At the second stage, the problem of its reflection from the interface (which has already been solved in the theory of plane waves) is solved for each plane wave, and then, at the third stage, the reflected wave is constructed on the basis of a set of reflected plane waves. Thus, at the first stage of solving the problem of reflection of a wave of any type from a plane surface there appears a problem of its decomposition on plane waves. At the second stage, for each plane wave, the problem of its reflection from the interface (which has already been solved earlier in the theory of plane waves) is solved for each plane wave from the obtained decomposition, and then, at the third stage, the construction of the reflected wave on the basis of a set of reflected plane waves is made. To illustrate the mathematical technique of the described method, let us consider the case of a spherical wave (the results for a cylindrical wave are obtained by analogy). Let the monochromatic wave emitter of frequency x be located in a homogeneous space characterized by the speed of sound c. As has been shown earlier (ratio (3.19)), the potential of the traveling monochromatic spherical wave emitted by the source at the point ð0; 0; 0Þ is set by u¼A

expðiðkr  xtÞÞ ; r

k ¼ x=c; r ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y 2 þ z2

ð4:29Þ

Since the constant amplitude factor A and exponential dependence on time expðixtÞ do not depend on spatial coordinates, the problem of potential decomposition (4.29) by plane waves is reduced to the decomposition of the value expðikrÞ=r. In general, a single plane wave is described by the ratio Aðkx ; ky ; kz Þ expðiðkx x þ ky y þ kz zÞÞ

ð4:30Þ

where Aðkx ; ky ; kz Þ is the amplitude, but the components of the wave vector kx , ky and kz are linked by the ratio kx2 þ ky2 þ kz2 ¼ k2 ¼ x2 =c2

ð4:31Þ

that allows you to express a component kz through a component kx , ky kz ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2  kx2  ky2

ð4:32Þ

Thus, the whole set of possible plane waves is set by variation of two parameters, namely wave components kx , ky , so that the ratio (4.30) can be given the form

54

4

Aðkx ; ky Þ expðiðkx x þ ky y þ

Plane Sound Waves

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2  kx2  ky2 zÞÞ

ð4:33Þ

Now, using the ratio (4.33), we must form a spherical wave expðikrÞ=r. This “formation” is a simple summation of the expression (4.33) for all the different parameter values allowed. Since these wave components change continuously in the range ð1; þ 1Þ, the summation turns into integration in infinite limits, so that the formula of spherical wave decomposition on plane waves takes the form expðikrÞ ¼ r

Zþ 1 Zþ 1 Aðkx ; ky Þ expðiðkx x þ ky y þ 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2  kx2  ky2 zÞÞdkx dky ð4:34Þ

1

In order for decomposition (4.34) to be fully defined, it is necessary to know the amplitude factor Aðkx ; ky Þ. It can be determined by applying the (4.34) two-dimensional inverse Fourier transform 1 ð2pÞ2

Zþ 1 Zþ 1 ð. . .Þ expðiðkx x þ ky yÞÞdxdy 1

ð4:35Þ

1

where by “ð. . .Þ object” we mean an object that is undergoing transformation. And then, it follows from (4.34, 4.35) 1 Aðkx ; ky Þ ¼ 2 4p

Zþ 1 Zþ 1 1

1

expðikrÞ expðiðkx x þ ky yÞÞdxdy r

ð4:36Þ

The implementation of integration in (4.36) results in Aðkx ; ky Þ ¼

i 2pkz

ð4:37Þ

Now, substitution (4.37) in (4.34) gives the final expression for the decomposition of the spherical wave on plane waves qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zþ 1 Zþ 1 expðiðkx x þ ky y þ k2  k2  k2 jzjÞÞ x y expðikrÞ i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ dkx dky r 2p 2  k2  k2 k x y 1 1

ð4:38Þ

The ratio (4.38) allows to consider the problem of reflection of a spherical wave emitted by a point monopole from the surface of the interface between two media. The schematic representation of the reflection process is shown in Fig. 4.3. Let a separate plane wave

4

Plane Sound Waves

55

Fig. 4.3 Spherical wave reflection

expðiðkx x þ ky y þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2  kx2  ky2 jzjÞÞ= k2  kx2  ky2

expðiðkx x þ ky y þ kz jzjÞÞ=kz from the decomposition (4.38) be reflected from the interface z ¼ z0 . At some point Sðx0 ; y0 ; z0 Þ, the amplitude of the wave equals to expðiðkx x0 þ ky y0 þ kz z0 ÞÞ=kz

ð4:39Þ

As a result of the process of reflection from the interface, the reflected wave at the same point Sðx0 ; y0 ; z0 Þ will have an amplitude Rðkx ; ky Þ expðiðkx x0 þ ky y0 þ kz z0 ÞÞ=kz

ð4:40Þ

where Rðkx ; ky Þ is the amplitude coefficient of reflection of a plane wave with components kx , ky . At the observation point Pðx; y; zÞ, the reflected plane wave will be described by the potential, which is the product of the amplitude (4.40) at the interface and the actual potential of the plane wave propagating S in the direction of P: expðiðkx ðx  x0 Þ þ ky ðy  y0 Þ  kz ðz  z0 ÞÞÞ. The minus sign before the wave number kz is conditioned by the fact that for the geometry considered in Fig. 4.3 the reflected wave propagates in the negative direction of the axis z. As a result, the considered plane reflected wave at the point Pðx; y; zÞ under consideration is described by the potential Rðkx ; ky Þ expðiðkx x0 þ ky y0 þ kz z0 ÞÞ=kz  expðiðkx ðx  x0 Þ þ ky ðy  y0 Þ  kz ðz  z0 ÞÞÞ Rðkx ; ky Þ expðiðkx x þ ky y  kz ðz  2z0 ÞÞÞ=kz

ð4:41Þ

56

4

Plane Sound Waves

The sum of all reflected plane waves determines the value of the reflected sound field potential at the point Pðx; y; zÞ as i 2p

Zþ 1 Zþ 1 Rðkx ; ky Þ 1

1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðiðkx x þ ky y  k2  kx2  ky2 ðz  2z0 ÞÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  dkx dky k2  kx2  ky2

ð4:42Þ

The result shows that the reflected field at the point Pðx; y; zÞ can be represented as a falling field of an imaginary source Q0 , which is a mirror reflection relative to the surface of the real source interface Q. In contrast to the real source, the imaginary source is not an isotropically emitting monopole, but a monopole with an amplitude pattern of directivity Rðkx ; ky Þ in the phase space of wave numbers kx , ky . In the case when the interface is a free surface of water, in zero approximation Rðkx ; ky Þ  1, and then the expression (4.42) for the reflected wave is transformed into a ratio of i  2p

Zþ 1 Zþ 1 dkx dky 1

1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðiðkx x þ ky y  k2  kx2  ky2 ðz  2z0 ÞÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  k2  kx2  ky2

ð4:43Þ

expðikr 0 Þ r0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 0 ¼ x2 þ y2 þ ðz  2z0 Þ2 is the distance from the imaginary source of that observation point. As a result, the potential of the full field (falling plus reflected) of the monopole in the presence of the interface in this case takes the form ¼

u ¼ A expðixtÞ

expðikrÞ expðikr 0 Þ  r r0

 ð4:44Þ

It should be emphasized that the ratio (4.44) is only valid when the density of the medium above the interface is negligibly small in comparison with the density of the medium under it. In the case when the source is located near the surface (this means that its depth is significantly less than the wavelength of the emitted wave: kz0  1), the ratio (4.44)

4

Plane Sound Waves

57

can be simplified. To do this, enter the distance R from the point of mirror symmetry on the surface of the partition to the observation point Pðx; y; zÞ, as shown in Fig. 4.3. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Because R ¼ q2 þ ðz þ z0 Þ2 , where q2 ¼ x2 þ y2 , pffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðikrÞ expðikRÞ  expðikð q2 þ z2  RÞÞ r R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðikRÞ expðikð R2  2zz0  RÞÞ  R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðikRÞ 2zz0 expðikRð 1  2  1ÞÞ ¼ R R expðikRÞ ikzz0 expðikRÞ expð expðikz0 cos hÞ Þ¼  R R R

ð4:45Þ

Similarly, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðikr 0 Þ expðikRÞ expðikð q2 þ ðz þ 2z0 Þ2  RÞÞ  r0 R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðikRÞ  expðikð R2 þ 2zz0  RÞÞ R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðikRÞ 2zz0 ¼ expðikRð 1 þ 2  1ÞÞ R R expðikRÞ ikzz0 expðikRÞ expð expðikz0 cos hÞ Þ¼  R R R

ð4:46Þ

Substitution (4.45, 4.46) in expression (4.44) for the acoustic field potential results in (4.47) expðikRÞ sinðkz0 cos hÞ R expðikRÞ cos h  2iAkz0 expðixtÞ R

u  2iA expðixtÞ

ð4:47Þ

which describes the field of a point dipole [see ratio (3.63)] located on the surface of the interface, which corresponds to reality, because when the source approaches the surface, the imaginary source also approaches it and in the limit forms an acoustic dipole with a real source.

Reference 1. Brekhovskikh LM, Goncharov VV (1982) Introduction to continuum mechanics. (Appendix to the theory of waves) M.: Science, 336 p

5

Geometric Acoustics

All previous studies have been allowed to obtain constructive results when considering a certain type of sound wave (plane, spherical, or cylindrical). In reality, arbitrary sound waves do not possess the properties of the above types, and their consideration is more difficult. At the same time, there are situations when in the local area of space the sound wave is similar to a plane wave, and in each area, the length of which is comparable with the wavelength, the wave can be roughly considered plane. If such a condition is met, it is possible to introduce the concept of sound rays, tangent to which in each point coincides with the direction of the wave propagation. This approach to the study of sound properties is called the approximation of geometric acoustics, which is true for the case k ! 0. Thus, we will consider such a sound field, which is locally described by the potential of the plane wave u ¼ Aðr; tÞ expðiwðr; tÞÞ:

ð5:1Þ

The amplitude Aðr; tÞ is considered to be a slowly varying function compared to the function wðr; tÞ called eikonal (angle). In the representation of the potential u by the ratio (5.1), the function of amplitude and eikonal cannot be set in any arbitrary way, because the potential itself must, at least approximately, satisfy the wave equation—one must never forget about the physical nature of the phenomenon. Substitution (5.1) in the wave equation @ 2 u=@t2  c2 Du ¼ 0 leads to the expression of h

i A00tt þ 2iA0t w0t þ iAw00tt  Aðw0t Þ2 h i  c2 DA þ 2irArw þ iADw  AðrwÞ2 ¼ 0

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_5

ð5:2Þ

59

60

5

Geometric Acoustics

Within the framework of the adopted model—slow changes in space and time of amplitude A—in Eq. (5.2), it is necessary to group the terms by the same degrees of smallness. It is necessary to take into account that since (5.1) must describe the plane wave approximately, albeit locally, the eikonal must satisfy the approximate relations rw  k;

w0t  x;

ð5:3Þ

where k, x—the local (in space and time) wave vector and frequency of the sound wave, connected by the ratio k2 ¼ x2 =c2 . As a result, two important ratios from Eq. (5.2) follow ðw0t Þ2  c2 ðrwÞ2 ¼ 0 2rArw þ ADw ¼ 0;

ð5:4Þ

which are called eikonal and transfer equations, respectively. Nowhere else had any assumptions been made about the constancy of sound velocity in space. Therefore, the obtained equations (5.4) are convenient for describing the wave propagation along with the medium with changing characteristics. If the properties of the medium through which the sound wave propagates do not change over time, the frequency of the wave remains constant along the ray. Let’s see what the full derivative of frequency is for. dx @x @x @x _ ¼ þ r_ þ k; dt @t @r @k

ð5:5Þ

where the dot above denotes the total time derivative. Since the change of frequency along the ray is related to the change of the wave vector by the ratio resulting from Formula (5.3)   @x @2w @ @w _ ¼ ¼ ¼ k; @r @r@t @t @r

ð5:6Þ

and frequency variations at change of a wave vector is nothing else, as group speed @x=@k which along a ray coincides with speed of a point on a ray r_ , that is r_ ¼

@x ; @k

ð5:7Þ

substitution (5.6, 5.7) in (5.5) results in dx @x @x @x @x @x @x ¼ þ     : dt @t @r @k @k @r @t

ð5:8Þ

5

Geometric Acoustics

61

As in stationary case @x=@t ¼ 0, then from (5.8) follows dx=dt ¼ 0 that it was necessary to prove. In order to finally describe the process of sound propagation in an inhomogeneous environment, it is necessary to define the shape of the sound rays or at least the ratio to which it (the form) satisfies. Let’s turn to Eqs. (5.6, 5.7) and rewrite them in a slightly different forms, allowing to introduce the characteristics of the ray. From Eq. (5.3) and the eikonal equation, the dispersion relation x ¼ cjrwj ¼ ck; follows, the substitution of which in (5.6) gives @x k_ ¼   rðckÞ  krc: @r

ð5:9Þ

In (5.9), the term crk is thrown away because the heterogeneity of the medium primarily affects the change in sound velocity and then, indirectly, the change in the wave vector. For the same reason, it follows from (5.7) r_ ¼

@x @ðckÞ @k k ¼ ¼c ¼ c ¼ cs; @k @k @k k

k s¼ ; k

ð5:10Þ

where s is a unit tangent vector to the ray. Knowledge of the law of tangent change fully determines the law of ray change. Therefore, it is enough to determine the evolution s to answer the question about the behavior of the sound ray. Because k ¼ ks, so   _ ¼ k_s þ ks _ ¼ k_s þ x_ s ¼ k_s  x c_ s k_ ¼ ðksÞ c c2   k @c k k ð3:18Þ  r_ s  k_s  ðrc  r_ Þs ¼ k_s  ðrc  csÞs ¼ k_s  c @r c c ð3:17Þ

¼ k_s  kðrc  sÞs ¼ krc: The use of the last equality after the contraction k leads to the s_ ¼ ðrc  sÞs  rc:

ð5:11Þ

Introduction of an element of ray length dl is passed by sound in time dt, so that dl ¼ cdt leads to the final appearance of the equation for the shape of the rays ds s rc ¼ ðrc  sÞ  : dl c c

ð5:12Þ

62

5

Geometric Acoustics

From the differential geometry, it is known that the tangent derivative to the curve along the length of its arc is equal to the ratio of the unit normal vector N to the curve to the local radius of curvature R, i.e., ds N ¼ : dl R

ð5:13Þ

The right side (5.12) is a covariant derivative of the sound velocity in the direction of the tangent to the ray, which is equal to the projection of the sound velocity gradient to the normal direction, i.e. s rc N ¼  ðN  rcÞ: ðrc  sÞ  c c c

ð5:14Þ

Equations (5.13) and (5.14) allow to write Eq. (5.12) as 1 1 ¼  ðN  rcÞ; R c

ð5:15Þ

where it follows that the ray curves toward the reduction of the speed of sound. In previous discussions of wave trajectories, it was assumed that the diameter of the ray tube along which the sound oscillations propagate tends to zero. In a real physical situation, the sound energy can be concentrated in some finite area of space, i.e., the wave is a wave packet. Suppose that the wave packet under study is such that its time spectral expansion includes only such components whose frequencies are weakly different from some mean frequency of the packet x, and the components of the spectral expansion along the wave vectors are also close to the mean wave vector k. Let the package be described at the beginning of time by the function u ¼ f ðrÞ expðik  rÞ:

ð5:16Þ

The amplitude function of the packet f ðrÞ is different from zero in some limited area of space, the size of which is larger than the average wavelength 1=k. The decomposition of this function in the Fourier integral on spatial harmonics has the form Zþ d f ðrÞ ¼

f ðkÞ expðiDk  rÞdk: d

But then the wave function of the packet itself can be represented as decomposition by components of the form, uðkÞ  expðiðk þ DkÞ  rÞ:

ð5:17Þ

5

Geometric Acoustics

63

The frequency corresponding to this spectral component is a function k þ Dk. Thus, at the moment of time t, the spectral component (5.17) will look like uðkÞ  expðiðk þ DkÞ  r  ixðk þ DkÞtÞ:

ð5:18Þ

Use of smallness jDkj in comparison to jkj allows to decompose the frequency xðk þ DkÞ in a row @x Dk þ    ; @k

xðk þ DkÞ  xðkÞ þ

substitution of which in (5.18) allows to give the spectral component uðkÞ in the following form uðkÞ  expðiðk  r  xtÞÞ expðiDkðr  @x=@ktÞÞ:

ð5:19Þ

Producing the inverse Fourier transform (5.19), we obtain the expression for the wave packet function at the time t u ¼ expðiðk  r  xtÞÞf ðr  @x=@ktÞ:

ð5:20Þ

Comparison of (5.20) and (5.16) shows that during the time t, the picture of amplitude distribution in the package has moved by a value @x=@kt. Therefore, the propagation speed of the wave packet is equal to U¼

@x @k

ð5:21Þ

and it’s called group speed. It should be noted that the movement of the wave packet without changing its shape, as follows from (5.20), is an obvious approximation. In reality, the wave packet “smears” through the space as it moves, and the area it occupies expands. To make sure of this, it is enough to substitute the representation of the wave packet function (5.16) in the wave equation Du þ k2 u ¼ 0: Because,  Du ¼ Dðf ðrÞ expðik  rÞÞ ¼

 @2f @f 2  k þ 2ik  f ; @r2 @r

Then, there is an equation describing the change in the distribution of the amplitude of the package in space

64

5

@2f @f ¼ 0: þ 2ik  @r2 @r

Geometric Acoustics

ð5:22Þ

Since the main amplitude changes take place in the transverse direction to the group velocity of the packet, the expression for @ 2 f =@r2 can be roughly replaced by a transverse Laplacian D? f , in which the derivatives are taken only in two coordinate directions locally orthogonal to wave vector k. The scalar product k  @f =@r allows an obvious identical replacement k@f =@l, where l is the arc length of the average trajectory of the wave packet. As a result, (5.22) takes the form D? f þ 2ik

@f ¼0 @l

ð5:23Þ

Since the arc length vector l is orthogonal to the transverse coordinate directions included in the transverse Laplacian, Eq. (5.23) is parabolic in its type. In fact, this is the diffusion equation of the distribution function of the amplitude of the wave packet in space. The properties of the solutions of such an equation are the spreading in the space of initial distribution. The diffusion coefficient has the value 1=2k ¼ k=4p. The obtained result proves the “smearing” of the wave packet space. So far, all the problems of radiation and sound propagation in the marine environment studied have been considered, with rare exceptions, for cases of homogeneous and isotropic liquids. In reality (sea, ocean), fluid is heterogeneous, giving rise to new phenomena and, consequently, new problems in describing the acoustic processes in such tasks. The radiation theory, the basis of which was described above, is quite effective, though approximate, in studying the processes of emission and propagation of sound waves in heterogeneous media. To understand the basic ideas of the application of ray theory in heterogeneous media, let’s consider a liquid, the parameters of which depend on only one coordinate (for certainty, it will be the coordinate z). The sound source is considered monochromatic frequency x. In all relations, the time dependence expðixtÞ is omitted to reduce the records. As long as the horizontal direction of the environment characteristic is unchanged, the sound velocity is a function of only one vertical coordinate z: c ¼ cðzÞ. Earlier, it was said that the application of radiation theory is justified in k ! 0. In order to clarify this ratio and to put into it a physical meaning, it is necessary to remember that when spreading in an inhomogeneous medium, the ray approach is justified only in the case when the relative change in the speed of sound at the length of the sound wave is rather small. Let’s assume that the sound velocity is equal to c1 ¼ cðz1 Þ at the point with the coordinate z1 and the sound velocity is equal to c2 ¼ cðz2 Þ at the point with the coordinate z2 . Record the speed of the sound at the point z2 through the speed at the point z1

5

Geometric Acoustics

65

c2 ¼ cðz2 Þ ¼ cðz1 þ ðz2  z1 ÞÞ ¼ cðz1 Þ þ

dcðz1 Þ ðz2  z1 Þ þ    dz1

ð5:24Þ

Then, the relative change in sound velocity will be determined by the value jc2  c1 j jc1 þ ðz2  z1 Þdcðz1 Þ=dz1 þ     c1 j ¼ c1 c1 jðz2  z1 Þdcðz1 Þ=dz1 j  : c1

ð5:25Þ

The resulting value, as mentioned above, should be much less than one at wavelength. The words “at wavelength” mean that the difference between the vertical coordinates z1 and z2 should be equal to the wavelength k, i.e., jz2  z1 j ¼ k: Then, the expression smallness condition (5.25) takes on the final form of the applicability condition of the ray theory   k dc  1: c  dz 

ð5:26Þ

Let us now consider the problem of the distribution of a single ray in the conditions of the proposed model of an inhomogeneous medium. Let, at any point z of the medium, the angle of incidence of the ray is determined by the value hðzÞ. At this point, the speed of sound is equal to cðzÞ. In a nearby point z þ dz, the angle of incidence is determined by the quantity hðz þ dzÞ and the speed—by the quantity cðz þ dzÞ. Since the horizontal components of the wave number k ¼ 2p=k at levels z and z þ dz must coincide, the following condition must be valid sin hðzÞ sin hðz þ dzÞ ¼ : cðzÞ cðz þ dzÞ

ð5:27Þ

Let’s write the right part of the ratio (5.27). sin hðz þ dzÞ sin hðzÞ þ cos hðzÞðdhðzÞ=dzÞdz þ oðdzÞ ¼ cðz þ dzÞ cðzÞ þ ðdcðzÞ=dzÞdz þ oðdzÞ sin hðzÞ ¼ ½1 þ cot hðzÞðdhðzÞ=dzÞdz þ oðdzÞ cðzÞ   1 ðdcðzÞ=dzÞdz þ oðdzÞ 1 cðzÞ     sin hðzÞ 1 ¼ 1 þ cot hðzÞðdhðzÞ=dzÞ  ðdcðzÞ=dzÞ dz þ oðdzÞ : cðzÞ cðzÞ

66

5

Geometric Acoustics

Substitution of this result in (5.27) leads to the     sin hðzÞ sin hðzÞ 1 ¼ 1 þ cot hðzÞðdhðzÞ=dzÞ  ðdcðzÞ=dzÞ dz þ oðdzÞ ; cðzÞ cðzÞ cðzÞ Where from   1 cot hðzÞðdhðzÞ=dzÞ  ðdcðzÞ=dzÞ dz þ oðdzÞ ¼ 0: cðzÞ In order to achieve this equality to the extent possible, it is necessary to put the member oðdzÞ in brackets in zero cot hðzÞðdhðzÞ=dzÞ 

1 ðdcðzÞ=dzÞ ¼ 0: cðzÞ

In the equivalent record, this expression looks like d lnðsin hðzÞ=cðzÞÞ ¼ 0; dz whose integration leads to the final result for the law of sound ray propagation in a vertically heterogeneous environment: sin hðzÞ ¼ const: cðzÞ

ð5:28Þ

It can be seen from (5.28) that the higher the sound velocity at the observation point, the closer the direction of propagation of the ray to the horizontal one. That is, the ray, with its propagation in a heterogeneous environment, curves. This phenomenon is called refraction. Find the connection between the curvature of the ray and the sound velocity gradient. For this purpose, let’s rewrite the ratio (5.28) in the form sin hðzÞ ¼ qcðzÞ;

ð5:29Þ

where q is the same constant in Eq. (5.28), which is a characteristic parameter of the studied ray in a given medium. We differentiate the ratio (5.29) by z. cos hðzÞ

dhðzÞ dcðzÞ ¼q : dz dz

If to enter a ray length element ds using its definition ds ¼ dz= cos hðzÞ, the last equation can be rewritten as

5

Geometric Acoustics

67

dh dc ¼q ; ds dz

ð5:30Þ

that is, the curvature of the ray dh=ds on a given horizon z is proportional to the sound velocity gradient. This increases the angle of incidence of the ray as the speed of sound with depth increases. This is equivalent to reducing the slip angle with depth if the speed of sound increases. In order to determine the phase characteristics of a ray when it propagates in an inhomogeneous environment, it is necessary to know the length of the path passed by the sound ray. Let the horizontal coordinate of the ray be set by a value r. Its small change dr at a small vertical shift dz is determined by the ratio dr ¼ jtan hdzj. When the ray passes from point z1 to point z2 , the horizontal offset will be the value   z  Z 2    r ¼  tan hðzÞdz:  

ð5:31Þ

z1

Such a simple formula is only valid when the ray from z1 to the z2 derivative of the speed of sound dc=dz does not change its sign all the way. A sign change occurs when the ray passes the turning point, as shown in Fig. 5.1. In this case, for correct definition of the horizontal distance passed by the ray, it is necessary to divide the segment ½z1 ; z2  into two segments ½z1 ; z0  and ½z0 ; z2 , in each of which the sign of the derivative velocity of sound on the vertical coordinate is constant. And then the ratio (5.31) is easy to modify   z   z0  Z 2  Z     r ¼  tan hðzÞdz þ  tan hðzÞdz:  0   z1

ð5:32Þ

z

In case of a larger number of turning points, the full segment ½z1 ; z2  should be divided into a greater number of segments, for each of which Formula (5.31) will be valid. Fig. 5.1 The turning point

68

5

Geometric Acoustics

In order to determine the phase incursion along the ray, it is necessary first to determine the time the ray has passed its curved path. The run time of the ray of the small element of the arc of its trajectory is equal to dt ¼

  ds  dz  ¼ : c c cos h

Then, the total travel time will be determined by the ratio   z  Z 2   dz :  t¼  cðzÞ cos hðzÞ  

ð5:33Þ

z1

Now, the phase raid can simply be defined as the product of the frequency of sound for the duration of the run of the ray of its trajectory Du ¼ xt

ð5:34Þ

where t is defined by the expression (5.33). As it has already been shown earlier, the vector of group speed in the geometrical theory of acoustics is directed on a tangent to a trajectory of a ray, that is the stream of sound energy occurs along the ray. Let us now study the question of how energy is distributed in space if the transmitter is a point source. Consider the system of rays coming out of the source within the angles ½h0 ; h0 þ dh0 , as shown in Fig. 5.2. Keep the transmitter at the point of r ¼ 0. The distance traversed by the ray at an angle h0 is a function of the angle of exit from the source, i.e., rðh0 Þ. It is clear that the section of the ray tube of the selected rays in the plane of the drawing is equal to j@r=@h0 j cos hdh0 , where h is the angle of inclination of the ray at the point of observation.

Fig. 5.2 The system of emitted ray’s

5

Geometric Acoustics

69

Since the entire radiation pattern is axially symmetrical with respect to the axis z, the total cross-sectional area of the system of rays coming out of the source in the angle ½h0 ; h0 þ dh0  range is    @r  dS ¼ 2pr   cos hdh0 : @h0

ð5:35Þ

The power dW emitted by the source in the angle range dh0 flows through this section. Since near the source, when vertical stratification is negligible, this interval emits part of the total energy W determined by the ratio of the solid angle dh0 of the interval (this value is equal 2p sin h0 dh0 ) to the total solid angle of radiation 4p. So that way dW ¼ W

2p sin h0 dh0 W ¼ sin h0 dh0 : 2 4p

ð5:36Þ

Since the density of the sound intensity flux is defined as the ratio of the power passing through the selected section to the value of this section, we obtain the desired value of (5.36) by (5.35) I¼

dW W sin h0 ¼ : dS 4pr j@r=@h0 j cos h

ð5:37Þ

If the transmitter is non-isotropic and is described by a directional diagram Dðh0 ; u0 Þ, the expression (5.37) should be multiplied by this value. If the medium was homogeneous and the source was point and isotropic, a spherical wave would spread in the medium, the intensity of which is known: I0 ¼ W=ð4pðr 2 þ z2 ÞÞ. Let’s introduce the ratio of intensity I to this value I0 f ¼

ðr 2 þ z2 Þ sin h0 : r j@r=@h0 j cos h

ð5:38Þ

The ratio defined in this way is called the focus factor of the ray tube. At the f \1, the rays in the tube disperse, its cross section grows, and the intensity of the sound wave drops. When the ratio is reversed, the rays thicken and, as a consequence, the intensity increases—the rays are focused. At f  1 and f 1, the ray theory becomes inapplicable and it is necessary to address the wave description of a sound field. Of particular interest is the highlighted case f ¼ 1. It takes place either in case r ¼ 0 or @r=@h0 ¼ 0. In the first case, we get to the source, where the intensity of the finite value is concentrated in the point, because it has the beginning of all the rays.

70

5

Geometric Acoustics

Fig. 5.3 Sound velocity profile cðzÞ, the ray and caustic system K  K

The second case defines the curve in the space described by equation @rðh0 ; zÞ ¼ 0: @h0

ð5:39Þ

Since the equation of each of the rays is set in the form r ¼ rðh0 ; zÞ, the ratio (5.39) describes the envelope family of rays. The geometric points where the focus factor turns to infinity are called caustics. So, the caustic matches the envelope. In Fig. 5.3, the approximate ray and caustic path are presented. In the area of the drawing above the caustic, two rays pass through each point of the space. Caustic is a family of points through which only one ray passes. Below the caustic, in the shadow zone, no rays at all. Thus, caustic is a sharp boundary between the area where there are sound oscillations and the area where they do not occur. It is clear that such a description does not correspond to the reality and the definition of the sound field should be carried out by other methods near the caustic. A detailed description of the geometric ray approach is given in [1].

Reference 1. Lighthill J (1981) Waves in liquids. Mir, Moscow, 598 p

6

Ray Description of the Sound Field in Inhomogeneous Media

The ray approach to sound phenomena developed in the previous section, together with the imaginary source method described in Chap. 4, is applicable to the problem of field representation in a liquid layer. Figure 6.1 shows a schematic representation of the problem. Let’s assume that there is a point source of monochromatic frequency x sound in the layer of thickness h ¼ h þ  h at the depth z = 0. Different areas of the environment are characterized by corresponding densities qi and sound speeds ci in them. The z ¼ h and z ¼ h þ are interfaces characterized by reflection coefficients V and V þ when the wave from the region z 2 ½h ; h þ  falls on them. It is necessary to determine the sound field value at the point Pðr; zÞ. Within the framework of the ray approach, a system of imaginary sources appears in space, organized as follows. In the first step of designing this system, we will reflect the real source relative to the planes z ¼ h and z ¼ h þ . Since the intensity of the imaginary source ½1 obtained by the reflection relative to the plane z ¼ h is equal to V , the distance from it to the observation point is the same R21 ¼ r 2 þ ðz  2h Þ2 , accordingly, the intensity of the imaginary source ½1 þ  obtained by the reflection relative to the plane z ¼ h þ is equal to V þ , and the distance from it to the observation point is R21 þ ¼ r 2 þ ðz  2h þ Þ2 , then to the field
expðikR0 Þ=R0 (k2 ¼ x2 c20 , R20 ¼ r 2 þ z2 ), from the real source at the point Pðr; zÞ it is necessary to add the fields of these imaginary sources V expðikR1 Þ=R1 and V þ expðikR1 þ Þ=R1 þ . At first glance, it seems that this can be stopped, but in reality it is not, because the boundary conditions on the interfaces z ¼ h and z ¼ h þ are not satisfied. In fact, the actual source and the imaginary source ½1 together satisfy the boundary conditions on the surface z ¼ h , while the actual source and the imaginary source ½1 þ  together satisfy the boundary conditions on the surface z ¼ h þ . But when all three sources ½1 þ  are considered at the same time, the presence of an imaginary

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_6

71

72

6 Ray Description of the Sound Field in Inhomogeneous Media

Fig. 6.1 Radiation picture of the sound field in the layer

source ½1 þ  violates the boundary conditions at z ¼ h , and similarly, the presence of an imaginary source ½1 violates the boundary conditions at z ¼ h þ : In order to eliminate the violation of the boundary conditions at z ¼ h þ ; it is necessary to introduce an additional imaginary source ½2, obtained by reflection of the imaginary source ½1 þ  with respect to the plane z ¼ h þ : For the same reasons, at the z ¼ h another imaginary source ½2 þ  obtained by reflection of the imaginary source ½1 relative to the plane z ¼ h is needed to eliminate disturbances in boundary conditions. The distances from these new imaginary sources to the observation point Pðr; zÞ will be equal R22 ¼ r 2 þ ðz  2ðh  h þ ÞÞ2 and R22 þ ¼ r 2 þ ðz  2ðh þ  h ÞÞ2 , respectively. Their intensities will be determined by the values V þ V and V V þ ; of course, they will be equal. The fields they create in a point Pðr; zÞ are specified by the values V þ V expðikR2 Þ=R2 and V V þ expðikR2 þ Þ=R2 þ . Now, the real source field plus imaginary source fields will meet the boundary conditions at the lower boundary and the real source field plus imaginary source fields at the upper boundary. However, the fields of all five sources (real plus four imaginaries) will not meet the boundary conditions. But the degree of divergence will be reduced, as the contribution of the sources ½2 þ  and ½2 at the lower and upper bounds determining the degree of divergence will be smaller, as they are more distant from the respective bounds than the sources ½1 þ  and ½1. We can continue the process of generation of imaginary sources, creating, by analogy, sources ½3, ½3 þ , ½4, etc., gradually reducing the sample rate under boundary conditions. In order to precisely meet the boundary conditions, the series of auxiliary imaginary sources should be infinite. Now, let’s present in analytical form the result of adding up the fields of sources—the real plus all imaginary. In its formal expression, this amount looks like

6

Ray Description of the Sound Field in Inhomogeneous Media

expðikR0 Þ expðikR1 Þ expðikR1 þ Þ þ V þ Vþ R R1 R1 þ  0  expðikR2 þ Þ expðikR2 Þ þ V V þ þ V þ V R2 þ R2   expðikR3 þ Þ expðikR3 Þ þ V þ V V þ þ V V þ V R3 þ R3   expðikR4 þ Þ expðikR4 Þ þ V V þ V V þ þ V þ V V þ V þ... ¼ R4 þ R4 expðikR0 Þ expðikR1 Þ expðikR1 þ Þ þ V þ Vþ ¼ R0 R1 R1 þ  expðikR2 þ Þ expðikR2 Þ þ V þ V þ R2 þ R2  expðikR3 þ Þ expðikR3 Þ þ Vþ þ V R3 þ R3  expðikR4 þ Þ expðikR4 Þ þ V 2þ V2 þ R4 þ R4  expðikR5 þ Þ expðikR5 Þ þ Vþ þ V þ... ¼ R5 þ R5 expðikR0 Þ expðikR1 Þ expðikR1 þ Þ þ V þ Vþ ¼ R0 R1 R1 þ  1 X expðikR2n þ Þ expðikR2n Þ þ ðV þ V Þn þ R2n þ R2n n¼1  expðikRð2n þ 1Þ þ Þ expðikRð2n þ 1Þ Þ þ V þ Vþ ; Rð2n þ 1Þ þ Rð2n þ 1Þ

73

u¼

ð6:1Þ

    where R2m ¼ r 2 þ ðz  2ðmh  mh ÞÞ2 , m ¼ m2 ", m ¼ m2 #. The symbol ½ " means rounding to the nearest larger whole and ½ # to the nearest smaller whole. The obtained ray representation of the field is valid when the reflection coefficients do not depend on the value of the angle of incidence of the ray on the interfaces. In the case of such a dependence, this fact should be taken into account in an explicit form, which naturally leads to certain changes in the formula (6.1), expressed in the fact that each reflection coefficient at each level of iteration of imaginary sources must be multiplied by a certain geometric coefficient. In a real situation, the use of a ray representation (6.1) requires limiting the number of members used in it. The idea of such a restriction is that as the index increases in total, the distance from the imaginary source to the observation point increases Rn , which leads to a decrease in the contribution from the source in the total field. From the obvious geometric consideration of the problem, it follows:

74

6 Ray Description of the Sound Field in Inhomogeneous Media

The greater the value of the relationship r=h, the greater the number of members to be taken into account in the representation (6.1). If the source is not monochromatic but impulse, in (6.1) the view expressions expðikRi Þ=Ri should be replaced by the value f ðt  Ri =cÞ=Ri , where f ðtÞ is the source function. An exhaustive description of the processes of sound wave propagation in layered structures is given in the book [1]. Spreading in the oceans and seas, sound waves are reflected from the bottom and at the water–air interface. Reflection of the sound from the water–atmosphere boundary is “soft”—the phase of the sound signal changes to the opposite, resulting in the total sound pressure at the water–air interface becoming zero. The water–air boundary in the majority of acoustic models is a completely reflective boundary. The surface of the sea, as a rule, is agitated, which leads to a diffuse reflection of sound waves (along with the mirror waves) and increases their attenuation. The presence of a large number of air bubbles in the near-surface layer of the ocean also increases the attenuation of sound. The ocean floor and the seafloor reflect sound waves differently, and its reflective properties are characterized by a reflection coefficient—the ratio of the sound pressure in the reflected wave to the sound pressure in the incident wave. The general ideas for the application of the ray approach to propagation problems in heterogeneous environments will be considered using the following physical situation as an example. Let the sound source be located in a plane-parallel homogeneous layer, on the boundaries of which the speeds of sound are equal, and at the source level the speed of sound is lower than at the boundaries. In terms of ray theory, all rays emanating from the source in the range (from the vertical) of angles jhj [ arcsinðcr =cb Þ will not be able to go beyond the layer boundaries, as they will be fully reflected internally. Here, cr is the speed of sound at the transmitter level, and the cb is the speed of sound at the boundaries. The rays emanating from the source in the remaining range of angles will partially extend beyond the homogeneous layer and partially re-reflect inside it. Schematically, this situation is shown in Fig. 6.2. In Fig. 6.2, the solid lines show the total internal reflection of the rays, and dotted lines show the rays partially reflected and partially removed from the layer.

Fig. 6.2 Ray stroke system in the simplest sound channel

6

Ray Description of the Sound Field in Inhomogeneous Media

75

At a certain horizontal distance from the source, the rays partially passing into the external environment will practically not contribute to the intensity of the sound field inside the homogeneous layer. Thus, all the energy of the acoustic field enclosed in the layer will be enclosed in the rays that experience full internal reflection. Since there is no energy dissipation in the system, these rays can spread horizontally over an infinite distance from the source. The result is a simple sound channel. Within the framework of the radiation theory, the energy concentrated inside the sound channel coincides with the energy of the rays emitted within the angles jhj [ arcsinðcr =cb Þ. For a point isotropic source, the ratio of this energy to the total radiated energy is the same as the corresponding body angles and is called the energy capture coefficient

K¼

2

R p=2 R 2p ~ h

0

sin h dh du

4p

4p ¼ 4p

Zp=2

sin h dh ¼ cos hj~h ¼ cos ~ h; p=2

ð6:2Þ

~h

where ~h ¼ arcsinðcr =cb Þ: So, the main reason for the formation of the sound channel is that there is an area in space where the speed of sound is less than in the surrounding areas. In our specific case, the velocity distribution with height is described by the simplest dependence shown in Fig. 6.3. The path of the sound rays in the underwater sound channel can be calculated using a ray theory based on the assumption that the sound energy in the medium propagates along certain ray lines. If the transmitter is placed on the axis of the sound channel, under certain conditions, the rays coming from this source at different sliding angles will be re-assembled at the same point, i.e., focused. And so, the underwater sound channel acts on the sound rays as a collector lens. The speed of sound c in the ocean changes in horizontal direction significantly less than the vertical one, and it can be assumed that the acoustic properties of the environment change only along axis z in the ocean. The ocean is a layered, heterogeneous environment. How does sound spread in such environments?

Fig. 6.3 Sound velocity distribution

76

6 Ray Description of the Sound Field in Inhomogeneous Media

For plane acoustic waves that fall on the boundary of the two media at an angle hi , the Snellius law is followed ci sin ht ¼ ct sin hi , where ht is the angle of refraction applied. This formula is the basis for ray acoustics. If sound waves move from the medium with a slower speed ci to the medium with a faster speed ct ðct [ ci Þ, they deviate upward from the direction of propagation; i.e., there is a positive refraction. If the propagation takes place under the condition ct ðct [ ci Þ, then the rays deviate down from their direction of propagation and then indicate negative refraction. In the first case (with positive refraction), there is a limit angle hi for which sin hi ¼ ci =ct , and if hi [ hi , the reflection is full and sound waves do not pass in the second medium. When sound comes from the air into the water, sin hi  330=1500 ¼ 0:22 and hi ¼ 12:70 . Sound from the air into the water will not go away at the hi [ 12:70 . If we break up the sea medium into thin layers with c ¼ const within each of them, we can show by calculation that the sound ray will deviate toward a lower rate of propagation when spreading sound in a layered inhomogeneous environment. As mentioned above, the speed of sound in seawater depends on its temperature, salinity and pressure (or depth), i.e., c ¼ f ðT; S; pÞ. The existing vertical distribution in the ocean T, S and hydrostatic pressure p on the depth leads to the formation of vertical distribution cðzÞ, characterized by the presence of a minimum located at some depth (in the Atlantic at depths from 1200 to 2000 m, in the Pacific Ocean—from 500 to 700 m, in polar latitudes—near the surface at depths of 100 m and less). An example of such a distribution cðzÞ is shown in Fig. 6.4. It turned out that for the propagation of sound in the ocean the most important thing is not the absolute value c, but the profile of the curve cðzÞ, i.e., the type of vertical distribution c, i.e., the positions of extrema on this curve, the ratio of values cðzÞ at the bottom, at the ocean surface and at extreme points, the distribution of gradients dc ðzÞ=dz with depth, etc. The profile cðzÞ essentially defines the conditions of sound propagation in the ocean. With one type of speed of sound profile cðzÞ, the sound propagation range can reach hundreds and thousands of kilometers,

Fig. 6.4 Vertical distribution of sound velocity most frequently found in deep-sea seas and oceans (a), and formation of sound channel (b)

6

Ray Description of the Sound Field in Inhomogeneous Media

77

with another profile—only a few kilometers. In the course of the experiments, it was repeatedly noted that sound propagation at a distance of up to 10,000 km was observed. The deep ocean profile is typical of the profile shown in Fig. 6.4a, i.e., a profile with a minimum located at some depth zm . Above the level zm speed c increases due to growth Tw , below the level zm value c increases due to pressure p growth. As it was mentioned above, the presence of the minimum speed of sound leads to the formation of a very interesting phenomenon, namely the emergence of an underwater sound channel. As a result, the conditions are created for the long-distance propagation of sound because sound waves are not scattered on the surface of the ocean and are not absorbed in the bottom ground. The only factor that weakens sound pressure is attenuation in seawater. For low frequencies, for which water absorption is low, the range of sound propagation can be several thousand kilometers. The sound would have been able to bypass the entire globe if it had not been interrupted by the mainland. Such sound propagation is called waveguide, and underwater sound channel is a particular case of natural waveguide. If the speed of sound is lower than the channel axis due to increased hydrostatic pressure, the sound channel is called hydrostatic. In some places of the ocean, the appearance of the underwater channel is caused by the presence of warm water masses of increased salinity below the axis of the channel. Channels of this type are called thermal. A typical channel of this type is the underwater sound channel in the Black Sea. The almost universal existence of the sound channel was used to create a hydroacoustic system to rescue people caught in the so far sea disaster. The pilots on the sea surface, the sailors in distress, dropped special depth bombs into the water, which exploded at the depth of the sound channel. The sound of the explosion, recorded at several coastal stations, made it possible to accurately determine the position of those in distress. It should be noted that acoustic waveguides exist not only in the ocean but also in the atmosphere. Let’s consider several typical variants of sound channel formation in water. 1. Sound speed is constant (does not change with depth). In this case, there is no refraction, and the sound rays will be straightforward. Such a picture is observed in shallow water bodies during the autumn–winter period, when convective circulation, covering the whole water body, develops and the temperature and salinity of water are equalized in depth: Tw ¼ const and S ¼ const. Due to the shallow water conditions, the influence of pressure is not significant. 2. Temperature Tw ¼ const and S ¼ const, but the sea is deep and the speed of sound is affected by depth (i.e., hydrostatic pressure). The distribution cðzÞ is shown in Fig. 6.5. In this case, refraction will be positive, the surface will reflect the rays and a kind of near-surface waveguide is formed in the water near the interface, which increases the range of sound rays. This type of waveguide is found in tropical and temperate zones, where the upper ocean layer has approximately the same temperature and salinity due to wind mixing.

78

6 Ray Description of the Sound Field in Inhomogeneous Media

Fig. 6.5 Distribution of speed of sound typical of deep-sea areas at constant depth of temperature and salinity during the daytime hours of spring–summer (a) and corresponding positive refraction of sound rays (b)

3. Shallow spring–summer months, when the sun’s rays warm up from the surface. Distributions of the speed of sound during day and night hours are shown in Figs. 6.6 and 6.7. In the daytime, there will be negative refraction, the rays will be deflected downward and their range will be small, and there will be an area of shadow in which the rays will not be able to penetrate. At night, however, the situation will be different: A subsurface sound channel will be formed near the surface, and the range of sound rays will be increased. The natural question arises: How will the sound field be formed if instead of a jump in the speed of sound at the boundaries of space there is a smooth distribution of sound? for example, in Fig. 6.4a. In this case, there are no sharp boundaries from which the rays could be reflected, as in the previous case, and at first glance, it seems that the underwater sound channel cannot be formed in such conditions. In fact, there is a mechanism that does not allow a part of the rays to go beyond a certain area. And this

Fig. 6.6 Distribution of the speed of sound with depth in the surface layer of shallow sea during the daytime hours of spring–summer (a) and the corresponding distribution of acoustic ray paths (negative refraction) (b)

6

Ray Description of the Sound Field in Inhomogeneous Media

79

Fig. 6.7 Depth distribution of the speed of sound during spring–summer night hours (a) and corresponding formation of a subsurface sound channel (b)

mechanism is the phenomenon of refraction of rays at their propagation in the environment, which is heterogeneous in terms of speed of sound, studied in the previous chapter. Let’s get to the bottom of this. Let the point source be located at the coordinate point z ¼ f. On this horizon, the speed of sound is equal to cðfÞ. Consider a single ray coming out of the source at an angle h to the vertical. According to the theory of refraction of sound rays, the constant characteristic of a given ray (ray invariant) is sin hðfÞ sin hðzÞ ¼ ¼ const: cðfÞ cðzÞ

ð6:3Þ

According to Fig. 6.4, the ray will gradually depart from the vertical when propagating to the region z [ f, as it is in this region cðzÞ [ cðfÞ. That is, the angle hðzÞ will be approaching to p=2. It may be that the angle of the ray reaches the value p=2 while the speed of sound is less than cb . After that, the ray will turn and spread in the opposite direction of the axis. At a given angle of radiation hðfÞ, it is possible on the basis of (6.3) to determine the level z at which the angle of turning is located. Since it is at the turning point sin h sinðp=2Þ ¼ 1, z satisfies Eq. (6.3) c1 ðz Þ ¼

sin hðfÞ : cðfÞ

ð6:4Þ

Rays coming out of the source at different angles will pass their turning point at different levels, according to (6.4). The part of the rays for which the condition ~ h\ arcsinðcðfÞ=cb Þ is valid, although it will experience the phenomenon of refraction, will not be able to reach the turning point and will go beyond the wave channel. As a result, an extremely complex radial pattern is created in space, in which caustic surfaces can also be observed. The specific view of the sound field picture depends on the specific dependence of the speed of sound cðzÞ on the vertical coordinate.

80

6 Ray Description of the Sound Field in Inhomogeneous Media

It should also be noted that this picture, even with a fixed distribution of sound velocity cðzÞ, can vary significantly depending on the location of the source, because its (the source) own characteristic, namely the value of the speed of sound cðfÞ at the level of the source, is included in Eq. (6.4), which determines the geometry of the ray trajectory. For this reason, it is impossible to give a characteristic ray picture of the sound field for all occasions. Each specific case requires a separate review. But any ray approach to such problems is based on the use of the ratio (5.28) for the ray invariant, as well as the dependencies (5.32, 5.33), which determine the total trajectories of the rays coming from the source and, consequently, the ray picture of sound propagation in the CCD. When caustics are formed in a CCD, entire extended zones may occur in which the focusing factor f may be greater than one. These zones are called convergence zones. When propagated horizontally from the source, these areas alternate with areas where sound is virtually absent and the focus factor is much less than one. The dependence of speed of sound on water temperature is being used in acoustic thermometry and acoustic tomography techniques, which are increasingly being used. The basic principle of ocean acoustic tomography is to restore the structure of oceanic inhomogeneities by measuring the time of signal propagation along individual rays. From 1992 to 1995, the most famous and major experiment in the acoustic thermometry of Acoustic Thermometry of Ocean Climate (ATOC) was carried out. The main objective of the experiment was to observe the large-scale variability in water temperature in the North Pacific over a long period of time by measuring the propagation time of acoustic signals between several radiators and receivers located at considerable distances from each other. The analysis of the data obtained in the course of the first stage of the Russian– American experiment on the trans-American acoustic propagation allowed to reveal the warming of the Atlantic water layer along the Svalbard–Beaufort Sea route on average at the 0.30 °C relatively to the average climatic data. Studies have shown that acoustic thermometry can be a powerful tool for obtaining unique data on changes in the average temperature of deep waters of the oceans with high accuracy and continuously over an almost unlimited period of time. We looked at sound propagation in an environment whose characteristics were unchanged in the horizontal direction. This situation is not always realized in the real ocean. For example, if sound propagates in a coastal wedge, the nature of the propagation is significantly influenced by the changing depth of the environment. Another example of horizontal heterogeneity is the intersection by acoustic waves of frontal oceanic zones, such as the Gulf Stream and the Kuroshio Current. Also, the influence of horizontal inhomogeneities should not be neglected in the propagation of sound over long distances, when the change in its speed in the horizontal direction is accumulated.

6

Ray Description of the Sound Field in Inhomogeneous Media

81

In the most general case, it is impossible to develop a theory of sound propagation because of the practical unsolvability of acoustic equations in both wave and ray variants. However, for a number of cases where the characteristics of the medium change in the horizontal direction rather slowly, approximate approaches associated with the integration of equations with a small parameter are possible [2]. Such a small parameter for the coastal zone (ocean depth change) is the small angle of inclination of the bottom. If there is a horizontal speed of sound gradient, it is acceptable to assume that the gradient is small compared to the vertical gradient. At present, methods of cross section, parabolic equation and ray method are developed for such situations. Since it is simply impossible to describe all the details of the application of these methods in a short course of lectures, we will dwell on the main ideas contained in them. Let’s consider the case of a slow change in the speed of sound in the horizontal direction with a possible simultaneous change in depth. Since the speed of sound is now a function of all three coordinates, i.e., c ¼ cðz; rÞ, where r ¼ fx; yg, the wave equation for sound pressure in the medium takes the form Dp þ k2 ðz; rÞp ¼ 0;


k2 ðz; rÞ ¼ x2 c2 ðz; rÞ:

ð6:5Þ

Let’s cross section of the medium with a vertical plane r ¼ r0 . Let’s call by waveguide of comparison in this plane of such a waveguide, the distribution of the speed of sound in which changes only with the height c ¼ cðz; r0 Þ, the depth of its unchanged is h ¼ hðr0 Þ. The schematic explanation is shown in Fig. 6.9. In Fig. 6.8, the bold lines show the surface and bottom of the medium, and a thin line shows the bottom of the comparison waveguide. In the section r ¼ r0 , the wave Eq. (6.5) is rewritten in the form of locally determined (i.e., in the small vicinity of the section) wave equation Dp þ k2 ðz; r0 Þp ¼ 0;


k2 ðz; r0 Þ ¼ x2 c2 ðz; r0 Þ:

ð6:6Þ

Since the depth of the waveguide is finite, the solution (6.6) can be represented in the form of a series of expansions of the problem’s own functions, including the Fig. 6.8 True waveguide and waveguide comparisons

82

6 Ray Description of the Sound Field in Inhomogeneous Media

Fig. 6.9 Possible speed of sound distribution in the antiwaveguide

equation itself (6.6), plus the boundary conditions on the free surface and the comparison waveguide bottom. Let a similar boundary problem possess a system of eigen-strained orthonormalized functions fwn g. Then, the pressure in the section r ¼ r0 can be represented as pðz; r0 Þ ¼

X n

An wn ðz; r0 Þ:

ð6:7Þ

Considering that the medium parameters and its depth change with distance r rather slowly, we extrapolate the ratio (6.7) to the whole space, considering the coefficients An not as constants but as slowly changing coordinate functions of r; i.e., the solution of the initial Eq. (6.5) is presented as pðz; rÞ ¼

X n

An ðrÞwn ðz; rÞ:

ð6:8Þ

Since fwn g are the eigenfunctions of the comparison waveguide, they satisfy the equation  @ 2 wn  2 þ k ðz; r0 Þ  n2n ðr0 Þ wn ¼ 0: 2 @z

ð6:9Þ

Let’s put now (6.8) in (6.5) taking into account (6.9). Because @2p @2p @2p @2p Dp ¼ 2 þ 2 þ 2 ¼ r2r p þ 2 ; @x @y @z @z

rr ¼

@ @ ; ; @x @y

6

Ray Description of the Sound Field in Inhomogeneous Media

83

then there is a correlation Dp þ k2 ðz; rÞp ¼

X

@ 2 wn @ z2 n X  þ An r2r wn þ 2rr An rr wn þ wn r2r An An

n 2

þ k ðz; rÞ

X n

An wn ¼ 0;

from which, taking into account (6.9), it follows X n

X    wn r2r þ n2n An ¼  2rr An rr wn þ An r2r wn ;

ð6:10Þ

n

Using the orthonormalization of eigenfunctions fwn g expressed by the 

Zh wn wm qðzÞ dz ¼ dnm ;

dnm ¼

0

1; n ¼ m ; 0; n ¼ 6 m

the multiplication of Eq. (6.10) by qðzÞwm and integration of the result by z within ½0; h 

 r2r þ n2m Am 0 1 Zh Zh X @2rr An qðzÞwm rr wn dz þ An qðzÞwm r2 wn dzA: ¼ r n

0

ð6:11Þ

0

This equation for the required distance r dependence of amplitudes fAn g lies at the heart of the cross-sectional method. The right side (6.11) describes the interaction of normal modes of the waveguide comparison. Since there are operators in it rr and r2r , the value of the right part is small because of the small gradients of the speed of sound and low slope of the bottom. If we neglect the right part, fAn g, we obtain an equation for the amplitudes, which forms the basis of the so-called adiabatic approximation. Let’s continue the consideration of the problem in adiabatic approximation, and in addition, let’s assume a cylindrical symmetry of the problem in relation to the axis z. In this case, Eq. (6.11) is reduced to

1@ @Am r þ n2m Am ¼ 0: r @r @r By replacing Bm ðrÞ ¼

pffiffi r Am ðrÞ, Eq. (6.12) is reformed into

ð6:12Þ

84

6 Ray Description of the Sound Field in Inhomogeneous Media



@ 2 Bm 1 2 þ n þ Bm ¼ 0: m 4r 2 @ r2

ð6:13Þ

Since we are interested in solutions at the nm r 1, we can write down the following in the approach of the WKB 0 r 1 Z Cm Bm  pffiffiffiffiffiffi exp@i nm dr A; nm

ð6:14Þ

0

where Cm is a constant value. And now, the final result for sound pressure looks like 0 r 1 Z X Cn pffiffiffiffiffiffiffi wn ðz; rÞ exp@i nm dr A: pðz; rÞ ¼ nn r n

ð6:15Þ

0

The approximation used in the WKB solution is closely related to the ray description of the sound propagation process. As it is known from the ray theory, when a ray spreads in an inhomogeneous medium, there is a phenomenon of refraction, which is directly related to the concept of a turning point. In the trace variable, the waveguide in the path of the ray may encounter turning points with different vertical coordinates z. In contrast to the vertically stratified medium, where the value acts as an invariant cos hðzÞ=cðzÞ, in the waveguide trace the invariant is the value Zz2 I¼ z1

cos hðzÞ dz ¼ const; cðzÞ

ð6:16Þ

where z1 , z2 are the vertical coordinates of two consecutive turning points of the ray. The trajectory of the ray between these points is called the ray cycle. Sometimes, the ratio (6.16) is written down as I I¼

cos hðzÞ dz ¼ const cðzÞ

ð6:17Þ

and the integration is done by a ray cycle. In addition to the ray invariant, one can enter the cycle time T and cycle length of the sound wave D I T¼

dz ; cðzÞ cos hðzÞ

I D¼

tanhðzÞdz:

ð6:18Þ

6

Ray Description of the Sound Field in Inhomogeneous Media

85

According to Snellius’ law, sin hðzÞ sin hðz0 Þ ¼ ; cðzÞ cðz0 Þ where the starting point z0 of the ray turning is selected as the starting point. We will use the identity that follows from Snellius’ law cos hðzÞ 1 sin hðz0 Þ ¼  tanhðzÞ cðzÞ cðzÞ cos hðzÞ cðz0 Þ and integrate it into the ray cycle, resulting in I ¼T D

sin hðz0 Þ : cðz0 Þ

ð6:19Þ

Thus, using the concept of the ray invariant we can get the law of changing the angle of the ray inclination at the point of rotation from the distance r. In the previous chapter, the formation of an underwater sound channel was considered as a phenomenon which, when propagated, prevented the sound rays from leaving the waveguide region. This phenomenon was due to the fact that in the area the sound channel was observed at least in the distribution of the speed of sound vertically. Otherwise, when there is an area where the maximum is reached, the so-called antiwaveguide propagation of sound is observed—the rays emitted at the local maximum of the speed of sound leave this area and never come back. The possible distribution of the speed of sound in the antiwaveguide is shown in Fig. 6.9. The calculation of the sound field in ray approximation for such situation does not differ in any way from the calculation of the field in the case of an underwater sound channel.

References 1. Dzhrbashyan MM (1966) Integral transformations and representations of functions in the complex domain. Moscow, “Science”, 672 p 2. Lighthill J (1981) Waves in liquids. Moscow, Mir, 598 p

7

Wave Description of the Sound Field in Inhomogeneous Media

The ray description of sound phenomena contains inevitable approximations that do not allow to describe acoustic processes adequately in certain cases, especially when it is necessary to reveal important subtle characteristics of the sound field, for example, near caustic surfaces. An alternative to the ray approach is the wave description [1]. As an example of the wave representation of the acoustic field, let us consider, already studied in the ray approximation, the problem of sound propagation in the liquid layer, the scheme of which is shown in Fig. 6.1. Let’s begin with the fact that, in general terms, for a monochromatic source, the pressure field in space satisfies the wave equation @2P dðrÞ dðzÞ expð
i x tÞ
c2 ðzÞDP ¼ P0 @ t2 8 2pr > < cþ ; z [ hþ cðzÞ ¼ c0 ; h
\z\h þ : > : c
; h
[ z

ð7:1Þ

Since the response from a harmonic source is also harmonic, the solution (7.1) can be represented by P ¼ Pðr; zÞ expð
i x tÞ, a substitution of which in (7.1) rewrites the equation in the form of DP þ j2 ðzÞP ¼


P0 dðrÞ dðzÞ; c20 2pr

j2 ðzÞ ¼ x2 =c2 ðzÞ:

ð7:2Þ

For convenience of the further calculations, we will enter designations: P ¼ P þ at z [ h þ , P ¼ P at h
\z\h þ , and P ¼ P
at h
[ z. Since there are interfaces in the problem, in order to obtain solutions of Eq. (7.2), it is necessary to supplement it with boundary conditions © Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_7

87

88

7 Wave Description of the Sound Field in Inhomogeneous Media

P ¼ P þ jz¼h þ ; P ¼ P
jz¼h
;

@P=@z ¼ @P þ =@zjz¼h þ

ð7:3Þ

@P=@z ¼ @P
=@zjz¼h
:

The geometry of the problem is such that the sound field is axially symmetrical in relation to the axis z. This allows to present the pressure field as an integral of Bessel Z1 Pi ¼

Pi ðk; zÞkJ0 ðkrÞdk;

ð7:4Þ

0

substitution of which in Eqs. (7.2) and (7.3) reduces the set problem to the form d2 P P0 þ ðj2 ðzÞ
k2 ÞP ¼
2 dðzÞ dz2 c0    P ¼ P þ z¼h þ ; @P=@z ¼ @P þ =@zz¼h þ   P ¼ P
z¼h
; @P=@z ¼ @P
=@zz¼h
:

ð7:5Þ

In the area z [ h þ , the sound field is a system of waves running in the positive direction of the axis z, as these waves are from the h
\z\h þ Pþ

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼ A þ exp i j2þ
k2 z ;

j2þ ¼ x2 =c2þ ;

z [ hþ :

ð7:6Þ

Similarly, in the area z\h
, the sound field is a system of waves running in the negative direction of the axis z 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  P
¼ B
exp
i j2

k2 z ;

j2
¼ x2 =c2
;

z\h
:

ð7:7Þ

In the intermediate area, i.e., in the layer h
\z\h þ , there is a source, so the field is represented as h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i P ¼ A exp i j2
k2 z þ B exp
i j2
k2 z #ðzÞ h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i þ C exp i j2
k2 z þ D exp
i j2
k2 z #ð
zÞ;

ð7:8Þ

where #ðzÞ is the Heaviside step function, j2 ¼ x2 =c20 . Since the radial wave number k can vary from zero to infinity, negative numbers may appear in the subordered terms in expressions (7.6)–(7.8). In order to satisfy the conditions of radiation at infinity at z ¼ 1, the root sign should be chosen

7

Wave Description of the Sound Field in Inhomogeneous Media

positive, so as Im

89

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2þ
k2 ; Im j2

k2 [ 0 to ensure attenuation of P þ at

z ! þ 1 and P
at z !
1. The solution (7.8) is built in this form for the following reasons. In the area above the source (the term with the multiplier #ðzÞ), the field represents the waves emitted by the source upward (the term with the amplitude A), plus the waves reflected from the surface z ¼ h þ and going downward (the term with the amplitude B). In the area under the source (term with a multiplier #ð
zÞ), the field represents the waves emitted by the source downward (term with amplitude D), plus the waves reflected from the surface z ¼ h
and going upward (term with amplitude C). The complete solution (7.6)–(7.8) of the problem (7.5) contains six unknown amplitudes, while the boundary conditions are only four. In order to obtain the necessary additional relations, it is necessary to substitute the solution (7.8) for the region h
\z\h þ in Eq. (7.5) with the source on the right side. For this purpose, it is necessary to calculate the second derivative d2 P=dz2 of the solution (7.8). Let’s do it consistently. But first, in order to reduce the size of the computations, we will pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi introduce a sign k ¼ j2
k2 by which the solution record (7.8) is reduced and takes the form P ¼ ½A expðikzÞ þ B expð
ikzÞ#ðzÞ þ ½C expðikzÞ þ D expð
ikzÞ#ð
zÞ;

ð7:9Þ

and Eq. (7.5) is rewritten as d2 P P0 þ k2 P ¼
2 dðzÞ: dz2 c0

ð7:10Þ

Then, the first derivative dP=dz is determined by dP ¼ ik½A expðikzÞ
B expð
ikzÞ#ðzÞ dz þ ½A expðikzÞ þ B expð
ikzÞdðzÞ þ ik½C expðikzÞ
D expð
ikzÞ#ð
zÞ
½C expðikzÞ þ D expð
ikzÞdðzÞ

:

ð7:11Þ

 ikf½A expðikzÞ
B expð
ikzÞ#ðzÞ þ ½C expðikzÞ
D expð
ikzÞ#ð
zÞg þ ðA þ B
C
DÞdðzÞ On the basis of (7.11), the second derivative d2 P=dz2 is given by the expression

90

7 Wave Description of the Sound Field in Inhomogeneous Media

d2 P ¼
k2 f½A expðikzÞ þ B expð
ikzÞ#ðzÞ dz2 þ ½C expðikzÞ þ D expð
ikzÞ#ð
zÞg þ ikðA
B
C þ DÞdðzÞ þ ðA þ B
C
DÞd0 ðzÞ

ð7:12Þ


k2 P þ ikðA
B
C þ DÞdðzÞ þ ðA þ B
C
DÞd0 ðzÞ Substitution of (7.12) in (7.10) results in the equation ikðA
B
C þ DÞdðzÞ þ ðA þ B
C
DÞd0 ðzÞ ¼


P0 dðzÞ: c20

ð7:13Þ

In order for this equation to be satisfied, it is necessary to put A þ B
C
D ¼ 0;

A
B
C þ D ¼ iP0 =kc20

ð7:14Þ

Expressing from (7.14), the amplitude B and C reflected inside the wave layer through the amplitudes A and D radiated by the source C ¼A


iP0 ; 2kc20

B ¼ Dþ

iP0 2kc20

and substituting this result in (7.9), we will give the final form to the solution in the field h
\z\h þ

   iP0 P ¼ A expðikzÞ þ D þ expð
ikzÞ #ðzÞ 2kc20

   iP0 expðikzÞ þ D expð
ikzÞ #ð
zÞ þ A
2kc20

ð7:15Þ

¼ A expðikzÞ þ D expð
ikzÞ iP0 ½expð
ikzÞ #ðzÞ
expðikzÞ#ð
zÞ: þ 2kc20 For the convenience of further calculations, we will rewrite here solutions (7.6) and (7.7) in the same form P þ ¼ A þ expðik þ zÞ; P
¼ B
expð
ik
zÞ;

kþ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2þ
k2 ;

z [ hþ :

ð7:16Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2

k2 ;

z\h
:

ð7:17Þ

k
¼

So, we have a solution record that contains four unknown amplitudes A, S, A þ , and B
. To determine them, let’s substitute the obtained representations (7.15)– (7.17) in the boundary conditions of the problem (7.5).

7

Wave Description of the Sound Field in Inhomogeneous Media

91

Cross-linking of solutions (7.15) and (7.16) at the upper boundary z ¼ h þ gives rise to relations A expðikh þ Þ þ D expð
ikh þ Þ þ

iP0 expð
ikh þ Þ 2kc20

¼ A þ expðik þ h þ Þ k½A expðikh þ Þ
D expð
ikh þ Þ


ikP0 expð
ikh þ Þ 2kc20

ð7:18Þ

¼ k þ A þ expðik þ h þ Þ: A similar cross-linking of solutions (7.15) and (7.17) at the lower boundary z ¼ h
gives rise to relations A expðikh
Þ þ D expð
ikh
Þ


iP0 expðikh
Þ 2kc20

¼ B
expð
ik
h
Þ ikP0 expðikh
Þ k½A expðikh
Þ
D expð
ikh
Þ
2kc20

ð7:19Þ

¼
k
B
expð
ik
h
Þ: The solution of the system (7.18) and (7.19) is A ¼ }ðk þ k þ Þ expð
ikh þ Þðk
cosðkh
Þ þ ik sinðkh
ÞÞ D ¼
}ðk þ k
Þ expðikh
Þðk þ cosðkh þ Þ
ik sinðkh þ ÞÞ A þ ¼
2}k expð
ik þ h þ Þðk cosðkh
Þ þ ik
sinðkh
ÞÞ

ð7:20Þ

B
¼
2}k expðik
h
Þðk þ cosðkh þ Þ
ik sinðkh þ ÞÞ: }¼

iP0 1 ; 2kc20 D

h ¼ h þ
h


D ¼ kðk þ k þ Þ cosðkhÞ
iðk þ k
þ k2 Þ sinðkhÞ: Substitution of the obtained result (7.15)–(7.17), (7.20) in the integrative expression of the ratio (7.4) determines the integral representation of the sound field of the point source, located in the liquid layer. This solution is at the same time a Green function for a harmonic source with an arbitrary pressure distribution over its surface, provided that it is entirely in the layer h
\z\h þ . Exactly, the same ideological approach is taken when applying wave theory to describe the processes in the underwater sound channel. But before proceeding to the wave description of acoustic processes in the underwater sound channel, it is necessary to define some distribution of the speed of sound in space. As in the ray approximation problem, the speed of sound is considered as a function of vertical coordinate z only. Let the distribution cðzÞ is defined by the view shown in Fig. 7.1.

92

7 Wave Description of the Sound Field in Inhomogeneous Media

Fig. 7.1 Speed of sound distribution in space

Thus, the zone z
\z\z þ where the speed of sound changes with a coordinate is limited to semi-infinite areas z  z
and z  z þ where the speed of sound is constant and equal to c
and c þ , respectively. The source of sound is a monopoly located at a point z ¼ f whose position on the axis is z not yet specified. The source oscillations are monochromatic at the frequency x. In the presence of a source, the wave equation for pressure looks like e @2 P e ¼ P0 dðrÞ dðz
fÞ expð
i x tÞ:
c2 ðzÞD P 2 @t 2pr

ð7:21Þ

Writing the right part—the source term—of Eq. (7.21) already takes into account the axial symmetry of the problem relative to the axis z. The solution (7.21) is sought in the form of harmonic waves, so that a repree ¼ Pðr; zÞ expð
i x tÞ. Substitution of this representation is used for pressure P sentation into Eq. (7.21) and division of the result into the square of the speed of sound transforms the wave equation to the form DPðr; zÞ þ

x2 dðrÞ Pðr; zÞ ¼
P0 dðz
fÞ: 2prc2 ðfÞ c2 ðzÞ

ð7:22Þ

Since in the cylindrical coordinate system, taking into account the axial symmetry of the problem, the Laplace operator D looks like D¼

@2 @2 1 @ ; þ þ r@r @ z2 @ r 2

Then, the solution (7.22) can be searched for using the method of integral transformations, using for the function Pðr; zÞ the spectral expansion of the 2 eigenfunctions of the operator @@r2 þ 1r @@r, which, in our case, of the point source located at the point with the coordinate r ¼ 0, has the form

7

Wave Description of the Sound Field in Inhomogeneous Media

93

Z1 Pðr; zÞ ¼

Pðk; zÞkJ0 ðkrÞdk:

ð7:23Þ

0

Substitution of (7.23) in (7.22) reduces the wave equation to the equation for spectral components Pðk; zÞ

2  d2 Pðk; zÞ x P0 2
k þ Pðk; zÞ ¼
2 dðz
fÞ: dz2 c2 ðzÞ c ðfÞ

ð7:24Þ

Even for known distributions cðzÞ, Eq. (7.24) is practically insoluble, not to mention the general case. In order to obtain a constructive result, it is necessary to use any reasonable approximations. Let’s consider the dependence chart 1=c2 ðzÞ shown in Fig. 7.2. Since the differences in the speed of sound for real-world environments are relatively small (usually within the range ), piecemeal linear approximation of the inverse square of the sound velocity is acceptable c2 ðfÞ ¼ am ðz
zm
1 Þ þ bm c2 ðzÞ

2  1 c ðfÞ c2 ðfÞ
am ¼ ; zm
zm
1 c2 ðzm Þ c2 ðzm
1 Þ

c2 ðfÞ : bm ¼ 2 c ðzm
1 Þ

ð7:25Þ

In this case, in areas with constant sound velocity, such as, for example, areas z  z
, z  z þ or zi
1  z  zi (for this particular Fig. 7.2), the coefficient ai is zero. The view (7.25) divides the medium space into plane-parallel layers in which the inverse square of the sound velocity changes linearly with the velocity characteristic of each of the layers. Introducing the designation j2f ¼ x2 =c2 ðfÞ and using the representation (7.25) to distribution of the inverse square of the speed of sound, we reduce Eq. (7.24) to a form convenient for further analysis

Fig. 7.2 Bite-linear approximation 1=c2 ðzÞ

94

7 Wave Description of the Sound Field in Inhomogeneous Media

i d2 Pðk; zÞ h 2 þ jf ðam ðz
zm
1 Þ þ bm Þ
k2 Pðk; zÞ 2 dz P0 ¼
2 dðz
fÞ; z 2 ½zm
1 ; zm : c ðfÞ

ð7:26Þ

Since the area of Eq. (7.26) is a straight line broken down into segments, in each of which the properties of the medium differ from those of the adjacent segments, it is necessary to supplement the set of Eq. (7.26) with boundary conditions at z ¼ z
, z ¼ z1 , …, z ¼ zi , …, z ¼ zN , z ¼ z þ ; …, as well as radiation conditions at z ! 1. The boundary conditions on the layer interfaces are the equality of the pressures and normal oscillatory velocities in the adjacent layers. Since the oscillatory velocity of the medium is determined through its potential by the ratio v ¼ ru; e is associated with the potential for equality and pressure P e ¼
@ u P @t e ¼ i xu the equality of which, in the case of harmonic oscillations, looks like P normal velocity components vz at the layer interface is reduced to the equality of pressure derivatives by coordinate z. Thus, the boundary conditions for the set of Eq. (7.26) take the form  Pm 

z¼zm

 ¼ Pm þ 1 

z¼zm

;

  @Pm  @Pm þ 1  ¼ ; @z z¼zm @z z¼zm

ð7:27Þ

where Pm is the spectral component in the layer zm
1  z  zm . Now, the task of determining the sound field in the underwater sound channel is fully set. Having determined the spectral components for each of the layers with the help of (7.26) and (7.27) and substituting the results in (7.23), we get an expression for the sound field in the whole space. Let’s get down to the task at hand. First of all, let us pay attention to Eq. (7.26). Since there is only one source, it is located in one layer, and the rest of the layers do not have one. Therefore, the right side of Eq. (7.26) is different from zero only for one layer. Let it be a layer defined by the ratio zn
1 \f\zn . In Fig. 7.3, for the convenience of understanding the solution technique, the division of space into layers is presented. First, we will find a solution (7.26) for the areas z  z
and z  z þ . In the area z  z
, the speed of sound is constant, there is no source, and therefore, Eq. (7.26) is simplified and takes the form

7

Wave Description of the Sound Field in Inhomogeneous Media

95

Fig. 7.3 Splitting the space into plane-parallel layers

d2 Pðk; zÞ 2 þ j

k2 Pðk; zÞ ¼ 0; 2 dz

j2
¼

x2 : c2


ð7:28Þ

Since there are no sources in the area under consideration, the solution can only be the waves running in the negative direction of the axis z, so  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  P
ðk; zÞ ¼ B
exp
i j2

k2 z ;

ð7:29Þ

and a square root with a positive mimic is selected to meet the radiation conditions z !
1 at k [ j
. In expression (7.29), B
is the amplitude of the wave is still uncertain. In the area z  z þ ; the speed of sound is also constant, there is no source, and therefore, Eq. (7.26) takes on a form similar to (7.28)

d2 Pðk; zÞ 2 þ j þ
k2 Pðk; zÞ ¼ 0; 2 dz

j2þ ¼

x2 : c2þ

ð7:30Þ

Since there are no sources in this area, the solution is the waves running in the positive direction of the axis z, so  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  P þ ðk; zÞ ¼ A þ exp i j2þ
k2 z ;

ð7:31Þ

and a square root with a positive imaginary part is selected to meet the radiation conditions z ! þ 1 at k [ j þ : In the expression (7.31), A þ is also uncertain wave amplitude so far. Let’s look at some layer now zi
1  z  zi . In general, this layer can be either homogeneous ðai ¼ 0Þ or heterogeneous ðai 6¼ 0Þ. Let’s look at these cases

96

7 Wave Description of the Sound Field in Inhomogeneous Media

separately. Let the layer be homogeneous. Then, Eq. (7.26) has the same form as Eqs. (7.28) and (7.30)

d2 Pðk; zÞ 2 þ ji
k2 Pðk; zÞ ¼ 0; dz2

j2i ¼

x2 x2  : c2 ðzi
1 Þ c2 ðzi Þ

ð7:32Þ

Although there is no source in the layer, the sound waves are re-reflected from the layer boundaries and the solution of Eq. (7.32) is represented as a sum of the waves running in both positive and negative directions of the axis z.  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Pi ðk; zÞ ¼ Ai exp i j2i
k2 z þ Bi exp
i j2i
k2 z

ð7:33Þ

As in previous cases, the imaginary part of the square roots is chosen non-negative. The amplitudes Ai and Bi definitions have not yet been determined. Let the layer in question be heterogeneous, i.e., ai 6¼ 0. In this case, Eq. (7.26) looks like i d2 Pðk; zÞ h 2 þ jf ðai ðz
zi
1 Þ þ bi Þ
k2 Pðk; zÞ ¼ 0 2 dz

ð7:34Þ

In order to find a solution to this equation, it is necessary to bring it to a canonical form, which uses the replacement of the variable form  
2=3 h i ni ¼ j2f ai j2f ðai ðz
zi
1 Þ þ bi Þ
k2 by which (7.34) is converted to a form d2 Pðk; ni Þ þ ni Pðk; ni Þ ¼ 0 dn2i

ð7:35Þ

We have before us the Airy equation, the general solution of which looks Pi ðk; ni Þ ¼ Ai Aið
ni Þ þ Bi Bið
ni Þ;

ð7:36Þ

where Ai, Bi—functions of Airy of the first and second order, respectively, defined by integral representations 1 Aið
xÞ ¼ p Bið
xÞ ¼

1 p

Z1

cosðt3 =3
xtÞdt;

0

Z1 0



expð
t3 =3
xtÞ þ sinðt3 =3
xtÞ dt:

7

Wave Description of the Sound Field in Inhomogeneous Media

97

B (7.36) amplitudes Ai and Bi are also not yet defined. The above solutions (7.29), (7.31), (7.33), and (7.36) are own waves in the considered areas without a source. Now, it remains to find a solution for the area zn
1  z  zn where the source is located. As with the area zi
1  z  zi , there are two cases: an ¼ 0—the layer is homogeneous in terms of sound velocity and an ¼ 0—the layer is heterogeneous. Let’s look at these cases separately again. For a homogeneous layer, Eq. (7.26) takes the form

d2 Pðk; zÞ 2 P0 þ jn
k2 Pðk; zÞ ¼
2 dðz
fÞ dz2 c ðfÞ x2 x2  2 : j2n ¼ 2 c ðzn
1 Þ c ðzn Þ

ð7:37Þ

Since there is a sound source inside the layer under consideration, the solution of Eq. (7.37) is sought in the form of Pn ðk; zÞ ¼ ½A expðikn zÞ þ B expð
ikn zÞ#ðz
fÞ þ ½C expðikn zÞ þ D expð
ikn zÞ#ðf
zÞ;

kn ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2n
k2 :

ð7:38Þ

Since Eq. (7.37) is of the second order, and in the representation (7.38) of the solution, there are four unknown constants A, B, C, and D, it is necessary to substitute (7.38) in (7.37) to determine the relationship between the mentioned constants and reduce their number. We need to calculate the second derivative. First, let’s calculate the first derivative. We need to calculate the second derivative d2 Pn ðk; zÞ=dz2 . First, let’s calculate the first derivative. dPn ðk; zÞ ¼ ikn ½A expðikn zÞ
B expð
ikn zÞ#ðz
fÞ dz þ ½A expðikn zÞ þ B expð
ikn zÞdðz
fÞ þ ikn ½C expðikn zÞ
D expð
ikn zÞ#ðf
zÞ
½C expðikn zÞ þ D expð
ikn zÞdðf
zÞ

ð7:39Þ

 ikn ½A expðikn zÞ
B expð
ikn zÞ#ðz
fÞ þ ikn ½C expðikn zÞ
D expð
ikn zÞ#ðf
zÞ þ ½ðA
CÞ expðikn fÞ þ ðB
DÞ expð
ikn fÞdðz
fÞ The expression (7.39) contains the Dirac delta function dðz
fÞ. When calculating the derivative of (7.39) (we need to substitute in (7.38) the second derivative!) in the desired ratio will appear derivative of the delta function d0 ðz
fÞ. When substituting the result in Eq. (7.38) dðz
fÞ, in which the right part is only

98

7 Wave Description of the Sound Field in Inhomogeneous Media

present, there will be a need to zeroize the coefficient at the d0 ðz
fÞ. This can be done already now, which leads to a correlation between the entered constants ðA
CÞ expðikn fÞ þ ðB
DÞ expð
ikn fÞ ¼ 0:

ð7:40Þ

In this case the equation (7.39) is simplified and takes the form dPn ðk; zÞ ¼ ikn ½A expðikn zÞ
B expð
ikn zÞ#ðz
fÞ dz þ ikn ½C expðikn zÞ
D expð
ikn zÞ#ðf
zÞ: Now, let’s calculate the second derivative d2 Pn ðk; zÞ ¼
k2n ½A expðikn zÞ þ B expð
ikn zÞ#ðz
fÞ dz2
k2n ½C expðikn zÞ
D expð
ikn zÞ#ðf
zÞ þ ikn ½ðA
CÞ expðikn fÞ
ðB
DÞ expð
ikn fÞdðz
fÞ

ð7:41Þ

¼
k2n Pn ðk; zÞ þ ikn ½ðA
CÞ expðikn fÞ
ðB
DÞ expð
ikn fÞdðz
fÞ Substitution of (7.41) in the initial Eq. (7.37) leads to the ratio ikn ½ðA
CÞ expðikn fÞ
ðB
DÞ expð
ikn fÞdðz
fÞ P0 ¼
2 dðz
fÞ; c ðfÞ from which it follows ðA
CÞ expðikn fÞ
ðB
DÞ expð
ikn fÞ ¼

iP0 : kn c2 ðfÞ

ð7:42Þ

Sharing (7.40) and (7.42) allows the expression of two constants C and D through two others A and B C ¼A


iP0 expð
ikn fÞ ; 2kn c2 ðfÞ

D ¼ Bþ

iP0 expðikn fÞ : 2kn c2 ðfÞ

ð7:43Þ

Substitution of (7.43) in the submission of the solution (7.38) leads to the final result

7

Wave Description of the Sound Field in Inhomogeneous Media

Pn ðk; zÞ ¼ An expðikn zÞ þ Bn expð
ikn zÞ iP0 ½expðikn ðf
zÞÞ#ðf
zÞ : þ 2kn c2 ðfÞ
expðikn ðz
fÞÞ#ðz
fÞ

99

ð7:44Þ

The obtained ratio is nothing else but the sum of own waves of the layer under consideration An expðikn zÞ þ Bn expð
ikn zÞ and the Green function of the monopole located in this layer iP0 ½expðikn ðf
zÞÞ#ðf
zÞ
expðikn ðz
fÞÞ#ðz
fÞ: 2kn c2 ðfÞ Amplitudes An and Bn not yet defined. A similar technique is used in the study of a heterogeneous layer, where an 6¼ 0, it leads to a result that is given here in its final form Pn ðk; zÞ ¼ An Aið
nn Þ þ Bn Bið
nn Þ pP0  2 
1=3 þ 2 j an Aið
nf ÞBið
nn Þ#ðnn
nf Þ 2c ðfÞ f

þ Bið
nf ÞAið
nn Þ#ðnf
nn Þ

ð7:45Þ

 
2=3 h i where nf ¼ j2f an j2f ðan ðf
zn
1 Þ þ bn Þ
k2 . As in the homogeneous case, (7.45) is nothing but the sum of the natural waves of the layer under consideration An Aið
nn Þ þ Bn Bið
nn Þ and the Green function of the monopole located in this layer pP0  2 
1=3 j an Aið
nf ÞBið
nn Þ#ðnn
nf Þ 2c2 ðfÞ f

þ Bið
nf ÞAið
nn Þ#ðnf
nn Þ : Amplitudes An and Bn are not yet defined too. Thus, the relations expressing spectral amplitudes Pðk; zÞ for all layers, as well as for the upper and lower half-spaces, were obtained. In order to complete the solution of the problem on the wave representation of the sound field in the underwater sound channel, it is necessary to determine the values of the sets of constants fAm g, fBm g, A þ , and B
, which are still undefined, as well as.

100

7 Wave Description of the Sound Field in Inhomogeneous Media

It is the use of boundary conditions (7.27) that allows us to calculate the required values unambiguously. The merging of the obtained solutions for layers on the boundaries generates an inhomogeneous system of linear algebraic equations, the solution of which determines all the necessary constants. System heterogeneity is due to the presence of the Green function of the source part (7.44) or (7.45). The mentioned system of equations is very cumbersome, the technique of its solution is standard, but very volumetric, and therefore neither the system, nor the technique of its solution are not given here. Substitution of the obtained spectral components in (7.23) leads to a solution in the quadrature of the problem. The underwater sound channel is not a mere speculation. Its existence in different areas of the world’s oceans is confirmed by numerous in situ measurements. The formation of the underwater sound channel is determined by the specific features of the hydrological structure of sea and oceanic waters. Often, real water structures are characterized by both spatial and interseasonal contrasts of the sound velocity field. Moving deep into the water column, several main layers of water structure can be identified. For example, in the border seas of the Pacific Ocean zone of the Far East [2] in winter at shallow depths, there is a subsurface layer, in which the sound velocity at first approximation can be put unchanged, because its vertical gradients are minimal. Below the surface layer, there is tachocline, which has the highest sound velocity gradients. Underneath the tachocline, there are layers with more moderate positive gradients than in the tachocline. In summer, a sound channel is formed in the same zone due to the fact that high negative gradients of the sound velocity are observed in the near-surface layer, the layer with the minimum deviations of velocity is located below, and the same positive gradients as in winter are observed even lower. It is important to note that such a distribution of the sound velocity field is typical for sufficiently deep-sea zones of seas and oceans, because in shallow waters, the structure of waters is quite homogeneous in terms of sound velocity. Table 7.1 shows the data [2] on the water structure in the region of the Kuril Islands and adjacent water areas in warm seasons. For the thermoline area, the maximum variation in sound velocity is given. On the basis of the presented experimental data, the group of researchers has made the following conclusions [2]: For the Pacific subarctic structure of waters, the formation of the sound velocity field is largely related to the Kuril Current, where the axis of the sound channel, as the studies have shown, coincides with the current core and the zone of the minimum temperature of the cold intermediate layer. The type of sound waveguides being formed is thermal. In the hunting seawater structure, the negative values of the minimum water temperature in the cold intermediate layer determine the formation of a pronounced underwater sound channel. It was found that in the sound velocity field here, as well

7

Wave Description of the Sound Field in Inhomogeneous Media

Table 7.1 Water structure in the region of the Kuril Islands and adjacent water areas in warm seasons

Layer Pacific area Surface Tachocline Audio channel axis Okhotsk region Surface Tachocline Audio channel axis South Okhotsk region Surface Tachocline Audio channel axis Kuril Islands zone Surface Tachocline Audio channel axis Shallow water zones Surface—bottom

101

Speed of sound, m/s

Depth, m

1460–1465 Dc ¼ 10 1450–1455

0–50 50–75 75–100

1460–1475 Dc ¼ 15 1449–1450

0–30 30–75 75–100

1490 Dc ¼ 20 1449–1450

0–10 10–100 100–200

1455–1460 Dc ¼ 5 1450–1453

0–10 10–100 100–200

*1453

0–150

as for the core of the cold intermediate layer, there is a “breakage” of the plane layered waveguide at the crossing of the Prikurilsky front of the Sea of Okhotsk. In the structure of South Hawthomorie waters, the form of the vertical sound velocity curve is determined not only by the vertical temperature profile, but also by the non-monotonous distribution of the salinity profile due to the intrusion of warmer, more saline waters. The position of the sound channel axis is therefore slightly deeper than the position of the core of the cold intermediate layer. The sound channel type is no longer purely thermal. The peculiarity of the structure of the sound velocity field in this region is also the maximum range of change in the sound velocity from the surface to the axis of the sound channel, compared to other areas considered here. The water structure of the Kuril Straits zone is characterized by relatively small sound velocity values on the surface, smoothed extrema of the curve of the vertical profile of sound velocity and erosion of the sound channel axis. In the homogenized waters of the shallow water zone, the destruction of the sound channel until its disappearance is observed. The main characteristics of the RoW are subject to severe seasonal variability. Table 7.2 shows the characteristic parameters of sound velocity and width of the sound channel in another zone of the Pacific Ocean depending on the season. Such observed variability is explained by Brekhovskikh [3] by the following reasons: In the cold half of the year, the vertical structure of the sound velocity field of the area under study is almost homogeneous due to the development of convective

102

7 Wave Description of the Sound Field in Inhomogeneous Media

Table 7.2 Characteristic parameters of sound velocity and width of the sound channel in another zone of the Pacific Ocean depending on the season Characteristics

Winter The speed of sound on surface, m/s The speed of sound on axis of the USC, m/s Spring The speed of sound on surface, m/s The speed of sound on axis of the USC, m/s Width USC, m Summer The speed of sound on surface, m/s The speed of sound on axis of the USC, m/s Width USC, m Autumn The speed of sound on surface, m/s The speed of sound on axis of the USC, m/s Width USC, m

Prikamchatka zone

Bering-maritime zone

Pacific– Aleutian zone

the 1450–1456

1456–1460

1464–1466 1462

the 1449–1454

–

–

the 1458–1460

1461–1463

1466–1468 1463–1464

the 1450–1454

1456–1458

–

–

300–700

–

–

the 1490–1494

1483–1485

1486–1490 1482–1484

the 1451–1454

1456–1464

1462–1468 1467–1470

700–1200

1100–1200 1100–1500

the 1462–1466

1468–1470

1470–1472 1468–1470

the 1455–1460

1464–1466

1466–1470 1466–1470

700–1200

1100–1200 1100–1500

200–400

250–700

300–700

Kamchatsky and middle straits

–

mixing during autumn–winter cooling. The minimum sound velocity values are located in the surface layer. The distribution of sound velocity on the surface is also subject to seasonal variations. During the warm period of the year, there is a tendency to reduce the sound velocity from the moraine and oceanic areas to the ridge straits, where a zone of low water temperature is formed in summer. By the cold half-year, respectively, the temperature field adjustment, the zone of lowered values shifts to the Kamchatsky district. The minimum sound velocity distribution has its own peculiarities. During the whole warm half of the year, the lowest values are observed in the Kama region. In the summer period, an area of overestimated values relative to the adjoining water areas is noted in the field of the minimum sound velocity in the zone of the Aleutian Islands. In case of water homogenization observed in shallow islands and shallow straits, the minimum is not observed (destruction of the sound channel, up to its disappearance).

7

Wave Description of the Sound Field in Inhomogeneous Media

103

Thus, each hydrological structure of water is characterized by its own thermohaline characteristics. As a consequence, there is a corresponding hydro-acoustic structure, which is characterized by its shape of the curve of vertical distribution of sound velocity and characteristics (depth of position and numerical values of extrema, the type of sound channel and its values) [3].

References 1. http://elementry.ru/news/430411 2. Vinogradova MB, Rudenko OV, Sukhorukov AP (1990) Theory of waves. Science, 432 p 3. Brekhovskikh LM (1973) Waves in layered media. Science, 344 p

8

Sound Wave Reflection from the Ocean Floor

Let’s now move on to the study of reflection processes from the ocean floor. The study of the problem essentially depends on the mathematical model of the bottom, which, naturally, depends on the structure of the bottom sediments, the underlying surface, etc. At this level of consideration of this problem, we agree that the bottom is plane in the horizontal direction. Since it is impossible to describe all kinds of structural features of the bottom in a single approach, let’s focus on the most common variants. 1. A model of the liquid layered bottom. The simplest liquid layered bottom model divides the space occupied by the ocean floor system into three areas, as shown in Fig. 8.1. The upper half-space z
0 is occupied by water (1), d  z  0 followed by a layer of silt (2), below which, in the half-space z   d is the underlying environment (3). Let the plane sound wave, whose plane of fall coincides with the plane y ¼ 0, fall to the bottom of the water z ¼ 0: It allows to consider the problem in two-dimensional statement, in the coordinate system ðx; zÞ, as it is shown in Fig. 8.1. When a plane wave falls from the region (1), a reflected wave is formed in the same region, which passed the wave in the region (2), reflected from the lower boundary of the silt layer of the wave in the region (2), passed the wave in the region (3). In order to define the sound field in the presented geometry, we will enter the necessary designations. Let the velocity potential of the incident wave is described by the function u1þ ¼ expðiðk1x x þ k1z z  x tÞÞ:

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_8

ð8:1Þ

105

106

8

Sound Wave Reflection from the Ocean Floor

Fig. 8.1 Liquid layered bottom model—slimy deposits on the underlying surface

The reflected wave potential in the region (1) is set as u 1 ¼ A expðiðk1x x  k1z z  x tÞÞ;

ð8:2Þ

and the components of the wave vector k1x , k1z in the relations (8.1, 8.2) are the 2 2 þ k1z ¼ x2 =c21 , where c1 is the speed of same and satisfy the dispersion relation k1x sound in the region (1). The total speed potential in the region (1) is the sum of the potentials (8.1) and (8.2), so that for this region it is possible to write u1 ¼ expðiðk1x x þ k1z z  x tÞÞ þ A expðiðk1x x  k1z z  x tÞÞ

ð8:3Þ

In area (2), the corresponding speed potential looks like u2 ¼ B þ expðiðk2x x þ k2z z  x tÞÞ þ B expðiðk2x x  k2z z  x tÞÞ

ð8:4Þ

where B þ , B —amplitudes of waves running in the negative and positive directions of the axis z, respectively. The components of the wave vector k2 satisfy the dispersion equation. 2 2 k2x þ k2z ¼ x2 =c22 :

For the third area, the speed potential is given by the ratio u3 ¼ C þ expðiðk3x x þ k3z z  x tÞÞ; 2 2 þ k3z ¼ x2 =c23 . where C þ is the amplitude of the transmitted wave, k3x

ð8:5Þ

8

Sound Wave Reflection from the Ocean Floor

107

The dispersion relations for all three media can be combined by a common formula kix2 þ kiz2 ¼ x2 =c2i ;

i ¼ 1; 2; 3:

ð8:6Þ

If one introduce the concept of wave number ki ¼ x=ci , the components of wave vectors can be expressed through the angles hi shown in Fig. 8.1 kix ¼ ki sin hi ;

kiz ¼ ki cos hi

ð8:7Þ

Since the shape of the sound field is defined by the relations (8.2–8.4), it remains only to substitute them in the boundary conditions consisting in continuity at the boundaries of pressures qi @ ui =@ t and normal components of velocity @ ui =@ z: Given the fact that according to Snellius’ law k1x ¼ x

sin h1 sin h2 sin h2 sin h3 ¼x ¼ k2x ¼ x ¼x ¼ k3x ; c1 c2 c2 c3

ð8:8Þ

cross-links of physical fields on the borders of regions lead to equations: At z ¼ 0 q1 ð1 þ A Þ ¼ q2 ðB þ þ B Þ;

c2 cos h1 ð1  A Þ ¼ c1 cos h2 ðB þ  B Þ:

At z ¼ d q2 ðB þ expðik2z dÞ þ B expðik2z dÞÞ ¼ q3 C þ expðik3z dÞ c3 cos h2 ðB þ expðik2z dÞ  B expðik2z dÞÞ ¼ c2 cos h3 C þ expðik3z dÞ The resulting system of equations is conveniently represented in a matrix form 2

3 2 3 A  q1 6 B þ 7 6 c2 cos h1 7 7 6 7; ½ M 6 4 B 5 ¼ 4 5 0 0 Cþ where the matrix ½M  looks like 2

m11 6 m21 ½M  ¼ 6 4 m31 m41

m12 m22 m32 m42

m13 m23 m33 m43

3 m14 m24 7 7 m34 5 m44

ð8:9Þ

108

8

m11 ¼ q1 ;

m12 ¼  q2 ;

m21 ¼ c2 cos h1 ;

Sound Wave Reflection from the Ocean Floor

m13 ¼  q2 ;

m22 ¼ c1 cos h2 ;

m14 ¼ 0 m23 ¼ c1 cos h2 ;

m24 ¼ 0

m31 ¼ 0; m32 ¼ q2 expðiw2 Þ; m33 ¼ q2 expðiw2 Þ; m34 ¼  q3 expðiw3 Þ; m41 ¼ 0; m42 ¼ c3 cos h2 expðiw2 Þ m43 ¼ c3 cos h2 expðiw2 Þ; m44 ¼ c3 cos h2 expðiw3 Þ xd xd w2 ¼ k2z d ¼  cos h2 ; w2 ¼ k2z d ¼  cos h3 c2 c3 The solution of (8.9) may be presented in the form of A ¼

DA DB DB DC ; B þ ¼ þ ; B ¼  ; C þ ¼ þ ; D D D D

meanwhile D ¼ expðiw3 Þ½q1 c1 cos h2 F þ þ q2 c2 cos h1 F  DA ¼ expðiw3 Þ½ q1 c1 cos h2 F þ þ q2 c2 cos h1 F  F ¼ expðiw2 Þðq2 c2 cos h3 þ q3 c3 cos h2 Þ  expðiw2 Þðq2 c2 cos h3  q3 c3 cos h2 Þ

ð8:10Þ

DB þ ¼ 2q1 c2 cos h1 expðiðw2  w3 ÞÞðq2 c2 cos h3 þ q3 c3 cos h2 Þ DB ¼ 2q1 c2 cos h1 expðiðw2  w3 ÞÞðq3 c3 cos h2  q2 c2 cos h3 Þ DC þ ¼ 4q1 q2 c2 c3 cos h1 cos h2 Since the amplitude of the incident wave is assumed to be equal to unity, the reflection coefficient of the structure under consideration will be determined by the expression   DA 2   : R ¼ j A j ¼  D  2

ð8:11Þ

The formula (8.11) defines the value we are interested in the most general way. Using this ratio, two interesting limit cases can be considered. The first case is characterized by the condition that the thickness of the silt layer is stacked with an integer number of values k2 =ð2 cos h2 Þ, i.e., d¼

Nk2 ; 2 cos h2

k2 ¼ 2p c2 =x:

ð8:12Þ

Substitution (8.10) in (8.11), taking into account the condition (8.12) leads to the ratio

8

Sound Wave Reflection from the Ocean Floor

  q3 c3  q1 c1 2 R¼ ; q3 c3 þ q1 c1

109

ð8:13Þ

from which it follows that the reflection coefficient does not depend in any way on the parameters of the silt layer, as if the reflection would occur directly from the underlying surface. In the case of a normal fall ðcos h1 ¼ cos h2 ¼ cos h3 ¼ 1Þ, the condition (8.12) is converted to a form d ¼ Nk2 =2: The silt layer is called half-wave layer at N ¼ 1. In the second case, the condition that an integer number of values k2 =ð4 cos h2 Þ is stacked on the thickness of the silt layer is valid. In a N ¼ 1 normal fall, it is called a quarter-wave layer. Substitution of corresponding expressions in the formula for the reflection coefficient leads to the final result in the considered case R¼

 2 2 2 q2 c2 cos h1 cos h3  q1 q3 c1 c3 cos2 h2 : q22 c22 cos h1 cos h3 þ q1 q3 c1 c3 cos2 h2

ð8:14Þ

From (8.14), we can see that q22 c22 cos h1 cos h3 ¼ q1 q3 c1 c3 cos2 h2 the reflection coefficient is zero. Zi ¼

qi ci cos hi

In terms of medium impedances determined by the ratio, the condition of equality to zero of the reflection coefficient is reduced to the execution of the ratio Z22 ¼ Z1 Z3 :

ð8:15Þ

The case when several layers are successively placed on the underlying surface is considered similar to the case of one layer. At the same time, physical fields are cross-linked at all borders separating the layers. The cumbersome expression for the reflection coefficient does not make sense here, because the ideology of its obtaining is the same as in the previous case. The concept of the liquid bottom, used in the case study, is related to the fact that both the layer and the underlying surface were considered in the model of the liquid medium and they were characterized only by densities and speeds of sound in them. In general, the elastic medium, as already mentioned in Chap. 2, is characterized by two elastic constants: k and l which are the Lamé coefficients of comprehensive compression and shear, respectively. The velocity field in the elastic medium is set by two potentials—scalar u and vector w by means of the

110

8

Sound Wave Reflection from the Ocean Floor

v ¼ ru þ r  w

ð8:16Þ

and the potentials themselves satisfy the wave equations @2u 2 @t2  cjj Du ¼ 0; 2 @ w 2 @t2  c? Dw ¼ 0;

c2jj ¼ k þq2l c2? ¼ lq

;

ð8:17Þ

in which differ the speeds of spreading of longitudinal and transverse oscillations. In order to simplify the calculations without losing commonality, let’s assume that the sound wave falling to the hard bottom is characterized by the plane of fall ðx; zÞ. The boundary of the partition coincides with the plane z ¼ 0. In this way, the wave characteristics do not depend on the coordinate y, moreover, in both environments, there is no oscillatory velocity component along this coordinate. When a sound wave falls to the solid bottom, a reflected wave is formed in the water, and in the bottom, there are two past (longitudinal and transverse) waves. As in the previous problem, let the potential of velocity in water be described by u ¼ expðiðk1x x þ k1z z  x tÞÞ þ A expðiðk1x x  k1z z  x tÞÞ

ð8:18Þ

Since there should not be a component of the velocity directed along the axis y in the solid bottom, the vector potential w2 can be represented in the form of w2 ¼ ð0; w2 ; 0Þ;

ð8:19Þ

so that the components of velocity in the solid day are determined by the relations v2x ¼

@ u2 @ w2  ; @x @z

v2z ¼

@ u2 @ w2 þ : @z @x

ð8:20Þ

The scalar and vector potentials themselves u2 and w2 , we will set in the form of potentials of plane longitudinal and transverse waves, respectively u2 ¼ B expðiðk2x x þ k2z z  x tÞÞ : w2 ¼ C expðiðj2x x þ j2z z  x tÞÞ

ð8:21Þ

The values of the wave numbers k1 , k2 and j2 , and those present in (8.18) and (8.21), are given by the expressions k12 ¼

x2 ; c21

k22 ¼

x2 ; c2jj

j22 ¼

x2 : c2?

ð8:22Þ

Let h1 is the angle of incidence of the sound wave, h2 is the angle of transmission of the longitudinal wave and c2 is the angle of the transverse wave transmission.

8

Sound Wave Reflection from the Ocean Floor

111

Since the tangent components of the wave vectors of all waves (Snellius law) should be equal at the interface, the relations are valid k1 sin h1 ¼ k2 sin h2 ¼ j2 sin c2 :

ð8:23Þ

Let’s now write down the boundary conditions formulated earlier. The equality of normal stress at the border means that rzz w ¼ kw nnn w ¼ knnn s þ 2lnzz s ¼ rzz s ;

ð8:24Þ

where rzz w , rzz s —normal components of stress tensors, nik w , nik s —components of deformation tensors for water and solid bottom, respectively. As previously mentioned kw ¼ q1 c21 . Since water is considered to be non-viscous, the equality of tangential stresses at the boundary leads to the condition rxz ¼ 2lnxz ¼ 0:

ð8:25Þ

Let’s present the conditions (8.24, 8.25) explicitly:    @ nx w @ nz w @ nx s @ nz s @n þ þ ¼k þ 2l z s @x @z @x @z @z   @ nx s @ nz s þ ¼ 0: @z @x q1 c21



ð8:26Þ

In (8.26), there are displacements of water particles and solid bottoms. At the same time, our description of wave fields is given through the potentials of velocities rather than displacements. In order to use the relations (8.26), it is necessary to rewrite these expressions through the velocity potentials (8.18) and (8.21). First of all, expressions (8.26) should be rewritten through the velocity of the environment volume elements. Since the velocity of a particle is a derivative of its time shift, differentiation (8.26) leads to     @ vx w @ vz w @ vx s @ vz s @ vz s þ þ q c21 ¼k þ 2l @x @z @x @z @z   @ vx s @ vz s þ ¼ 0: @z @x Substitution in these expressions of formulas (8.20), which determine the components of velocity in the solid bottom through potentials u2 and w2 , as well as relations v1x ¼

@ u1 ; @x

v1z ¼

@ u1 @z

112

8

Sound Wave Reflection from the Ocean Floor

for the water velocity components, results in the final form of boundary conditions recording at z ¼ 0 

 @ 2 u2 @ 2 w2 ¼ kDu2 þ 2l þ @ z2 @x@z 2 2 2 @ u2 @ w2 @ w2 þ 2  ¼ 0; @x @z @x2 @z2

q1 c21 Du1

ð8:27Þ

which, naturally, should be supplemented by the condition of equality of normal velocity components at the boundary @ u1 @ u2 @ w2 ¼ þ @z @z @x

ð8:28Þ

Thus, the form of potentials is given [ratios (8.18) and (8.21)], the boundary conditions are fully defined, and it remains only to substitute in them the expressions for potentials. This substitution leads to a system of equations relative to amplitudes A, B and C: k1z ð1  AÞ ¼ k2z B þ j2x C

 2  B þ j2x j2z C q1 c21 k12 ð1 þ AÞ ¼ kk22 B þ 2l k2z   2k2x k2z B þ j22x  j22z C ¼ 0

ð8:29Þ

The solution to this system is presented as DA DB DC ; B¼ ; C¼ D D D D ¼ c3jj c3? F þ ; DA ¼ c3jj c3? F   F ¼ cos h1 k þ 2l cos2 h2 cosð2c2 Þ þ l sinð2h2 Þ sinð2c2 Þ    q1 c1 cjj cos h2 cosð2c2 Þ þ c? sinð2h2 Þ sin c2 A¼

DB ¼ 2q1 c5jj c3? cos h2 cosð2c2 Þ;

ð8:30Þ

DC ¼ 2q1 c3jj c5? cos h1 sinð2h2 Þ

The reflection coefficient of the sound wave from the solid bottom is determined by the expression R ¼ j Aj 2 ¼



F Fþ

2

ð8:31Þ

The use of ratios for DB and DC makes it possible to determine the proportions of longitudinal and transverse waves in the solid bottom. In the case of a normal fall ðh1 ¼ h2 ¼ c2 ¼ 0Þ DC ¼ 0 and transverse (shear) waves are not formed in the bottom.

8

Sound Wave Reflection from the Ocean Floor

113

pffiffiffi  If sin h1 ¼ c1 = 2c? , so cosð2c2 Þ ¼ 0, as a consequence DB ¼ 0, the longitudinal waves (compression waves) will not be excited in the bottom. Let the characteristics of water and solid ground be such that the ratio c1 [ cjj [ c? is valid. Then, the reflection coefficient from such a solid bottom will be less than the reflection coefficient from the liquid bottom, which has the same density q2 and the speed of sound coincides with the speed of longitudinal oscillations cjj . In the case of ratios cjj [ c1 [ c? , as is often the case for real ground, and sin h1 ¼ c1 =cjj , it turns out that h2 ¼ p=2 from (8.23) follows. As a result DB ¼ DC ¼ 0, there is a complete internal reflection of the sound wave. When  the angle of incidence h1 increases, it follows that sin h2 ¼ cjj =c1    sin h1 [ cjj =c1 cjj =c1 [ 1. It is clear that such a result is impossible at the actual values of wave numbers used in the representations (8.18) and (8.21) for velocity potentials. As long as the wave numbers k1z , k2z and j2z are short and have imaginary parts, there should be waves concentrated near the boundary, the amplitudes of which should drop exponentially at a distance from the boundary z ¼ 0. This case is convenient to consider separately. Since the presence of the incident wave is not necessary for the existence of such surface waves, it makes sense to set the velocity potentials of the wave in the form u ¼ A expðiðkx  x tÞÞ expðj1 zÞ#ðzÞ þ B expðiðkx  x tÞÞ expðj2 zÞ#ðzÞ w2 ¼ C expðiðkx  x tÞÞ expðj3 zÞ#ðzÞ j21 ¼ k2  x2 =c21 ;

j22 ¼ k2  x2 =c2jj ;

ð8:32Þ

j23 ¼ k2  x2 =c2?

where #—a single function of Heaviside and at the same time there is Im k; ji ¼ 0, Re ji [ 0 that provides attenuation of waves at a distance from the border. As can be seen from the view (8.32), this wave propagates along the axis x without attenuation. Let’s mark its speed with a symbol cs . Because Im k; ji ¼ 0   x2 x2 x2 1 2 1 k  2 ¼ 2  2 ¼x  [0 c2s c21 cs c1 c1 ! x2 x2 x2 1 1 k2  2 ¼ 2  2 ¼ x2 2  2 [ 0 cs cjj cs cjj cjj   x2 x2 x2 1 1 k2  2 ¼ 2  2 ¼ x2 2  2 [ 0 cs c? cs c? c? 2

ð8:33Þ

Since (8.33) can only be performed at cs \c1 ; cjj ; c? , the speed of the surface wave should be less than the speed of sound in water and the speeds of longitudinal and transverse waves in the solid bottom.

114

8

Sound Wave Reflection from the Ocean Floor

All surface wave characteristics—pressure and speed—must be continuous at z ¼ 0. Analytical representation of this condition leads to a system of equations of the form j1 A    ¼ j2 B þ ikC   q1 c21 j21 k2 A ¼k j22  k2 B þ 2l j22 B þ ikj3 C 2ikj2 B  j23 þ k2 C ¼ 0

ð8:34Þ

We have a homogeneous system of linear algebraic equations with respect to unknown quantities A, B and C. The condition of its non-trivial solvability is zero of its main determinant: j1



     k2 þ j23 k j22  k2 þ 2lj22  4lk2 j2 j3      q1 c21 j21  k2 j2 k2  j23 ¼ 0:

ð8:35Þ

There is only one unknown value in this expression—k ¼ x=c. Since the values of the Lamé coefficients k and l can be expressed by means of the ratio (8.17) through the density of the bottom and the

velocity of the longitudinal and transverse waves: l ¼ q2 c2? , k ¼ q2 c2jj  2c2? , the Eq. (8.35) with the help of auxiliary parameters s ¼ c2s =c2? , q ¼ c2? =c2jj , r ¼ c2? =c21 can be reduced to q1 q2

rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi 1  sq 2 s ¼ 4 1  sq 1  s  ð2  sÞ2 : 1  sr

ð8:36Þ

The solution of this equation is relative s to the velocity of the surface wave, pffiffi because cs ¼ c? s. Thus, the surface wave at the boundary between water and the hard bottom is fully defined. This wave is called the Stoneley wave. Finding the analytical solution of Eq. (8.36) is a very non-trivial task, because depending on the material the value of the parameter q changes in a very wide range. Table 8.1 shows the characteristic values cjj , c? and q for some solids [1]. In the limiting case, when the density of the upper medium tends to zero, q1 ! 0, which is the case when the surface wave at the boundary of vacuum and elastic solid surface is considered, the ratio (8.36) is simplified

Table 8.1 The characteristic values c||, c⊥ and q for some solids

Substance

cjj (m/s)

c? (m/s)

q

Aluminum Tungsten Quartz (melt) Brass Polystyrene

6400 5174 5980 4280 2350

3130 2842 3760 2020 1120

0.24 0.30 0.40 0.22 0.23

8

Sound Wave Reflection from the Ocean Floor

4

pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi 1  sq 1  s  ð2  sÞ2 ¼ 0

115

ð8:37Þ

and comes down to the third degree equation s3  8s2 þ 8sð3  2qÞ þ 16ðq  1Þ ¼ 0:

ð8:38Þ

The solution to this equation determines the speed of the so-called Rayleigh wave.

Reference 1. http://www.pacificinfo.ru/climate/acvad/2/speed/

9

Scattering of the Sound by Surface and Ocean Floor Irregularities

So far, the problems of sound propagation in the ocean have been considered on the assumption that the surface and bottom were planes. In reality, both the bottom and the free surface are statistically uneven surfaces. In general, it is impossible to solve the problem of sound wave propagation for any free surface or bottom relief— boundary value problems are set on surfaces that do not allow the integration of equations with boundary conditions. But, in some cases, it is possible to describe the acoustic field in real situations with a good degree of approximation. Let’s start by studying the basic characteristics of uneven surfaces. If the surface has a small number of protrusions—individual swelling of the bottom—such surfaces cannot be called statistically uneven and we will not consider them. For a large number of irregularities, there are two main possible variants—stationary and non-stationary irregularities. Stationary irregularities are the deviations of the ocean floor from the plane. The category of unsteady roughness includes roughness of the free surface, as the sea agitation changes with time. We will only consider stationary irregularities. Surfaces with a large number of stationary irregularities are called rough surfaces. It is clear that when a sound wave falls on such surfaces, it is scattered on roughness. The nature of the scattering is determined by a variety of factors. In addition to the roughness and wavelength of the irradiating sound field, the size of the scattering surface, the way it is irradiated, reflecting the properties of the surface, etc., are important. Depending on the relationship between the different parameters, different scattering methods are used. Let the rough surface be given by the ratio z ¼ fðx; yÞ; the true appearance of which is unknown to us, and we do not need, because we are interested in random roughness. So we want to study the average statistical characteristics of scattering. This means that we will give the answer about the nature of scattering not for one particular rough surface, but for the whole ensemble, the elements of which are combined by the same statistical characteristics—the moments hfm i of surface

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6_9

117

9 Scattering of the Sound by Surface and Ocean Floor …

118

evasion from the undisturbed level z ¼ 0; as well as the moments of evasion gradients hjrfjm i determined by the relations hfm i ¼

ZZ

fm dxdy;

hjrfjm i ¼

S

ZZ

ðrfÞm dxdy

ð9:1Þ

S

For simplicity, we will consider on average plane surfaces, i.e., those for which hfi ¼ 0: The boundary conditions must be fulfilled on the surface itself. We will consider only two possible boundary conditions: an absolutely soft and absolutely rigid surface. For the velocity potential u of the monochromatic wave (time dependence expði x tÞ), in case of absolutely soft surface the boundary condition looks like ujz¼fðx;yÞ ¼ 0

ð9:2Þ

which means zero sound pressure at the bottom. In the case of absolutely rigid surfaces, the boundary conditions are defined by the ratio ðn  ruÞjz¼fðx;yÞ ¼ 0;

n¼

rF ; F ¼ z  fðx; yÞ jrF j

ð9:3Þ

where n is unit normal to a rough surface defined by equation F ¼ 0. Absolutely rigid boundary conditions express the equality to zero of the normal velocity component at the bottom. The environment, in which the problem of sound wave scattering is considered, is homogeneous, i.e., the speed of sound in it is constant everywhere. In this case, the wave equation for the velocity potential takes the form Du þ k2 u ¼ f ðrÞ;

k2 ¼ x2 =c2

ð9:4Þ

where the function f ðrÞ defines the distribution of sound sources in space. It is impossible to obtain a precise solution to the problem (9.2–9.4), so it is necessary to use approximate methods that describe the scattering characteristics with a certain degree of reliability only under certain restrictions imposed on the characteristics of the roughness. The solution to this problem is achievable in one of two cases: in the scale of roughness wavelengths k, either small and gentle, or smooth. The planeness of the surfaces means that the slopes of the surface are on average low, i.e., hðrfÞ2 i  r2f =l2f  1

ð9:5Þ

9

Scattering of the Sound by Surface and Ocean Floor …

119

where r2f  hf2 i is the average square of deviation from the undisturbed surface z ¼ 0 and lf —a characteristic size of irregularities. The smallness of irregularities mean that the moments hfm i are small compared to km , i.e., hfm i=km  1

ð9:6Þ

hf2 i=k2  r2f =k2  1

ð9:7Þ

in particular,

As a result, for small and gentle surfaces, it is possible to decompose both the solution and the boundary conditions by degrees of small parameters jf=kj  rf =k  1 and jrfj  rf =lf  1. The use of this technique is at the heart of the small disturbances method. In the case of smooth irregularities, the amount of deviation can be very large, and the main thing is that the radii of curvature of the surface would be large in comparison with the wavelength, i.e., R=k  1

ð9:8Þ

Thus, in the case of smooth roughness irregularities at f  k with roughness should have a large length in the directions x and y (the so-called large-scale roughness). In the case of smooth irregularities, the Kirchhoff method can be used to solve scattering problems. We will consider only small and gentle irregularities and, accordingly, study only the method of small disturbances. So, as it was mentioned above, in the method of small perturbations, there is a decomposition of the potential of the velocity field and boundary conditions by small parameters rf =k  1 and rf =lf  1. A similar approach was suggested by Rayleigh, who instead of a small parameter rf =k  1 used the parameter called Rayleigh parameter P ¼ 4prf cos h0 =k. At P  1, in case of unevenness, they are small and the value P  1 corresponds to large unevenness. We will use the parameter rf =k  1 in the small disturbances method. Let UðrÞ is the potential of the velocity field falling on the rough surface. Let’s present the solution of Eq. (9.4) in the form of uðrÞ ¼ UðrÞ þ

X

uðnÞ ðrÞ

ð9:9Þ

n¼0

 n  n where n—member of a row has an order of magnitude rf =k or rf =lf .

9 Scattering of the Sound by Surface and Ocean Floor …

120

The series (9.9) is a multiplicity of scattering. Now, we need to arrange the boundary condition in a row near the undisturbed surface z ¼ 0. Let’s consider first the condition (9.2) for an absolutely soft bottom. In this case 

    2   @u  @ u  z2  ujz¼fðx;yÞ ¼ ujz¼0 þ þ    zþ ¼0    @ z z¼0 @z2 z¼ 0 2 z¼fðx;yÞ

ð9:10Þ

It can be seen from (9.10) that the value rf is not included in the boundary condition, so the planeness requirement in case of absolutely soft bottom is superfluous and in the decomposition (9.9 and 9.10) only one small parameter—the parameter of lowness—is sufficient rf =k  1. Let’s substitute the decomposition (9.9) in the boundary condition (9.10) and separately equate the terms of one order of smallness with zero.  Ujz¼0 þ uð0Þ z¼0 ¼ 0    @ ð0Þ ð1Þ  u z¼0 þ ðu þ UÞ f ¼ 0 @z z¼0   ð1Þ  2   @u @ ð2Þ  ð0Þ  f þ 2 ðu þ UÞ u z¼0 þ  @z @z z¼0

ð9:11Þ 2

f ¼0 z¼0 2

Substitution of decomposition (9.9) into the wave Eq. (9.4) and equalization with zero terms of the same order of smallness leads to the system of equations DuðnÞ þ k2 uðnÞ ¼ 0

ð9:12Þ

Thus, the problem of n—times scattered field uðnÞ finding is reduced to the problem of a wave field that takes a known value on the plane z ¼ 0. Such tasks are solved by the Green function method, which in this case means that 1 u ðrÞ ¼  2p ðnÞ

Zþ 1 Zþ 1 1

1

   @ expði k  rÞ  u ðrÞ z¼0 dxdy @z r ðnÞ

ð9:13Þ

Since, as follows from (9.11),  uð0Þ ðrÞz¼0 ¼ Ujz¼0    @ uð1Þ ðrÞz¼0 ¼  ðuð0Þ þ UÞ f @z z¼0   ð1Þ  2   @u @ ð2Þ ð0Þ   f  2 ðu þ UÞ u ðrÞ z¼0 ¼   @z @z z¼0

ð9:14Þ 2

f z¼0 2

9

Scattering of the Sound by Surface and Ocean Floor …

121

substitution (9.14) in (9.13) determines the values of n—scattered potentials uðnÞ 1 u ðrÞ ¼ 2p ð0Þ

uð1Þ ðrÞ ¼

1 2p

Zþ 1 Zþ 1 Ujz¼0 1 1 Zþ 1 Zþ 1

1 1 Zþ 1 Zþ 1

  @ expði k  rÞ dxdy @z r

    @ ð0Þ @ expði k  rÞ ðu þ UÞ f dxdy @z r z¼0 @z

  ð1Þ    f2 1 @u  @2 ð0Þ  f þ ðu þ UÞ uð2Þ ðrÞ ¼  2 2p @z z¼0 @ z2 z¼0 1 1   @ expði k  rÞ dxdy @z r

ð9:15Þ

If the initial field is a plane wave, the potential uð0Þ will also describe a plane wave, but mirrored from the surface z ¼ 0. Thus, uð0Þ it is a zero approximation of the solution, for which there are no surface roughnesses. And if UðrÞ ¼ A expðiðkx x þ ky y þ kz zÞÞ, then uð0Þ ðrÞ ¼ A expðiðkx x þ ky y  kz zÞÞ

ð9:16Þ

A single scattered field uð1Þ is obtained by substituting a mirror-reflected field uð0Þ in (9.15), which is specified by the formula (9.16). Because 

  @ ð0Þ @  u þ U  ¼ A expðiðkx x þ ky yÞÞ ðexpðikz zÞ  expðikz zÞÞ @z @z z¼0 z¼0 ¼ 2ikz A expðiðkx x þ ky yÞÞ so there is an expression for potential uð1Þ ikz A u ðrÞ ¼ p ð1Þ

Zþ 1 Zþ 1 expðiðkx x þ ky yÞÞfðx; yÞ 1



1

 @ expði k  rÞ dxdy @z r

ð9:17Þ

Because hfi ¼ 0; so the average value of the field uð1Þ ðrÞ is zero. In the case, when the observation point is several wavelengths away from the plane z ¼ 0 and the ratio kz  1 is correct, the expression (9.17) is transformed into

9 Scattering of the Sound by Surface and Ocean Floor …

122

kkz Az u ðrÞ  p ð1Þ

Zþ 1 Zþ 1 expðiðkx x þ ky y þ k  rÞÞfðx; yÞ 1

1

dxdy r2

ð9:18Þ

The average intensity of the field calculated on the basis of the formula (9.18) assuming statistical homogeneity of fluctuations fðx; yÞ; is determined by I ¼

k2 kz2 z2 2 j Aj p2

Zþ 1 Zþ 1 Zþ 1 Zþ 1 1

1

1

vf ðq0  q00 Þ

1

expðiðk?  ðq0  q00 Þ þ k  ðr0  r00 ÞÞÞ

ð9:19Þ

2 0 2 00

d qd q r 02 r 002

where vf ðq0  q00 Þ ¼ hfðq0 Þfðq00 Þi is the function of correlation of irregularities, q0 ¼ ðx0 ; y0 Þ, q00 ¼ ðx00 ; y00 Þ are coordinates of two irregularities on the surface, d2 q0 ¼ dx0 dy0 , d2 q00 ¼ dx00 dy00 are differentials of areas in the vicinity of two selected irregularities, r0 ¼ ðq0 ; zÞ, r00 ¼ ðq00 ; zÞ is radius-vectors from points on the surface to the observation point, k? is the projection of the wave vector k on the plane z ¼ 0: To simplify further calculations, variables n ¼ q0  q00 and g ¼ ðq0  q00 Þ=2 are introduced. Since the correlation function vf ðnÞ drops quickly at jnj  lf , it is possible to decompose the correlation function r0 and r00 into Taylor rows in the integrative expression (9.19), so that rg ¼ r0 r00 ¼ ðq  g; zÞ;

r0  r00 ns? n; n ¼ jnj

where ns? is the projection on the surface z ¼ 0 of a unit vector ns ¼ ðq  g; zÞ=jðq  g; zÞj. Introduction of a symbol k? ¼ kni? and use of the fact that it makes it possible to reduce (9.19) to the form d2 q0 d2 q00 ¼ d2 nd2 g; Zþ 1 Zþ 1   d2 g I ¼ 4k4 niz 2 z2 j Aj2 Ff ðq? Þ 4 rg 1

ð9:20Þ

1

  where q? ¼ k ni?  ns? is the transverse component of the scattering vector, q ¼ kðni  ns Þ and the two-dimensional spatial spectrum of inhomogeneities Ff ðjÞ is given by the integral Ff ðjÞ ¼

1 4p2

Zþ 1 Zþ 1 1

1

vf ðnÞ expði j  nÞ d2 n

ð9:21Þ

9

Scattering of the Sound by Surface and Ocean Floor …

123

which represents the Fourier transform of the correlation function vf ðnÞ: As follows from (9.20), each surface element d2 g ¼ dr with the center at the point g gives the total intensity I of the contribution  2  2 d2 g d2 g dI ¼ 4k4 niz z2 j Aj2 Ff ðq? Þ 4 ¼ 4k4 niz nsz j Aj2 Ff ðq? Þ 2 rg rg that consequently means that in direction ns the sound field is scattered  by the definite harmonic of roughness to which the scattering vector q? ¼ k ni?  ns? corresponds. This is so-called Bragg condition.  2 The value rðq? Þ ¼ 4k4 niz nsz Ff ðq? Þ is the scattering cross section in the ns direction and then the ratio (9.20) may be presented in the form I ¼ j Aj2

Zþ 1 Zþ 1 1

1

rðq? Þ 2 d g rg2

ð9:22Þ

The result received shows in explicit from the incoherentness of waves scattered by separate elements of rough surface because the summation is done with respect to intensities but not to field’s amplitudes. In the case of absolutely rigid surface, the scattered field depends on not only parameter rf =k  1 but also on rf =lf  1. Because for absolutely rigid surface the boundary condition ðn  ruÞjz¼fðx;yÞ  ½ðr? f  r? uÞ  @ u=@ zjz¼fðx;yÞ ¼ 0

ð9:23Þ

is valid then the substitution of (9.9) into (9.23) leads to the system of consequent boundary conditions  @U=@zjz¼0 þ @uð0Þ =@zz¼0 ¼ 0     @2 @uð1Þ =@zz¼0 r? f  r? ðuð0Þ þ UÞz¼0 þ 2 ðuð0Þ þ UÞ f ¼ 0 @z z¼0

ð9:24Þ

So the problem of n—times scattered field uðnÞ finding is reduced to the problem of wave field which normal derivative has known value on the plane z ¼ 0. Such problems also may be solved by method of Green function which in this concrete case has a form 1 u ðrÞ ¼  2p ðnÞ

Zþ 1 Zþ 1 1

1

 expði k  rÞ dxdy @uðnÞ ðrÞ=@zz¼0 r

ð9:25Þ

9 Scattering of the Sound by Surface and Ocean Floor …

124

The substitution (9.24) into (9.25) leads to result 1 u ðrÞ ¼ 2p ð0Þ

1 u ðrÞ ¼ 2p ð1Þ

Zþ 1 Zþ 1 @U=@zjz¼0 1 1 Zþ 1 Zþ 1

1

1

expði k  rÞ dxdy r

     ð9:26Þ @ 2 ð0Þ ð0Þ   r? f  r? ðu þ UÞ z¼0 þ 2 ðu þ UÞ f @z z¼0

expði k  rÞ dxdy r Let as in the case of absolutely soft surface, the plane wave UðrÞ ¼ A expðiðkx x þ ky y þ kz zÞÞ falls on rough interface. Then, from the first relation of (9.26), it follows that scattered wave is mirror reflected but without phase incursion p;, i.e., uð0Þ ðrÞ ¼ A expðiðkx x þ ky y  kz zÞÞ

ð9:27Þ

The substitution of the result received into the second relation of (9.26) forms the scattering field in the form A u ðrÞ ¼ p ð1Þ

Zþ 1 Zþ 1 h 1

   2 i ik ni?  r? f þ k2 niz f

1

expðiðkx x þ ky yÞÞ

ð9:28Þ

expði k  rÞ dxdy r

The mean value of this field also equals to zero. The mean intensity of the scattered field is defined by ratio I ¼ j Aj

2

Zþ 1 Zþ 1 1

1

rðq? Þ 2 d g rg2

ð9:29Þ

where the scattering cross section of absolutely rigid place h  2 i rðq? Þ ¼ 4k4 1  niz nsz Ff ðq? Þ

ð9:30Þ

depends on the geometry of problem and spectral density Ff ðq? Þ of roughness. As it is evident from the relations received, the result of scattering strongly depends on the spectral density Ff ðq? Þ. In the case of isotropic Gauss correlation (this conditions sometimes is in a good agreement with characteristics of uneven bottom), the function Ff ðq? Þ has the form

9

Scattering of the Sound by Surface and Ocean Floor …

Ff ðq? Þ ¼

r2f l2f 2p

expðq2 l2f =2Þ

125

ð9:31Þ

For the sea surface of the experimental data on the wave is very small, but in some cases, the spectral density of the wave at a given value of the velocity onset U is presented in the form Ff ðq? Þ ¼

g1=2 S

pffiffiffiffiffiffiffiffiffiffiffi gjq? j

2jq? j3=2

K ðjq? j; aÞ

ð9:32Þ

where g is gravitational acceleration, K ðjq? j; aÞ is a function characterized the angle distribution of the sea roughness energy, a is the angle between the directions pffiffiffiffiffiffiffiffiffiffiffi of the mean wind and propagation of surface wave, S gjq? j is the Neiman-Pirson spectrum for fully developed waving which is defined by relation SðxÞ ¼

  C g 2 exp 2 ; 6 x xU

C ¼ 2:4 m2 =s5

ð9:33Þ

Substitution of (9.23) or (9.33) (depending on the problem under consideration) in the previously obtained expressions for the intensity of the scattered field allows to determine the desired scattering characteristics. In the most detailed presentation of the problem of wave propagation in statistically inhomogeneous media or in media with randomly rough boundaries are studied in [1]. Another source of sound, characterized in the description of statistical properties, is the noise of the sea. Figure 9.1 shows the spectrogram of sea noise of different origin. Area I is the area of noise generated by wind waves (two curves for waves with wind 1 point and 8 points on the Beaufort scale), i.e., the noise from traveling waves, their collision and destruction. Area II is the noise in the sea when heavy rain, snow and intense snow loads. The cause of noise in the area III (range from 10 to 103 Hz, maximum in the range 50–300 Hz) is the navigation because each vessel leaves a special sound trail in the ocean, which fades slightly and spreads over long distances. For example, there are more than 1000 vessels in the North Atlantic alone at the same time. Ship noises are caused by the action of propellers, vibration of the hull and cavitation effects. The area IV is the noise from earthquakes, seaquake, underwater explosions, volcanic eruptions, noises from a turbulent movements on the ocean—atmosphere boundary. The left part of the spectrogram enters the infrasonic region, which is not perceived by the human ear. The energy of these noises is very large (more than from other sources many thousands of times). Attenuation at them weak and, as a consequence of it, the most broad distribution on range. The spectrum of seismic is homogeneous throughout the ocean. The living organisms inhabiting the sea are also a source of noise: the shrimp click; roar, whistle and even tweet beluga’s (roars like a beluga); the black horse

126

9 Scattering of the Sound by Surface and Ocean Floor …

Fig. 9.1 Spectrogram of sea noises. I—noise generated by wind waves, II—noise by the heavy rains and intense snow loads, III—navigation noise, IV—noise from earthquakes a turbulent movements on the ocean—atmosphere boundary

mackerel barks; makes a sound of hooded horse hoof mullet; beeping and purring herring. Acoustic noise also occurs when marine organisms move like fish (shoal movement, jumps of individual large eczemas), pinnipeds, etc. Noises of biological origin are more intense only in certain areas of the ocean and at certain times. The ocean is not a world of silence, but a world without the Sun—this is how the wonderful explorer of the World Ocean Jacques-Yves Cousteau called him. A specific type of acoustic noise is the noise due to sea ice, which is due to the variety of causes causing this type of noise. The noise of ice has a very wide spectral range, but is localized only in the surface layer and high latitudes. Noise also includes thermal noise per MHz band, but the intensity of thermal noise is low. Proper underwater noise is the most important acoustic characteristic of the ocean. They contain information about the state of the ocean and its marine animals, the atmosphere above it, and some processes in the earth’s crust under the ocean, such as earthquakes. The use of noise is based on a passive location in which noise interpretation is used to determine the type and location of the source. Hydroacoustic noise of the ship allows you to determine its type. Hydroacoustic noise arising from bottom movements and submarine volcanic eruptions allows, in principle, to solve the problem of forecasting earthquakes and tsunamis. After the termination of the sound source in the sea during a certain time interval in an area of space, the acoustic field continues to exist, the intensity of which decreases with time. This is a reverb sound. In fact, reverb is an underwater echo. Reverberation of irregularities of the bottom and surface, volume inhomogeneities, such as accumulations of air bubbles due to waves, gas bubbles during decay, small-scale temperature inhomogeneities, suspended particles and small living organisms are generated. Depending on the degree of influence of certain factors, there are three types of reverberation in the sea: volume, surface and bottom. The predominance of one or another kind of reverberation depends on many conditions. The intensity of the volume reverberation is proportional to the radiation power, the duration of the signal and is reverse proportional to the square of time. Marine organisms that cause reverberation form sound-scattering layers, which can have a

9

Scattering of the Sound by Surface and Ocean Floor …

127

considerable length, depending on the biological productivity. During the reverberation, the time taken to reduce the sound intensity is taken 106 times. Reverberation prevents hydroacoustics from working, as it has a masking effect on the useful signal. A relatively recently discovered phenomenon, the opposite of reverb— pre-reverb. This phenomenon consists in the fact that when sound propagates in the sound channel at large distances from the sound source, before the arrival of individual components of the sound vibrations, a sound background appears, which is ahead of the arrival time of the main signal. Its level does not go away over time, but increases and reaches its maximum value at the end before the arrival of the main signal. The causes of this phenomenon are the reflection and scattering of a signal on internal waves, scattering from surface waves, scattering and diffraction on large gradients of the speed of sound and scattering on inhomogeneities of the marine environment. The theory of this phenomenon, in contrast to reverberation, is not fully developed. The reason for this is the uncontrollability of the conditions of the experiment and, as a consequence, some doubtfulness of the experimental data.

Reference 1. http://www.pacificinfo.ru/climate/acvad/1/speed/

Methodical Instructions and Tasks

The guidelines provide a methodology for solving tasks on the theme “Basics of Ocean Acoustics.” These tasks, as well as their various variations, can be used in exams and seminars to check the depth of learning of the course material. These detailed solutions deepen the understanding of the main course and are a good help for students in self-study. In drawing up the tasks, the authors gave preference to the tasks, first, fixing the foundations of the course being studied, second, those close to the problems of real physical experiments and third, developing the skills of theoretical research. Approximately half of the tasks belong to the category of increased complexity and require an informal approach to their solution. The tasks are intended for undergraduate and graduate students of physical, mechanical and mathematical and hydrometeorological faculties of universities, as well as for teachers in the preparation and conduct of classes in physics. Methodical Guidelines for Solving Tasks The following rules should be followed when solving problems: • First of all, carefully read the condition of the task and understand the problem. • The problem should be solved in general terms, and only then, if the conditions require it, substitute the numerical values in the final result. • Analyze the obtained solution, and check its performance in extreme cases. • Only after receiving the final result to read the proposed solution in the collection; • If the methods of decisions obtained independently and presented in the compendium do not coincide, it is necessary to check their possible equivalence. The authors hope that these recommendations will be useful in meeting the challenges.

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6

129

130

Methodical Instructions and Tasks

Tasks for Chapter 1 Task 1 The maximum pressure sensitivity of some hydrophone is 2
105 Pa. Within the framework of the plane sound wave model, to evaluate the absolute value of the oscillatory velocity corresponding to the minimum measured pressure. The speed of sound in water under measurement conditions p0 ¼ 1 atm, T = +20 °C is cw ¼ 1:49
103 m/s, the density of water—qw ¼ 103 kg/m3 . The solution. In a plane wave, the acoustic potential is determined by the ratio u ¼ f ðx  cw tÞ, where f—the arbitrary function, x—the direction of propagation of the plane wave. Pressure perturbation is defined by the expression ~ p ¼ qw @ u=@ t ¼ qw cw f 0 , where the bar indicates the derivative function

of the argument. Vibrant speed is set by the ratio v ¼ ru, so jvj ¼ jruj ¼
u0x
¼ jf 0 j. Thus, there is a plane wave j~pj ¼ qw cw jvj. Assuming that the minimum recorded pressure value coincides with the sensitivity of the hydrophone, we obtain an estimate of the vibrational velocity for this case: j vj ¼

2
105 Pa pj j~ ¼ 3 ¼ 1:34
1011 m/s qw cw 10 kg/m3
1:49
103 m/s

This result should be treated with great care, because the resulting oscillatory velocity is too small and is obviously under the level of thermal fluctuations of water parameters. Task 2 Determine the relative change in water density at the minimum recorded pressure in the sound wave 2
105 Pa. The solution. Since the variations in density and pressure in the sound wave are ~c2w , the relative change in water density is related by the ratio ~p ¼ q ~j 2
105 Pa jq j~pj ¼ ¼ ¼ 9:0
1015 2 qw qw cw 103 KC=M3 1:492
106 M2 =c2 Task 3 Determine the relative water compression at the bottom of the vertical column in height H ¼ 10 M. Do the same calculations for the air. The solution. The hydrostatic pressure of the water column is a value Dp ¼ qw gH. This additional pressure corresponds to a sealing of the medium by a value Dq ¼ Dp=c2w . As a result, the relative water compression at the bottom of the column is determined by q gH gH jDqj jDpj ¼ ¼ w 2 ¼ 2 ¼ 4:42
105 qw qw c2w qw c w cw

Methodical Instructions and Tasks

131

For air, the same value is determined by the value jDqj jDpj qa gH gH ¼ ¼ ¼ 2 ¼ 8:3
104 qa qa c2a qa c2a ca It can be seen that the compressibility of air is about 20 times greater than the compressibility of water. Task 4 The sensitivity of the human ear in terms of pressure is the magnitude of the order 104 Pa (in the case of particularly sensitive people, the order 5
105 Pa). The hammer anvil-stretch system transmits the signal from the eardrum to the inlet of the cochlea with a pressure gain of approximately k ¼ 90. Determine what kind of oscillatory velocity a plane sound wave should have in a person’s ear with a damaged eardrum so that they can pick up sounds at the same level as before the damage. At pressure p0 ¼ 1 atm and temperature T = +20 °C, the air density qa ¼ 1:204 kg/m3 is equal and the sound velocity in the air is equal ca ¼ 343 m/s. The solution. With a hearing aid working properly, the pressure sensitivity of the ear is related ~p ¼ k qa ca jvj to the vibrational speed at the inlet of the ear, k ¼ 1. Well, then. If the eardrum is damaged, the middle ear system does not work, so there is no gain. So as a result. jvj ¼

~p 104 Pa ¼ ¼ 2:4
107 m/s qa ca 1:204 kg/m3 343 m/s

Task 5 Determine the physical meaning of the quantity Du, where u—the acoustic potential, D is Laplace’s operator. The solution. One of the equations of linear acoustics looks like @@ q~t þ q0 Du ¼ 0. Let’s consider the perturbation of the density of some volume of selected particles. Let these particles take up volume before the arrival of the sound wave V1 , and after arrival—volume V2 . The change in the density of these particles DV DV 2 ~ ¼ Vm2  Vm1 ¼ m VV11V will be determined by the value q V2  m V 2 ¼ q0 V where m is the mass of the allocated volume, jDV j  V, V1  V. Substitution of this expression in the written out acoustic equation reduces it to the form Du ¼

@ DV 1 @ DV ¼ @t V V @t

which means that Du it is the speed of the relative change in the volume of the selected particles.

132

Methodical Instructions and Tasks

Task 6 To show that the system (1.1) of the linear acoustics equations taking into account the dissipation of wave energy does not allow to enter the acoustic potential with the help of which it is possible to define all physical fields unambiguously. @v 1 ¼  r ~p þ m D v þ ðl þ m=3Þrr  v @t q0 @ ~p @~ q þ q0 c2 r  v ¼ 0; þ q0 r  v ¼ 0 @t @t

ð1:1Þ

The solution. In order to enter the acoustic potential, the ratio r
v ¼ 0 must be met. Applying the operation r
to the Navier–Stokes equation of the system (1) results in ð@=@ t  m DÞr
v ¼ 0 The solution to this equation is presented as a sum of two solutions: v ¼ v1 þ v2 , where r
v1 ¼ 0, ð@=@ t  m DÞv2 ¼ 0. And if the first equation has a form solution v1 ¼ ru, the second equation has no solution of this kind, which automatically converts it into an identity. Simple discarding of the solution v2 is similar to the case with the Euler equation ð@ v2 =@ t ¼ 0Þ is unacceptable because the equation ð@=@ t  m DÞ v2 ¼ 0 has a solution changing both in time and space. Task 7 Based on the system of equations (1.2), derive (without using the acoustic potential) ~, ~p) separately. the equations for each physical field (v, q @v 1 ¼  r ~p; @t q0

@ ~p þ q0 c2 r  v ¼ 0; @t

@~ q þ q0 r  v ¼ 0 @t

ð1:2Þ

The solution. Let’s get the speed equation out first. Applying the operation 2 @=@ t to the first equation of the system (1.2) transforms it into a form @@ tv2 ¼  q1 @@t 0

r~p ¼  q1 r @@ ~pt . 0 Using the second equation of the system (1.2) allows you to write down the ratio r @@ ~pt ¼ q0 c2 rðr  vÞ. The combination of these two results leads to the equation @2v  c2 rðr  vÞ ¼ 0 @ t2 As it is known from vector analysis, the identity r
r
a ¼ rðr  aÞ  Da is correct for an arbitrary vector a. Because r
v ¼ 0 for the vibrational field of the sound wave there is an equality rðr  vÞ ¼ Dv The use of this equality leads to the final result @2v  c2 Dv ¼ 0 @ t2

Methodical Instructions and Tasks

133

In order to obtain the pressure equations, it is sufficient to apply the operation r (divergence) to the first equation of the system (1.2), then multiply it by q0 c2 , after which the result should be subtracted from the second equation of the system (1.2), previously differentiated by time. As a result, a wave equation with respect to pressure perturbation ~p is formed: @ 2 ~p  c2 D ~p ¼ 0 @ t2 ~ Exactly the same equation is applied to the behavior of density perturbation q because it (this perturbation) is related to the pressure perturbation by the ratio ~c2 . ~p ¼ q Thus, the acoustic wave represents a change in time and space of all physical fields of the environment. Task 8 Prove that linear approximation equity (1.2–1.4) requires conditions to be met: ~j  q0 , j~pj  q0 c2 , j v j  c. jq @v 1 ¼  r ~p; @t q0

@ ~p þ q0 c2 r  v ¼ 0; @t

@~ q þ q0 r  v ¼ 0 @t

@2u  c2 Du ¼ 0 @ t2 v ¼ ru;

~p ¼ q0

@u ; @t

ð1:2Þ

ð1:3Þ ~¼ q

~ p c2

ð1:4Þ

The solution. Equations (1.2–1.4) were derived from the full equations of hydrodynamics and the equation of state, recorded implicitly. The use of the equation of state is due to the small deviations of the thermodynamic fields of the medium from their equilibrium values q0 and p0 that allows to apply the decom~=q0 j and j~ position of this equation in the Taylor series on small parameters jq p=p0 j. As follows from the equations of linear acoustics, the relationship between pressure ~=q0 j  1, uh ~c2 is fair. Because jq and density perturbations ~p ¼ q 2 2 ~jc  q0 c . Finally, disregard for nonlinear terms in Euler’s equation j~pj ¼ jq pj  q0 c2 , the combinameans that the condition q0 v2 =2  j~pj is met. Because j~ tion of these results gives v2 =2  c2 where the last condition comes from j v j  c.

134

Methodical Instructions and Tasks

Tasks for Chapter 2 Task 1 On the basis of the conditions on the boundary of two contacting elastic media, for ð1Þ ð2Þ which slippage is allowed: ni rij ¼ ni rij , n  nð1Þ ¼ n  nð2Þ , where n is unit vector normal to the interface; rik ¼ k npp dik þ 2l nik - stress tensor; elements of strain   tensor are determined by the ratio nik ¼ 12 @@ xnki þ @@ nxki ; nik ¼ nki ; the upper indices ð1Þ ; ð2Þ indicate the corresponding characteristics of contacting media, to deduce the dynamic and kinematic boundary conditions on the surface of the interface of two liquids. The solution. Since the transition to a liquid medium assumes zeroing of shear stresses ðl ¼ 0Þ, the expression for the stress tensor is simplified: rik ¼ k npp dik . ð1Þ

ð2Þ

Substitution of this expression into a dynamic boundary condition ni rij ¼ ni rij transforms it into a form ð2Þ ð2Þ kð1Þ nð1Þ pp ni dij ¼ k npp ni dij

)

ð2Þ ð2Þ kð1Þ nð1Þ pp ¼ k npp

In order to continue the study of the obtained ratio, it is necessary to write an explicit trace of the deformation tensor npp . Deformation tensor elements are defined through the components of the deformation vector n by the expression   1 @ ni @ nk nik ¼ þ 2 @ xk @ xi The strain vector is defined through scalar and vector potentials by the formula n ¼ ruþr
w

ð2:2Þ

Let’s assume that the vector potential has the form w ¼ A ex þ B ey þ C ez where ei —are the corresponding Cartesian coordinate system orthoses. Then nx ¼ u0x þ Cy0  B0z ; Since nxx ¼ @@nxx , nyy ¼ expression

@ ny @y,

ny ¼ u0y þ A0z  Cx0 ; nzz ¼

@ nz @z,

nz ¼ u0z þ B0x  A0y

uh, the trail npp is determined by the

npp ¼ nxx þ nyy þ nzz 00 00 ¼ u00xx þ Cxy  B00xz þ u00yy þ A00yz  Cxy þ u00zz þ B00xz  A00yz

¼ u00xx þ u00yy þ u00zz ¼ D u00

Methodical Instructions and Tasks

135

As a result, the dynamic boundary condition takes the form kð1Þ D uð1Þ ¼ kð2Þ D uð2Þ As for liquid media kð1Þ ¼ q1 c21 , kð2Þ ¼ q2 c22 (here q1 , q2 —densities, c1 , c2 — sound velocity of contacting media), on the basis of the wave equation @2u  c2jj D u ¼ 0; @ t2

c2jj ¼

k þ 2l ; q

@2w l  c2? D w ¼ 0; c2? ¼ 2 @t q

ð2:3Þ

For scalar potentials, it is possible to give the form to the dynamic boundary condition q1

@ 2 uð1Þ @ 2 uð2Þ ¼ q2 2 @t @ t2

The potentials uð1Þ , uð2Þ used here are the potentials of the non-saline part of the deformation vector. The non-saline part of the velocity of liquid particles of contacting media is determined as a derivative of the time from the non-saline part of the deformation vector. This allows the introduction of non-solenoidal velocity potentials in environments with @ uð1Þ ð2Þ @ uð2Þ ; uv ¼ @t @t

uð1Þ v ¼

and use them to write a dynamic boundary condition ð1Þ

q1

ð2Þ

@ uv @ uv ¼ q2 @t @t

This is the final form of the desired condition, recorded via the velocity field potentials in the contact media. Naturally, expressions in the right and left parts of the condition are calculated at the boundary of the medium section. Let this boundary be set explicitly F ¼ z  fðx; y; tÞ ¼ 0: where the function fðx; y; tÞ describes the shape of the partition surface. Then, the strict record of the dynamic boundary condition looks like

ð1Þ ð2Þ @ uv

@ uv

q1 ¼ q2

@t
@t
z¼fðx;y;tÞ

z¼fðx;y;tÞ

As follows from the second equation (1.4) of linear acoustics, the pressure perturbations in the mediums are related to the acoustic potentials by ratios ~pð1Þ ¼ q1

@ uð1Þ ð2Þ @ uð2Þ ; p~ ¼ q1 @t @t

And then dynamic boundary conditions can be written down also through physical variables

136

Methodical Instructions and Tasks



~pð1Þ
z¼fðx;y;tÞ ¼ ~pð2Þ
z¼fðx;y;tÞ : Let’s now turn to the kinematic boundary condition on the partition boundary: n  nð1Þ ¼ n  nð2Þ . The vector of normal to the surface n is determined by the expression n¼

ez  f0x ex  f0y ey rF ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jrF j 1 þ f02 þ f02 x

y

Substitution of this result in the kinematic boundary condition leads to the ratio

0 ð1Þ ð1Þ
ð2Þ 0 ð2Þ 0 ð2Þ
 f n  f 0n ¼ n  f n  f n nð1Þ
y z x x y z x x y y
z¼fðx;y;tÞ

z¼fðx;y;tÞ

Taк кaк мы пpoвoдим иccлeдoвaния в paмкax линeйнoй мoдeли aкycтичecкиx пoтeнциaлoв, пoлyчeннoe выpaжeниe нyжнo yпpocтить, oтбpocив нeлинeйныe члeны, тaк чтo Since we carry out research within the framework of the linear model of acoustic potentials, the obtained expression needs to be simplified by rejecting nonlinear terms, so that


ð2Þ
nð1Þ ¼ n
z z
z¼fðx;y;tÞ

z¼fðx;y;tÞ

or

@ uð1Þ @ Bð1Þ @ Að1Þ

@ uð2Þ @ Bð2Þ @ Að2Þ

þ  þ  ¼ : @z @x @ y
z¼fðx;y;tÞ @z @x @ y
z¼fðx;y;tÞ ðiÞ

ðiÞ

As expressions @@Bx  @@Ay describe non-acoustic parts of velocity fields in adjoining media, they should be discarded at the description of exclusively acoustic processes, reducing kinematic conditions to a kind

@ uð1Þ

@ uð2Þ

¼ @ z
z¼fðx;y;tÞ @ z
z¼fðx;y;tÞ In order to pass from the obtained kinematic condition, recorded through the potential of the deformation vector to the velocity potential, it is necessary to differentiate this ratio by time, which gives !

2 ð1Þ
@ @ uð1Þ

@ 2 uð1Þ 0@ u
¼ þ ft @t @ z
z¼fðx;y;tÞ @ z@ t @ z2
z¼fðx;y;tÞ !

2 ð2Þ
@ @ uð2Þ

@ 2 uð2Þ 0@ u
þ ft ¼ ¼ @t @ z
z¼fðx;y;tÞ @ z@ t @ z2
z¼fðx;y;tÞ

Methodical Instructions and Tasks

137 ð1Þ

ð2Þ

Neglect by nonlinear members and the transition to uv , uv leads to a final result

ð1Þ ð2Þ @ uv

@ uv

¼

@z
@z
z¼fðx;y;tÞ

z¼fðx;y;tÞ

The use of the first equation (1.4) allows to write down this kinematic boundary condition through physical quantities

ð2Þ

vð1Þ z z¼fðx;y;tÞ ¼ vz z¼fðx;y;tÞ which means that the corresponding components of the velocity fields are equal on different sides of the partition boundary. Task 2 Show that the right side of equation (2.1) for the strain vector is elastic, unlike the right side of equation for the vibrational velocity of the system (1.1), which determines the dissipation of sound waves. The solution. For the sake of clarity, let’s write here Eq. (2.1), to which the deformation vector is subordinated: q

@2n ¼ l D n þ ðk þ lÞr r  n @ t2

Let’s present a solution to this equation in the form of Z1 n0 ðxÞ expðiðkðxÞ  r  x tÞÞ dx

n¼ 0

where kðxÞ—the wave vector corresponding to the frequency x. The use of the linearity property of both the equation and the integration operator allows us to consider the properties of a single strain vector harmonic: nðxÞ ¼ n0 ðxÞ expðiðkðxÞ  r  x tÞÞ Since @ @ nðxÞ ¼ n0 ðxÞ expðiðkðxÞ  r  x tÞÞ @t @t ¼ ixn0 ðxÞ expðiðkðxÞ  r  x tÞÞ; then @2 nðxÞ ¼ x2 n0 ðxÞ expðiðkðxÞ  r  x tÞÞ: @ t2

138

Methodical Instructions and Tasks

In addition, the following relations are valid r  n0 ðxÞ expðiðkðxÞ  r  x tÞÞ ¼ rðexpðiðkðxÞ  r  x tÞÞÞ  n0 ðxÞ ¼ iðkðxÞ  n0 ðxÞÞ expðiðkðxÞ  r  x tÞÞ rðr  n0 ðxÞ expði ðkðxÞ  r  x tÞÞÞ ¼ rði ðkðxÞ  n0 ðxÞÞ expðiðkðxÞ  r  x tÞÞÞ ¼ i ðkðxÞ  n0 ðxÞÞrð expðiðkðxÞ  r  x tÞÞÞ ¼ kðxÞðkðxÞ  n0 ðxÞÞ expðiðkðxÞ  r  x tÞÞ Dn0 ðxÞ expði ðkðxÞ  r  x tÞÞ ¼ k2 ðxÞn0 ðxÞ expði ðkðxÞ  r  x tÞÞ Substitution of these expressions in the equation for the vector of deformation and discarding in all terms of the value expði ðkðxÞ  r  x tÞÞ (because it does not turn to zero) leads to the equation of the form qx2 n0 ðxÞ  lk2 n0 ðxÞ  ðk þ lÞkðk  n0 ðxÞÞ ¼ 0 By multiplying the scalarly obtained equation by the wave vector k, as a result of which the previous equation is transformed into  2 qx  ðk þ 2lÞk2 ðk  n0 ðxÞÞ ¼ 0 This equation is satisfied in two ways: 1: k  n0 ðxÞ ¼ 0;

2: qx2  ðk þ 2lÞk2 ¼ 0

In the first case, we need to go back to the previous equation from which it follows: 1: qx2  lk2 ¼ 0: Since x, q, l—are non-negative real values, the vector k is also a real value. In the second case, k the real value is also non-negative, and k is the real vector. Thus, in the integral representation for the strain vector n, all wave vectors kðxÞ are real values. This means that Eq. (2.1) is an undissipative process. Task 3 How should the recording of the stress tensor be changed to obtain an equation for the evolution of the viscoelastic medium strain vector based on the general approach? The solution. The first equation of the system (1.1) gives the necessary hint for solving the problem. By rejecting a term with pressure in this equation, a parabolic-type equation is formed, which generates the existence of solutions with

Methodical Instructions and Tasks

139

dissipation. In this case, when discarding the viscoelastic medium’s elastic l D n þ ðk þ lÞr r  n (non-dissipative) part in the desired equation for the evolution of the deformation vector, the remaining terms should form a conditional equation @A ¼ g DA þ frr  A @t where g, f are some of the dissipative characteristics of the environment 2 Since there is a value @@ tn2 in the left side of equation (2.1), it is necessary to

choose a value A ¼ @@ nt as a value A. Thus, the general view of the sought equation is represented by the ratio q

@2n @n @n þ frr  ¼ l D n þ ðk þ lÞr r  n þ g D @ t2 @t @t

At the same time, we do not give a specific physical meaning to the dissipative coefficient g, f because its clarification is part of the task of another section of mechanics.

Tasks for Chapter 3 Task 1 Show that the fairness of the conservation law @E þ r  ð~pvÞ ¼ 0 @t

ð3:3Þ

~ j  q0 , j v j  c. requires compliance with the conditions j q The solution. Let’s calculate the value @@ Et included in the saving law (3.3): ~ @q ~ q @E @v ¼ c2 þ q0 v  q0 @ t @t @t Substitution of this expression in the law of preservation gives it the appearance c2

~ @q ~ q @v þ q0 v  þ ~p r  v þ v  r ~ p¼0 q0 @ t @t

If you use the nonlinearized kind of the Euler equations and continuity   ~ @v @q ~Þ r  v þ v  r q ~¼0 q0 þ ðvrÞ v ¼ r ~p; þ ðq0 þ q @t @t a part of the values included in the conservation law can be expressed by means of relations

140

Methodical Instructions and Tasks

q0

~ @v @q ~Þ r  v  v  r q ~ ¼ r ~p  q0 ðvrÞ v; ¼ ðq0 þ q @t @t

Substitution of these expressions in the law of conservation, taking into account ~c2 , leads to the result the relationship ~p ¼ q   ~ ~ q q ðvrÞ v v r þ ¼0 q 0 q0 c2 Since, in general, the field of vibrational velocity of the sound wave is not orthogonal to the value in brackets, the implementation of the conservation law requires that the value ~ ~ q q ðvrÞ v r þ q0 q 0 c2 had a second order of magnitude. This means that the conditions of the task have to be formulated. Task 2 Using expressions up ¼ u0 expðiðk  r  x tÞÞ; us ¼ u0

ð1Þ

uc ¼ u0 H0 ðkrÞ expðix tÞ

expðiðkr  x tÞÞ r

ð3:4Þ

for acoustic potentials of plane and spherical waves to determine the type of the corresponding source, which should stand in the right part of the wave equation (1.3). The solution. Let’s look at the plane wave first. To simplify the problem, we will assume that the entire space is divided into two halves by the plane ðx; yÞ, in one of which the plane wave propagates along the axis z, and in the other—against this axis. Let us also assume that the amplitudes of these waves are equal A þ and A , respectively. Thus, the full acoustic potential in the whole space is represented in the form u ¼ A þ eiðkzx tÞ hðzÞ þ A eiðkz þ x tÞ hðzÞ where hðzÞ—Heaviside’s single function. We need to find out which function Fðr; tÞ should be on the right side of the wave equation u00tt  c2 Du ¼ Fðr; tÞ that the solution of this equation is given by the potential presented above. With the help of the given potential, let’s calculate the left part of the wave equation. Double differentiation of the potential in time gives the result u00 tt ¼ x2 A þ eiðkzx tÞ hðzÞ  x2 A eiðkz þ x tÞ hðzÞ ¼ x2 u

Methodical Instructions and Tasks

141

Since the potential does not depend on spatial coordinates x and y, Du ¼ u00zz , it remains only to calculate the second derivative by z. Consistently acting, we’ll get u0z ¼ ikA þ eiðkzx tÞ hðzÞ þ A þ eiðkzx tÞ dðzÞ  ikA eiðkz þ x tÞ hðzÞ  A eiðkz þ x tÞ dðzÞ   ¼ ik A þ eiðkzx tÞ hðzÞ  A eiðkz þ x tÞ hðzÞ þ ðA þ  A Þeix t dðzÞ where dðzÞ—Dirac’s function. Next up:   u00zz ¼ k2 A þ eiðkzx tÞ hðzÞ  A eiðkz þ x tÞ hðzÞ þ ikA þ eiðkzx tÞ dðzÞ þ ikA eiðkz þ x tÞ dðzÞ þ ðA þ  A Þeix t d0 ðzÞ ¼ k2 u þ ikðA þ þ A Þeix t dðzÞ þ ðA þ  A Þeix t d0 ðzÞ Substitution of the obtained expressions in the wave equation, taking into account the dispersion relation x ¼ kc, gives the final result Fðr; tÞ ¼ c2 ðikðA þ þ A Þ dðzÞ þ ðA þ  A Þ d0 ðzÞÞ eix t It can be seen that the generation of plane monochromatic waves is caused by a source concentrated on a plane whose intensity varies harmoniously in time. The value ikc2 ðA þ þ A Þ ¼ ixcðA þ þ A Þ, standing at Dirac’s function dðzÞ, is interpreted as the volumetric density m of mass monopolies evenly distributed over the source plane. The value c2 ðA þ  A Þ derived from the Dirac function d0 ðzÞ is the surface density q of unidirectional mass dipoles evenly distributed over the source plane, with the dipole moments perpendicular to the source plane. At the given characteristics of a source, namely at known densities m and q, amplitudes of waves are defined on the basis of Aþ ¼ 

x q  imc ; 2x c2

A ¼

x q þ imc 2x c2

If only mass monopolies ðq ¼ 0Þ are located on the surface, then the wave amplitudes are equal A þ ¼ A and in phase, if there are only mass dipoles ðm ¼ 0Þ, then the amplitudes are equal but antiphase. The intermediate case is described by the above general relations. Let’s now turn to the source of the spherical wave, the potential of which is set by the expression u¼A

eiðkrx tÞ r

142

Methodical Instructions and Tasks

You can see that eiðkrx tÞ ¼ x2 u r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In our designations, r ¼ x2 þ y2 þ z2 . Since neither A, nor eix t do they ikr depend on the spatial coordinates, then Du ¼ Aeix t D er . Let’s sequentially calikr culate the value D er . Let’s start with the coordinate x derivative: u00 tt ¼ x2 A

  @ eikr 1 @ ikr @ 1 ik @ r @ 1 ikr e þ eikr ¼ þ ¼ e @x r r@x @xr r @x @xr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The function rðx; y; zÞ ¼ x2 þ y2 þ z2 is a homogeneous generalized function of the degree k ¼ 1 because pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðax; ay; azÞ ¼ a2 x2 þ a2 y2 þ a2 z2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a x2 þ y2 þ z2 ¼ arðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The function 1=rðx; y; zÞ ¼ 1= x2 þ y2 þ z2 is a homogeneous generalized function of the degree k ¼ 1, because pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=rðax; ay; azÞ ¼ 1= a2 x2 þ a2 y2 þ a2 z2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a1 1= x2 þ y2 þ z2 ¼ a1 1=rðx; y; zÞ @r @x

Dimension of the task space n ¼ 3. Since at k ¼ 1, 1  k 6¼ n, the derivatives and @@x 1r are calculated in the usual sense: @r x ¼ ; @x r

@ 1 x ¼ 3 @xr r

So, there has been @ eikr ¼ @x r



 ikx x  eikr r2 r3

Let’s calculate the second derivative by x:     @ 2 eikr @ ikx x ikx2 ikx x ikr ikr ikr @ ¼   e ¼ 4 ðikr  1Þe þ e @ x r2 r3 @ x r2 r3 @ x2 r r The function rx2 is a homogeneous function of the degree k ¼ 1. As in this case 1  k 6¼ n, the derivative is calculated in the usual sense: @ ikx r 2  2x2 ¼ ik @ x r2 r4

Methodical Instructions and Tasks

143

The function rx3 is a homogeneous function of the degree k ¼ 2. Since in this case 1  k ¼ n, the derivative is calculated in a generalized sense: Z @ x r 2  3x2 x dy dz ¼ þ dðx; y; zÞ @ x r3 r3 r5 C

where dðx; y; zÞ ¼ dðxÞdðyÞdðzÞ, C—the surface of a single sphere surrounding the origin of coordinates. Finally, 2 3 Z @ 2 eikr 4 k2 x2 r 2  3x2 x dy dz5 ikr ¼  3 þ ðikr  1Þ e  dðx; y; zÞ r3 @ x2 r r r5 C

Similar calculations show that 2 3 Z 2 ikr 2 2 2 2 @ e k y r  3y y dz dx5 ikr ¼ 4 3 þ ðikr  1Þ e  dðx; y; zÞ r3 @ y2 r r r5 C 2 3 Z 2 ikr 2 2 2 2 @ e k z r  3z z dx dy5 ikr ¼ 4 3 þ ðikr  1Þ e  dðx; y; zÞ r3 @ z2 r r r5 C

So that way 2 3 Z eikr 4 k2 x dy dz þ y dz dx þ z dx dy5 ikr ¼   dðx; y; zÞ e D r3 r r C  2  k þ 4pdðx; y; zÞ eikr ¼ r The last equality is conditioned by the fact that the integral is a solid angle under which a sphere from its center is visible, i.e., 4p. Substitution of the obtained results in the wave equation taking into account the dispersion relation x ¼ kc leads to the determination of the spherical wave source function: Fðr; tÞ ¼ 4pc2 A dðx; y; zÞ eix t Task 3 To show that the equivalence of force and mass sources (in terms of the identity of the physical fields they produce) in the system (3.5) q0

@v ¼ r ~p þ f; @t

~ @q þ q0 r  v ¼ m; @t

~ @q 1 @~ p ¼ @ t c2 @ t

ð3:5Þ

144

Methodical Instructions and Tasks

is achieved only when and when U and m the wave equation (1.3) is satisfied. Determine the relationship between U and m providing this equivalence. The solution. Exclude from the system (3.5) the density perturbation, resulting in the system taking the form q0

@v ¼ r ~p þ f; @t

@ ~p þ c2 q0 r  v ¼ m @t

If the first equation of the obtained system is differentiated by time, and Laplace operator is applied to the second equation, and then the difference in the results of differentiation is formed, the equation  2    @ @f @U 2 2 2  c D m ¼ D þc m q0 c D rv¼r @t @t @ t2 If we now multiply the first equation by c2 , and then apply the operation r(divergence) to it, then the second equation is differentiated by time and the difference of the obtained results is again formed, then the equation of the form      2  @ 2 2 @m 2 @m rf ¼c þ DU  c D ~p ¼ c @t @t @ t2 In order for there to be equivalence of power and mass sources, it is necessary that the right parts of these equations coincide with each other in cases U ¼ 0; m 6¼ 0 and U 6¼ 0; m ¼ 0. It follows from the first equation that when one type of source is replaced by another type of source, the condition must be met @U ¼ c2 m @t and from the second equation follows the condition @m ¼ DU @t The compatibility of these conditions leads to the fact that both mass and power sources must satisfy the wave equation  2   2  @ @ 2 2 c D U ¼ c D m¼0 @ t2 @ t2 and the necessary connection between the sources is defined by any of the above conditions. Task 4 Based on the asymptotic expression for the pressure field of a point acoustic dipole ~p  

  1 r €ðt  r=cÞ ; r  q_ ðt  r=cÞ þ q 3 4p r c

r ¼ jrj

Methodical Instructions and Tasks

145

where q—the dipole moment, to determine the field of the point acoustic quadrupole; the points above indicate differentiation in time. The solution. The acoustic quadrupole represents a set of two dipoles which dipole moments are identical in size, but are opposite on a sign. Let the dipoles be located at the points r0 and the pressure perturbation value created by them be calculated at the point r. The dottedness of the quadrupole means, among other things, the execution of the ratio jr0 j=jrj  1. In this case, the quadrupole field is determined by the expression   ðr þ r0 Þ j r þ r0 j €ðt  jr þ r0 j=cÞ _  jr þ r0 j=cÞ þ ~p   q  qðt c 4pjr þ r0 j3   ðr  r0 Þ j r  r0 j € _ q ðt  r  r þ þ  qðt  r þ r j j=cÞ j j=cÞ 0 0 c 4pjr  r0 j3 Here, r þ r0 —the vector directed from the point dipole with the moment q located in the point r0 , and r  r0 —the vector directed from the point dipole with the moment q located in the point þ r0 . For further calculations, we will need the values j r  r0 j, calculated taking into account the ratio jr0 j=jrj  1. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jr  r0 j ¼ ðr  r0 Þ2 ¼ r2  2r  r0 þ r20 ¼ r2  2r  r0 þ r20 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼ r 1  2r  r0 =r 2 þ r20 =r 2  r 1  r  r0 =r 2 Then 1 j r  r0 j

3



1 1 1  3 1  3r  r0 =r 2 3 3 r ð1  r  r0 =r 2 Þ r

Besides, it is necessary to calculate the values fðt  jr  r0 j=cÞ. Based on the above ratios   r r  r0   r r  r0  fðt  jr  r0 j=cÞ  f t  1  2 ¼f t  c c r rc r  r0 _  f ðt  r=cÞ  f ðt  r=cÞ rc Based on these auxiliary formulas, the quadrupole pressure field can be written as  r  r0  ðr  r0 Þ h r  r0 r r  r0   r  r0 vi € € _ ~p  1 þ 3 2 1  q þ q þ  q þ q 4pr 2 c r rc r2 rc h      r  r0 ðr þ r0 Þ r  r0 r r  r0 r  r0 vi € € _  13 2 1 þ q  q þ  q  q 4pr 2 c r rc r2 rc       1 r  r0 r r  r0 r  r0 €   6 2 r  2r0  q_ þ q 6 2 r0  2r  qv 2 2 2 4pr c r 4pr c r

146

Methodical Instructions and Tasks

The use of the fact that the vector r0 is orthogonal to the dipole moment q (i.e., r0  q ¼ 0) allows to simplify the obtained result:   1 r r2 v € _ ~p  q þ ð 2r  r Þ r  3 q þ 3 q : 0 4pr 4 c c2 Let the vector r cross the quadrupole plane at an angle u, and its projection on this plane is an angle h with the dipole moment direction q. Well, then 2r  r0 ¼ 2rr0 cos u sin h, where q ¼ j q j. If we lead a value s ¼ 2r0 q called quadrupole moment, the result for the pressure field can be represented as     sin h cos h 2 r r2 v Dðu; hÞ r r2 v ~p  € € cos u 3_ s þ 3 s ¼ 3_ s þ 3 s s þ s þ 4pr 2 c 4pr 2 c c2 c2 The value Dðu; hÞ ¼ sin h cos h cos2 u is a diagram of the direction of the quadrupole. When the characteristic time of the quadrupole torque change is significantly longer than the delay time r=c required for the signal to reach the observation point, the expression for pressure is significantly simplified: ~p 

3 s_ Dðu; hÞ 4pr 2

It should be noted here that the value t  r=c, which is omitted to reduce the record of results is the argument of functions q and S in the obtained expressions.

Tasks for Chapter 4 Task 1 The task space represents two semi-infinite regions with equal sound velocity c1 and c3 separated by a plane-parallel layer of thickness H. The speed of the sound in the layer is c2 . There is a relationship between the speeds c1 \c2 \c3 . Determine the limit angle of incidence of the plane wave from the first region, which, if exceeded in the third region, will not pass the sound. Will the result change if c1 \c3 \c2 ? The solution. There is a formal approach to solving the problem. Let the wave propagate at an angle h1 in the upper half-plane, at an angle in the layer h2 , and at an angle h3 in the third region. According to Snellius’ law, for this wave the relations sin h1 c1 ¼ ; sin h2 c2

sin h2 c2 ¼ sin h3 c3

Multiplication of the first ratio by the second one leads to the connection

Methodical Instructions and Tasks

147

sin h1 c1 ¼ sin h3 c3 In order for the wave not to pass to the third area, it is necessary to fulfill the condition c3 sin h1 1 c1 Hence, the limit value of the angle h1 above which the sound will not pass to the third area: h 1 ¼ arcsin c1 =c3 This result does not depend on the thickness of the layer H, nor on the ratio of velocities in the parallel layer and in the third medium. This result is correct, but the way it is obtained is incorrect. The fact is that if the condition cc21 sin h1 1 is met, there is no plane wave propagating at an angle h2 in the layer. At cc21 sin h1 \1, the above reasoning is true, and for such angles of decline h1 , this method of the solution remains valid. The correct way of solving a case cc21 sin h1 1 is given in the solution of task 5. Task 2 Calculate the reflection and propagation coefficients of a plane wave when it falls at an angle h on a plane-parallel plate of thickness d. The frequency of the wave is equal x, the density of the environment and the speed of sound in it are set by the values q1 , c1 . Similar characteristics of the plate are determined by the values q2 , c2 . The solution. The figure shows a schematic representation of the process of falling of a plane monochromatic wave on a plane-parallel plate.

148

Methodical Instructions and Tasks

The plane ðx; zÞ is aligned with the plane of the fall of the wave. Acoustic capacities in the highlighted areas are presented in the form: u1 ¼ A þ eiðkx þ j1 zx tÞ þ A eiðkxj1 zx tÞ u2 ¼ B þ eiðkx þ j2 zx tÞ þ B eiðkxj2 zx tÞ u3 ¼ Ceiðkx þ j1 zx tÞ where A þ , A are the amplitudes of the waves running in the region of “1” in the negative (incident wave) and positive (reflected wave) directions of the axis z, respectively; B þ , B are the amplitudes of the waves running in the region of “2” in the negative and positive directions of the axis z, respectively; C is the amplitude of the passed wave in the region of “3.” Components of wave vectors of all waves in the direction of the axis x are marked with a symbol k, and symbols j1 , j2 are marked with components of wave vectors along the axis z in the corresponding media. Since the incident, reflected and passed over the plate wave propagate in areas with the same characteristics, the reflection R and propagation coefficients T are determined by the jA j2

R¼

jA þ j

2

;

T¼

jC j2 jA þ j2

In order to determine the relations of the respective amplitudes, it is necessary to use the boundary conditions, which are



@ u1

@ u2

@ u1

@ u2

q1 ¼ q ; ¼ 2 @ t
z¼0 @ t
z¼0 @ z
z¼0 @ z
z¼0



@ u2

@ u3

@ u2

@ u3

q2 ¼ q ; ¼ 1 @t
@t
@z
@z
z¼d

z¼d

z¼d

z¼d

Substitution of representations for acoustic potentials in these expressions generates a system of algebraic equations for amplitudes A þ , A , B þ , B and C: q1 ðA þ þ A Þ ¼ q2 ðB þ þ B Þ; q2 ðB þ e

ij2 d

j2 ðB þ e

ij2 d

þ B e

ij2 d

 B e

Þ ¼ q1 Ce

ij2 d

j1 ðA þ  A Þ ¼ j2 ðB þ  B Þ ij1 d

Þ ¼ j1 Ceij1 d

Exclusion of amplitudes B from this system allows to define reflection and passage coefficients by values R¼

2a2 b2 ð1  cosð2j2 dÞÞ ; 4 a þ b4  2a2 b2 cosð2j2 dÞ

2

T¼

ð a2  b2 Þ 4 4 a þ b  2a2 b2 cosð2j2 dÞ

Methodical Instructions and Tasks

149

where a ¼ q1 j2 þ q2 j1 ;

b ¼ q1 j2  q2 j1

Now k, j1 , j2 it is necessary to express, through the task parameters. The wave number of sound waves in the regions “1” and “3” is set by the value x=c1 and therefore k ¼ x sin h=c1 , j1 ¼ x cos h=c1 . According to the Snellius law

c1 sin h sin h ¼ c2 , where h is the angle, which is the axis of the wave propagation z direction in the region of “2.” At the same time, j2 ¼ x cos h =c2 . As a result, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wave number j2 is set by the expression j2 ¼  c1xc2 c21  c22 sin2 h. Substitution

of the obtained values k, j1 , j2 , in the formulas for R and T determines the final result expressed through the data of the original problem. From the expressions for reflection and propagation coefficients, in particular, it follows that if the thickness of the plate d is stacked with an integer number of half-wave propagating in the region “2,” then there is no reflection from the plate (half-wave plate) R ¼ 0, T ¼ 1. And if an odd number of quarters of the wavelength is stacked at this thickness in the medium “2,” then there is a minimum of sound passing through the plate (quarter-wave plate). Task 3 Let the source of monochromatic spherical frequency wave is located in the center of the spherical cavity of the radius R, the density of the medium in which the speed of sound are set by the values q1 , c1 , and the source of monochromatic spherical frequency wave x. The cavity is surrounded by an unlimited environment with characteristics q2 , c2 . Determine the wavelength-to-space ratio of the cavity. The solution. Let’s write down the acoustic potentials for the inner and outer parts of the spherical cavity. Inside, the potential is defined by the expression u1 ¼ A þ

eiðk1 rx tÞ eiðk1 r þ x tÞ þ A ; r r

k1 ¼

x c1

which is the sum of the wave potentials traveling from the center (amplitude A þ ) and to the center (amplitude A ). In the external part, the potential is described by a spherical wave leaving the iðk rx tÞ cavity u2 ¼ B e 2r ; k2 ¼ cx2 The sound wave propagation coefficient is determined by the ratio of the energy density of the passed wave to the incident wave. The density of the sound wave energy flux S ¼ ~p v is determined by the value where the top line means averaging over the oscillation period. But this formula is written for the real values of pressure perturbations and oscillatory velocity. Since this is a complex description of physical fields, this formula needs to be modified: S¼

1 ~p v þ ~p v þ ~p v þ ~ p v

4

150

Methodical Instructions and Tasks

where the “star” stands for complex conjugate value. Since all physical values of the problem have a temporal dependence of the kind ix t e , then ~p v ¼ ~ p v ¼ 0, ~p v ¼ ~p v , ~p v ¼ ~p v, so 1 1 1 p vÞ S ¼ ð~p v þ ~p vÞ ¼ Reð~p v Þ ¼ Reð~ 4 2 2 In a falling wave @ eiðk1 rx tÞ eiðk1 rx tÞ Aþ ¼ ix q1 A þ @t r r @ eiðk1 rx tÞ eiðk1 rx tÞ Aþ ¼ Aþ ¼ ðik1 r  1Þ @r r r2

~p þ ¼ q1 vþ Then

! 1 ixq1 jA þ j2 ixq1 jA þ j2 ðik1 r  1Þ  ðik1 r  1Þ jS þ j ¼ 4 r3 r3 ¼

x k1 q1 jA þ j2 r2

Similar calculations show that in the past wave 1 ixq2 jBj2 ixq2 jBj2 ðik r  1Þ  ðik2 r  1Þ jSj ¼ 2 4 r3 r3 ¼

!

x k2 q2 jBj2 r2

so the coefficient of passage is determined by the value T¼

k2 q2 jBj2 c 1 q 2 j Bj 2 jSj ¼ ¼ jS þ j k1 q1 jA þ j2 c2 q1 jA þ j2

In order to determine the final pass rate, it is necessary to know the ratio jBj =jA þ j2 . To find it, you need to use the boundary conditions that look like



@ u1

@ u2

@ u1

@ u1

q1 ¼ q ; ¼ 2 @ t
r¼R @ t
r¼R @ r
r¼R @ r
r¼R 2

Substitution in these relations of potentials u1 and u2 generates a system of equations that connect the amplitudes A þ , A and B.  q1 A þ eik1 R þ A eik1 R ¼ q2 Beik2 R A þ eik1 R ðik1 R  1Þ  A eik1 R ðik1 R þ 1Þ ¼ Beik2 R ðik2 R  1Þ

Methodical Instructions and Tasks

151

An exception to this system of amplitude A : 2ik1 q1 R eik1 R A þ ¼ ðq2  q1 þ iRðk1 q2 þ k2 q1 ÞÞeik2 R B allows you to determine the desired ratio j B j2 =j A þ j2 , so that as a result the coefficient of passage is determined by the expression T¼

4q1 q2 c1 c2 x2 R2

ðq2  q1 Þ2 c21 c22 þ x2 R2 ðq1 c1 þ q2 c2 Þ2

Task 4 In the center of a spherical air bubble surrounded by water, a short sound impulse is emitted, the duration of which is significantly shorter than the time of the sound wave travel from the center to the border of the bubble. Estimate how long a sound wave generated in an air bubble of 1 mm diameter will emit a fraction of its energy q ¼ 0:5 into the surrounding water. The air density should be taken to be equal 1.22 kg/m3. Frequency of sound pulse filling f = 5
105 Hz. The solution. Since the wave is spherical, its phase front at the bubble boundary coincides with the boundary itself. In this case, the wave energy transfer coefficient for one act of fall on the boundary can be estimated approximately by the value x  103 (the result of task 2). During the first act of the wave falling to the border in the water will pass a fraction of energy x, and inside the bubble will remain a fraction of energy 1  x. During the second act, a fraction xð1  xÞ will go into the water and a fraction 1  x  xð1  xÞ ¼ ð1  xÞ2 will remain in the bubble. For the third act, the fraction xð1  xÞ2 will go into the water, and the fraction ð1  xÞ3 will remain in the bubble, etc. For N of falling waves into the water, the share of xð1 þ ð1  xÞ þ ð1  xÞ2 þ   Þ ¼ x

N 1 X

ð1  xÞi ¼ x

i¼0

1  ð1  xÞN 1  ð1  xÞ

¼ 1  ð1  xÞN ¼ q In this case, the number of sound wave impacts on the surface of the bubble required to emit the fraction q will be determined by N¼

ln ð1  qÞ ln ð1  xÞ

Between the falls, there is time DT ¼ d=c, where d—the diameter of the bubble, c is the speed of sound in the air. As a result, the required radiation time of the wave energy fraction q into water is T ¼ N DT ¼

d ln ð1  qÞ c ln ð1  xÞ

152

Methodical Instructions and Tasks

Substitution d ¼ 103 m, c ¼ 343 m/s, q ¼ 0:5, x ¼ 103 leads to evaluation T  2
103 s. The wave almost entirely emits its energy into the air ðq ¼ 0:99Þ for T  1:4
102 s. Task 5 Give solution of problem 1 for the case cc21 sin h1 1 on the basis of wave description of physical fields. The solution. Let’s introduce a coordinate system, in which the axis x is located in the plane of the wave fall and is directed along the surface of the first and second regions. The axis z is perpendicular to the first region. In the first region, the acoustic potential of the incident wave is described by  u1 ¼ A þ eikz þ A eikz eiðkxx tÞ ; Since

c2 c1

k¼

x sin h1 ; c1

k¼

x cos h1 c1

sin h1 1, in the second area, capacity is defined by the expression u2 ¼ ðB þ el z þ B el z Þ eiðkxx tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x sin2 h1  c21 =c22 l ¼ k2  x2 =c22 ¼ c1

Under the condition of the task c1 \c3 and therefore in the third area qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x sin2 h1  c21 =c23 u3 ¼ Dem z eiðkxx tÞ ; m ¼ k2  x2 =c23 ¼ c1 The use of pressure and vibrational velocity determination in the sound wave (~p ¼ q@ u=@ t, v ¼ r u) allows to record the physical fields of the task in the form of  ~p1 ¼ ixq1 A þ eikz þ A eikz eiðkxx tÞ

   v1 ¼ ik A þ eikz þ A eikz ex þ ik A þ eikz  A eikz ez eiðkxx tÞ ~p2 ¼ ixq2 ðB þ el z þ B el z Þ eiðkxx tÞ v2 ¼ ½ik ðB þ el z þ B el z Þ ex þ l ðB þ el z  B el z Þ ez eiðkxx tÞ ~p3 ¼ ixq3 Dem z eiðkxx tÞ v3 ¼ ½ik Dem z ex þ m Dem z ez eiðkxx tÞ Here, ex , ez are single orthoses along the axes x and z, respectively. On the basis of these relations, the acoustic energy fluxes density averaged over the period of oscillations in the incident, reflected waves in the first region, as well as in the second and third regions are calculated. It is done with the help of the 1 1 1 p vÞ S ¼ ð~p v þ ~p vÞ ¼ Reð~p v Þ ¼ Reð~ 4 2 2

Methodical Instructions and Tasks

153

Thus, in the incident and reflected waves Sþ ¼

xq1 jA þ j2 ðkex þ kez Þ; 2

S ¼

xq1 jA j2 ðkex  kez Þ 2

In Area 2, there is S2 ¼

 xq2 k  jB þ j2 e2l z þ jB j2 e2l z þ B þ B  þ B þ B ex 2 xq l  þ i 2 B þ B  B þ B  ez 2

And finally, in the third area: S3 ¼

xq3 k 2 2m z jDj e ex 2

The result for the third region immediately shows that the acoustic energy is not carried inside this region, and we should expect the incident wave to be entirely reflected from the boundary between the first and second regions. But that result still needs to be proven. Since the third region has no energy transfer along the axis z, the second region has no energy transfer along the axis at z ¼ H (i.e., at the boundary between the plane-parallel layer and the third region). This, as follows from the expression for the requirement: B þ B  B þ B  ¼ 0. In order to show the execution of this ratio S2 , it is necessary to use the boundary conditions at z ¼ H: ~p2 jz¼H ¼ ~p3 jz¼H ;

v2  ez jz¼H ¼ v3  ez jz¼H

Substitution in these conditions of the above-calculated expressions for the physical fields leads to the equations   q2 B þ el H þ B el H ¼ q3 Dem H ; l B þ el H  B el H ¼ m Dem H From these equations, it follows Bþ ¼

lq3 þ mq2 ðlmÞ H De ; 2lq2

B ¼

lq3  mq2 ðl þ mÞ H De 2lq2

Since q2 , q3 , l and m are real values, the requirement B þ B  B þ B  ¼ 0 is met, and in the second medium, there is no energy transfer along the axis z. But then in the first medium on the border of the first and second medium, there should be no energy flow along this axis. Of the expressions S for this means equality j A þ j2 ¼ j A j2 . Since the reflection coefficient is determined by the value 2

R ¼ jjSSþ jj ¼ jjAA jj2 , it can be seen that R ¼ 1—the incident wave will be fully þ

reflected, and this result does not depend on the thickness of the layer H, nor on the ratio between the velocities in the second and third regions.

154

Methodical Instructions and Tasks

Tasks for Chapters 5, 6 Task 1 In the two-dimensional case where the distribution of the sound velocity in the water depends only on the vertical coordinate z, to determine the limiting angle h to the vertical radiation of the beam from a point located at depth H, such that the beam will be reflected from the surface of the water, before reaching the point of rotation. The solution. In order to answer this question, one should use the concept of a hðzÞ radial invariant I ¼ sincðzÞ . In the initial point, this invariant takes on a value sin h at depth z ¼ H. If z 2 ½H; 0 at any point (z ¼ 0—surface IðHÞ ¼ cðHÞ coordinate) the beam is rotated, i.e., sin hðz Þ ¼ 1, the invariant is represented in the form Iðz Þ ¼ c ðz1 Þ. In order for the beam to reach the surface, the ratio of

IðHÞ\Iðz Þ;

8 z 2 ½H; 0

Hence, the angle h must meet the condition h\ arcsin

c ðHÞ ; c ðz Þ

8 z 2 ½H; 0

This condition defines the limit angle by the expression h ¼ arcsin

cðHÞ ; max cðzÞ

z 2 ½H; 0:

If the angle of radiation is less than this limit angle, the beam will reach the surface. Hence, the angle must meet the condition Task 2 The two-dimensional propagation of the sound beam in the plane ðx; zÞ is considered. The speed of sound depends only on the coordinate z. Calculate the path of the beam coming out of the point ðx0 ; z0 Þ at an angle to the vertical h, provided that the speed of sound at the length of the path of the beam changes by a small value, and the length of the path does not exceed the wavelength. The solution. Let’s consider differentially small segment of the trajectory, to which there correspond differentially small shifts of the beam dx and dz along the axes x and z, respectively. Using the current angle of inclination of the beam to the vertical hðx; zÞ allows you to record the ratio dx sin hðx; zÞ ¼ dz cos hðx; zÞ Along the beam, the ray invariant I is saved, which in the starting point ðx0 ; z0 Þ has the value

Methodical Instructions and Tasks

155

I¼

sin h c ðz0 Þ

Because I ¼ const there is an equality sin h sin hðx; zÞ ¼ c ðz0 Þ c ðzÞ From this equality, the values sin hðx; zÞ and cos hðx; zÞ are defined: cðzÞ sin h cðz0 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðzÞ c2 ðz0 Þ  sin2 h cos hðx; zÞ ¼ 1  sin2 hðx; zÞ ¼ cðz0 Þ c2 ðzÞ sin hðx; zÞ ¼

As a result, the trajectory equation takes the form dx sin h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : c2 ðz0 Þ dz  sin2 h c2 ðzÞ

Integration of this equation allows to connect the horizontal and vertical coordinates of the beam trajectory point: Zz x ¼ x0 þ z0

sin h d n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 c ðz0 Þ 2  sin h 2 c ðnÞ

This ratio is valid if the beam does not reach the turning point on its path, the vertical coordinate of which is determined by the equation c ðz Þ ¼

c ðz0 Þ : sin h

Let’s accept, for simplicity, that on the way the beam does not feel turn at the expense of refraction. Now, using the small variability of the sound velocity in the path of the beam, we will decompose the value c2 ðz0 Þ=c2 ðnÞ in a row: c2 ðz0 Þ c2 ðz0 Þ ¼ 2 2 c ðnÞ c ðz0 Þ  ðz0  nÞc0 ðz0 Þ þ ðz0  nÞ2 c00 ðz0 Þ=2 þ    1 ¼ 2 1  ðz0  nÞc0 ðz0 Þ=c ðz0 Þ þ ðz0  nÞ2 c00 ðz0 Þ=2c ðz0 Þ þ     1þ

2ðz0  nÞc0 ðz0 Þ c ðz0 Þ

156

Methodical Instructions and Tasks

The last equality is conditioned by the fact that in radial approximation, the condition of applicability of the radial theory kc jr cj  1 is fulfilled, which in this

0
ðz0 Þ
case looks like
ðznÞc
 1 it is true, because j z  n j\k. c ðz0 Þ Substitution of the resulting decomposition in the integrand expression determines the relationship between the coordinates of the formula Zz

dn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ðz Þ 0 cos2 h þ 2ðz0 nÞc z0 c ðz0 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! c ðz0 Þ sin h 2ðz0  zÞc0 ðz0 Þ 2 hþ cos h  cos ¼ x0 þ c0 ðz0 Þ c ðz0 Þ

x  x0 þ sin h

The obtained ratio expresses the horizontal coordinate of the beam trajectory point through its vertical coordinate. It is often more convenient to have feedback, which, according to the formulae presented, looks like it: z ¼ z0 þ ðx  x0 Þctg h 

c0 ðz0 Þðx  x0 Þ2 2c ðz0 Þ sin2 h

It can be seen that a square term ðx  x0 Þ describes the phenomenon of refraction in the local vicinity of the radiation point. Task 3 Get the solution of Task 2, provided that the size of the beam propagation area can exceed the wavelength of the wave. The solution. The general course of the solution repeats the approach described above, the only difference is the approximation of the value c2 ðz0 Þ=c2 ðnÞ. Let the vertical coordinate of the beam in the considered area be limited by some value z ¼ f. Since the deviation of the sound velocity is assumed to be small, linear approximation is acceptable c2 ðz0 Þ ¼ a ðn  z0 Þ þ 1; n 2 ½z0 ; f: c2 ðnÞ   2 ðz0 Þ 1 1 2 2  Here, a ¼ cfz 2 2 c ðfÞ c ðz0 Þ —the average gradient of the value c ðz0 Þ=c ðnÞ 0 in the segment n 2 ½z0 ; f. You can see that the approximated value at the ends of the segment takes the necessary values. Thus, there is an expression Zz x ¼ x0 þ z0

sin h d n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos h þ a ðn  z0 Þ

the integration of which leads to a ratio

Methodical Instructions and Tasks

x ¼ x0 þ

157

 ffi 2 sin h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 h þ a ðz  z0 Þ  cos h : a

Feedback is determined by the formula z ¼ z0 þ ðx  x0 Þctg h þ

a ðx  x0 Þ2 : 4 sin2 h

The validity of the approaches to the solution of Tasks 2, 3, based on the small variations of the sound velocity, is confirmed by extensive measurement data. Task 4 Calculate the phase charge arising from the propagation of a beam emitted by a monochromatic frequency source x under the conditions of Task 3. Present the result in two ways: as a function of the vertical coordinate z of the beam point and as a function of the horizontal coordinate x. The solution. In order to calculate the phase raids on the movement of the sound beam, it is necessary to take into account the length of the curvilinear path passed by the beam and the distribution of phase velocity along this path. For this purpose, 2 we will use the eikonal equation ðw0 t Þ c2 ðrwÞ2 ¼ 0, which, when passing into the coordinate system associated with the beam itself, looks like d wðzÞ x ¼ d lðzÞ cðzÞ where d lðzÞ is the differential small shift along the beam near the point of its trajectory with coordinates ðx; zÞ (the dependence is only determined by the presence of the connection between the horizontal and vertical coordinates of the point on the beam). Since the arc differential is determined by the ratio s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d x cðz0 Þ dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d l ¼ d x2 þ d z2 ¼ þ1dz ¼ 2 dz cðzÞ c ðz0 Þ  sin2 h c2 ðzÞ

then d wðzÞ ¼

x c ðz0 Þ dz x c2 ðz0 Þ dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ 2 2 ðzÞ 2 c ðz0 Þ c2 ðz0 Þ c ðzÞ c ðz Þ c 2 0  sin h  sin2 h c2 ðzÞ

¼

c2 ðzÞ

x a ðz  z0 Þ þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d z c ðz0 Þ a ðz  z0 Þ þ cos2 h

Thus, the phase shift in the beam path is determined by the integral

158

Methodical Instructions and Tasks

wðzÞ ¼

x c ðz0 Þ

Zz z0

a ðn  z0 Þ þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d n a ðn  z0 Þ þ cos2 h

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2x ðz  z0 Þ a ðz  z0 Þ þ cos2 h ¼ 3 c ðz0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2x þ ð1 þ 2 sin2 hÞ a ðz  z0 Þ þ cos2 h  cos h : 3 a c ðz0 Þ Using the link between the horizontal and vertical coordinates of the beam point (see Task 3) allows you to get a representation of the phase attack as a function of the horizontal coordinate: ! x x  x0 aðx  x0 Þ aðx  x0 Þ2 ctgh þ wðx) = 1þ : cðz0 Þ sin h 2 12 sin2 h Task 5 Taking as a basis the condition of Task 3, calculate the value of the sound beam attenuation by setting the model of spherical divergence of sound in the environment. The solution. To solve this problem, we need to calculate the length of the curvilinear path L traversed by the beam. Then, the weakening is equal 1=L. The length of the path is determined by the ratio Zz LðzÞ ¼

Zz lðnÞ d n ¼

z0

Zz ¼ z0

z0

cðz0 Þ dn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðnÞ c2 ðz0 Þ  sin2 h c2 ðnÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ðn  z0 Þ þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d n: a ðn  z0 Þ þ cos2 h

In order to integrate this ratio, a variable y is entered, such that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ðn  z0 Þ þ 1 y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : a ðn  z0 Þ þ cos2 h 2

2

2

h1 2y sin h In this case, n ¼ z0 þ y acos ð1y2 Þ and accordingly d n ¼  a ð1y2 Þ dy. Substitution of these substitutions in the integral determines the length of the path by

Methodical Instructions and Tasks

Zy þ

2 sin2 h LðzÞ ¼  a sin2 h ¼ 2a

y Zy þ

y

Here, y ¼ cos1 h, y þ ¼

159

y2 d y ðy2  1Þ2 ! 1 1 1 1 d y:  þ þ y  1 y þ 1 ðy  1Þ2 ðy þ 1Þ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ðzz0 Þ þ 1 a ðzz0 Þ þ cos2 h.

Elementary calculation of tabular integrals leads to the final result: LðzÞ ¼

 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða ðz  z0 Þ þ 1Þða ðz  z0 Þ þ cos2 hÞ  cos h a sin2 h ðy þ  1Þð1 þ cos hÞ ln :  2a ðy þ þ 1Þð1  cos hÞ

Task 6 Determine the maximum radius of curvature of the beam for the given distribution cðzÞ. How does the maximum radius of curvature of the beam relate to the turning point? The solution. The radius of curvature of the beam trajectory is determined by the formula R1 ¼ 1c ðn  rcÞ, where n—the unit vector is normal to the trajectory. In order to specify this expression, it is necessary to define the vector n. If the trajectory is set by an expression f ðx; zÞ ¼ 0, then n ¼ rf =jrf j. As it was shown in Task 2, the coordinates of the beam point are connected by Zz

sin h dn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : c2 ðz0 Þ 2  sin h 2 c ðnÞ

x ¼ x0 þ zl0

Thus, the trajectory equation looks like: Zz f ðx; zÞ ¼ x  x0  z0

sin h d n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0: c2 ðz0 Þ 2  sin h c2 ðnÞ

Since r f ¼ fx0 ex þ fz0 ez and r c ðzÞ ¼ c0z ez , the radius of curvature in this case is determined by the expression 1 1 1 ðfx0 ex þ fz0 ez Þ  c0z ez pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ðn  rcÞ ¼  02 02 R cðzÞ cðzÞ fx þfz ¼

fz0 c0z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: cðzÞ f 02x þ f 02z

160

Methodical Instructions and Tasks

It is not hard to calculate the necessary derivatives: fx0 ¼ 1;

sin h fz0 ¼  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : c2 ðz0 Þ 2 c2 ðzÞ  sin h

As fx02 þ fz02 ¼ 1 þ

sin2 h c2 ðz0 Þ c2 ðzÞ

 sin2 h

¼ c2 ðz Þ 0

c2 ðzÞ

c2 ðz0 Þ ; 2  sin2 h c ðzÞ 1

we come to the final result: 1 sin h c0z ¼ : R c ðz0 Þ h Since the value csin ðz0 Þ is a ray invariant, the maximum radius of curvature of the beam is reached at the minimum of the derivative c0z . So we do:

Rmax ¼

c ðz0 Þ : sin h ðc0z Þmin

In the case when the beam is radiated vertically upwards (h ¼ 0), the beam trajectory is a straight line with R ¼ 1 regardless of the type of function c ðzÞ and the radius of curvature at the same time is the maximum value of all possible. Another possible case of the radius of curvature turning to infinity is achieved at the point where c0z ¼ 0 this is the point of local homogeneity of the medium and the beam in the vicinity of this point also goes in a straight line. In general, the maximum radius of curvature is not related to the turning point. If it is known that the maximum radius of curvature is reached exactly at the turning point, it imposes certain conditions on the function cðzÞ. Indeed, at the turning point, the condition is fulfilled c ðz0 Þ ¼ cðzÞ sin h This is a consequence of the equation for the ray invariant. Thus, in the vicinity of the turning point R ¼ ccðzÞ 0 . The presence of an extreme radius of curvature at the z

turning point leads to the requirement that the distribution of sound velocity in its vicinity is determined by the expression cðzÞ ¼ c0 expðaz þ bÞ where c0 , a and b—some of them are permanent.

Methodical Instructions and Tasks

161

Tasks to Chapters 7, 8 Task 1 On the plane boundary between water (density qw , sound velocity c) and elastic body (density q, longitudinal and transverse velocities are cjj , c? , respectively), a plane monochromatic wave of unit amplitude-frequency x falls (from water) at an angle h. Determine the amplitude of the Stoneley wave excited at the interface. The solution. For the existence of the Stoneley wave, the following conditions must be met k[

x ; c

k[

x ; c?

k[

x cjj

where k—the wave number is the wave number of this wave. The method of the Stoneley wave excitation described in the problem is characterized by the fact that the wave vector k must coincide with the projection on the boundary of the wave vector interface of the incident plane wave. k ¼ xc sin h. Thus, it means that the first of the above conditions is never met. Therefore, Stoneley’s wave cannot be excited in this way, i.e., its amplitude is zero. Task 2 Determine all possible structures of physical fields in the water and seabed if a plane monochromatic wave falls from the water to the plane horizontal bottom at an angle h to the vertical. The speed of sound in the water—c1 . The density is q1 . The bottom is a plane horizontal layer of thickness H and density q, with the velocities of longitudinal and transverse waves cjj and c? , accordingly (cjj [ c? ), which is located on a semi-infinite “liquid” substrate characterized by the velocity of sound c2 and density q2 . Is it possible in this production to excite the Stoneley wave? If possible, indicate on which surface and under which conditions it is excited. The solution. To describe all physical fields, it is enough to know the potentials (scalar and vector) in all areas of the task. In the first area (above the plate) everything is described by one scalar potential  u1 ¼ eia z þ Aeia z eiðkxx tÞ ;

k¼

x sin h; c1

a¼

x cos h c1

where A is the amplitude of the reflected wave and the amplitude of the incident wave is assumed to be equal to one. One scalar potential is also sufficient in the area below the plate, the general view of which is presented below: u2 ¼ Feij z eiðkxx tÞ Substitution of this expression in the wave equation the definition of the wave number j:

@ 2 u2 @ t2

 c22 Du2 ¼ 0 leads to

162

Methodical Instructions and Tasks

j2 ¼

  x2 x2 c21 2 2  k ¼  sin h c22 c21 c22

If cc12 [ sin h, the value is purely valid, but if cc12 \ sin h—it is purely imaginary. Thus, there are two options for the structure u2 : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 c1 ij z iðkxx tÞ x I. c2 [ sin h; u2 ¼ Fe e ; j ¼  c1 c12  sin2 h; 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 II. cc12 \ sin h; u2 ¼ Fej z eiðkxx tÞ ; j ¼ cx1 sin2 h  c12 ; 2

Two potentials must be described in the plate: scalar u and vector w. Due to the fact that the problem is two-dimensional (there is no dependence on the coordinate y), we will present the vector potential in the form w ¼ w ey , so it is necessary to describe the structure of only two scalar functions u and w. The general view of these functions is shown below:   u ¼ B þ eib z þ B eib z eiðkxx tÞ ; w ¼ D þ eic z þ D eic z eiðkxx tÞ Substitution u and w in the wave equations @2u  c2jj Du ¼ 0; @ t2

@2w  c2? Dw ¼ 0 @ t2

leads to ratios x2 x2 b ¼ 2  k2 ¼ 2 cjj c1 2

c21  sin2 h c2jj

!

  x2 x2 c21 2 2 c ¼ 2 k ¼ 2  sin h c? c1 c2? 2

Depending on the relationship between c1 , cjj , c? , there are three cases: 1. c1 [ cjj [ c? c1 [ cjj [ c?  u ¼ B þ eib z þ B eib z eiðkxx tÞ ;  w ¼ D þ eic z þ D eic z eiðkxx tÞ ; 2. cjj [ c1 [ c?  u ¼ B þ eb z þ B eb z eiðkxx tÞ ;  w ¼ D þ eic z þ D eic z eiðkxx tÞ ;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c21  sin2 h c2jj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x c21 c¼  sin2 h c1 c2?

x b¼ c1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 sin2 h  21 cjj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x c21 c¼  sin2 h c1 c2?

x b¼ c1

Methodical Instructions and Tasks

163

3. cjj [ c? [ c1  u ¼ B þ eb z þ B eb z eiðkxx tÞ ; w ¼ ðD þ ec z þ D ec z Þ eiðkxx tÞ ;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 sin2 h  21 cjj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x c2 c¼ sin2 h  21 c1 c?

x b¼ c1

Thus, there are 2
3 ¼ 6 possible situations that are described by a combination of one of the two field structures below the plate and one of the three field structures in the plate. The Stoneley wave is a surface wave, the characteristics of which decrease exponentially at a distance from the boundary along which it propagates. It is clear that such a wave cannot exist near the upper boundary of the plate (z ¼ 0), because the field u1 does not have the required properties. However, it can exist near the lower boundary (z ¼ H) because the set of solutions (II, 3) allows the existence of such a wave. Task 3 In the formulation of the previous problem to consider the exact and approximate existence of one wave of Stoneley in the region z\0. The solution. As it was shown in the solution of the previous problem, the Stoneley wave appears in the variant (II, 3). The condition of the existence of this wave alone at z\0 leads to the necessary requirement B þ ¼ D þ ¼ 0. Thus, the required structure of the solution should look like  u1 ¼ eia z þ Aeia z eiðkxx tÞ ; z [ 0 u ¼ B eb z eiðkxx tÞ ; u2 ¼ Fej z eiðkxx tÞ ; 2

2

w ¼ D ec z eiðkxx tÞ ;

0[z[  H

H [ z

2ikbB þ ðk þ c ÞD ¼ 0;

2ikbB ebH þ ðk2 þ c2 ÞD ecH ¼ 0

Substitution of the obtained expressions in boundary conditions on both boundaries leads to a system of equations defining the sought amplitudes. Without giving the system of equations in full form, we will write down only the conditions of absence of tangential stresses at the plate boundaries (the two last equations). The condition of non-trivial solvability of these equations with respect to amplitudes B and D is expððc  bÞHÞ ¼ 1: Because b 6¼ c, this condition can never be precisely fulfilled, which means that the plate contains waves of another type (described by terms with amplitudes B þ and D þ ) other than the Stoneley wave. However, this condition can be approximately satisfied if j c  b jH  1.

164

Methodical Instructions and Tasks

As
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x

c2 c2

jc  bjH ¼ H
sin2 h  21  sin2 h  21
c1
cjj c?
then the requirement j c  b jH  1 means that the ratio cx1 H  1 is fulfilled. Thus, in order to excite at least one Stoneley wave in the region z\0, it is necessary that the length of the falling plane wave significantly exceeded the thickness of the plate.

Bibliography

1. Egorov NI (1974) Physical oceanography. Gidrometeoizdat, 455 p 2. Egorov NI (1978) In: Doronin YP (ed) Ocean physics, 296 p 3. Kistovich AV, Pokazeev KV (2006) Introduction to the acoustics of the ocean. Tutorial. LLC MAX Press, 136 p 4. Anisimova EP, Pokazeev KV (2002) Introduction to the physics of the hydrosphere. Physical Faculty of Moscow State University, 171 p 5. Trukhin VI, Pokazeev KV, Kunitsyn VE, Shreyder AA (2002) Fundamentals of environmental geophysics. Lan Publishing House, 384 p 6. http://www.calc.ru/617.html 7. Rytov SM, Kravtsov YA, Tatarsky VI (1978) Introduction to statistical radiophysics. Part II. Random fields. Science, 464 p

© Springer Nature Switzerland AG 2020 A. Kistovich et al., Ocean Acoustics, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-030-35884-6

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