The Method of the Generalised Eikonal: New Approaches in the Diffraction Theory 9783110311129, 9783110311297, 9783110383010, 9783110311303

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Michael V. Vesnik The Method of the Generalized Eikonal

De Gruyter Studies in Mathematical Physics

| Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 29

Michael V. Vesnik

The Method of the Generalized Eikonal | New Approaches in the Diffraction Theory

Physics and Astronomy Classification Scheme 2010 Primary: 42.25.Fx; Secondary: 42.25.Gy, 02.60.Lj, 02.60.Pn Author Dr. Michael V. Vesnik Russian Academy of Sciences Kotel’nikov Institute of Radio Engineering and Electronics of RAS Mokhovaya 11–7 125009 Moscow, Russia [email protected]

ISBN 978-3-11-031112-9 e-ISBN (PDF) 978-3-11-031129-7 e-ISBN (EPUB) 978-3-11-038301-0 Set-ISBN 978-3-11-031130-3 ISSN 2194-3532 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin, Protago-TEX-Production GmbH, Berlin Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

| Dedicated to my family

Preface This book was inspired by the awareness that achievements of modern diffraction theory are not used to full extent for the solution of applied problems, the main reason being that only few simple and practical analytical solutions are available. While not many rigorous analytical solutions have been obtained so far, new ones are becoming increasingly complex in their form and efforts required. For example, more than a hundred years passed between the diffraction solutions for a half-plane and an angular sector were obtained. New analytical solutions are becoming available from time to time, however, they can be very cumbersome and require high qualification of researchers in order to be used for practical purposes. Recent explosive development of computational hardware and numerical methods entailed many new numerical solutions to diffraction problems. Nevertheless, obtaining and reproducing numerical solutions can also require availability of qualified personnel, as well as modern computational platforms and software, which can be limited and need to be allocated between many problems. In view of the above reasons, direct application of available rigorous solutions to practical problems encounters significant difficulties. To overcome these difficulties, heuristic solutions based on the physical concepts, though not rigorous from the mathematical standpoint, can be very helpful. Application of heuristic approaches allows one to derive analytical formulas which are simple, yet accurate enough. Another way in which heuristic formulas can be used is to interpret the behavior of rigorous solutions. We overlay the rigorous solution by a “stencil” of our physical concepts of field behavior, and then find out their similarities and differences. Applied problems are sometimes solved by “engineering” formulas which just imitate the rigorous solution and are based, for example, on spline interpolations. As compared to the engineering formulas, heuristic formulas bear more physical meaning because they are based on fundamental physical principles. Among the latter ones are, for example, the principle of field locality, Babinet principle, etc. The following analogy is appropriate. Heuristic solutions are based on the general physical concepts of diffraction process. Therefore, the “stencil” has a simple geometrical shape (disk, triangle, square). The real solution can be thought of as a “patch” of a complex irregular shape. We place the stencils available in hand over the patch. If the agreement is good, we can use our stencil instead of the real patch when solving applied problems. In doing so, we assume that physics of the real process corresponds to the phenomena available in the stencil. If, however, the agreement is poor, we can either think of creating a new stencil, or adjust (for example, by engineering formulas) the shape of one of existing stencils to the shape of the patch; however, in this case we can use the adjusted stencil only for the solution of this particular problem in the future.

VIII | Preface

Approximate three-dimensional solutions are often obtained by the method of physical optics (PO) which allows one to derive simple and universal (and, therefore, widely used) formulas, despite the fact that their accuracy is insufficient. If necessary, PO solution can be refined by the method of edge waves (MEW); to this end, a rigorous two-dimensional solution must be first obtained, then it is compared with the physical optics solution in the two-dimensional case, after which a heuristic threedimensional solution can be constructed by adding appropriate terms to the physical optics solution. Application of MEW can often be very efficient. It is known the in the Stealth project, MEW allowed the funds allocated to calculations to be cut by about 10 billion dollars. In this book, refinement of PO solutions is achieved by additional multipliers, rather than additive terms. These multipliers are also obtained from the comparison of rigorous and physical optics solutions in the case of a perfectly conducting scatterer, after which they are extended to scatterers with other types of boundary conditions. In this way, we do not have to find out two-dimensional solutions for each type of boundary conditions. We are searching for the solutions to more complex problems in the class of functions pertaining to solutions of simpler problems. In the majority of cases, the accuracy of our approaches corresponds to that of MEW, or exceeds it. It is of no question that derivation of new rigorous solutions is a very important task. But it is equally important that, having some solutions in hand, we could apply them to practical problems in the most efficient way. An analogy seems relevant: a fisherman knows how to catch a fish, a chef knows how to cook it, however, what is important for a restaurant client is that both specialists did their job in the best possible way. Similarly, in this book we do not obtain new rigorous solutions, rather, we offer the ways in which known analytical formulas can be applied to the solution of new problems. At the initial stage, heuristic formulas represent a hypothesis on field behavior. In order to obtain a heuristic solution, one has to perform its comprehensive verification (checking and “fine-tuning”) against a rigorous solution. Only after that we can conclude that the hypothesis became an heuristic solution verified within its established applicability limits. In support of the approach proposed and developed in this book, we quote two prominent scientists: “Everything should be done as simple as possible but not any simpler” — A. Einstein “What is important, is not what is rigorous but what is correct” — A. N. Kolmogorov

Author would like to express his deep gratitude to P. Ya. Ufimtsev for the academic school and help in work; to Yu. A. Kravtsov, B. G. Kutuza, V. F. Kravchenko, and V. V. Shevchenko for their interest and support; to B. Z. Katsenelenbaum and V. A. Borovikov for useful discussions.

Contents Preface | VII Introduction | 1 1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.2 1.2.1 1.2.2

2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 3 3.1 3.1.1

Method of Generalized Eikonal | 15 Integral representation of solution | 15 Statement of the problem | 15 Construction of “auxiliary” domain and generalized geometrical optics function | 16 Boundary conditions | 20 Features of the solution | 23 Asymptotic calculation of contour integrals by method of stationary phase | 24 General solution | 25 Solution of diffraction problem for plane and cylindrical waves by the method of generalized eikonal | 27 Solution of Two-dimensional Problems by the Method of Generalized Eikonal | 35 Introduction | 35 Diffraction by half-plate | 36 Solution on the given curve r d0 | 40 Power normalization | 42 Solution by method of successive diffractions (MSD) | 46 Results of calculations | 48 Diffraction by a truncated wedge | 56 Schwarz–Christoffel integral | 59 Features of solution for half-plate | 60 Solution by method of successive diffractions | 62 Principles for the construction of heuristic solutions for diffraction by truncated wedge | 63 Solution with generalized Fresnel integral | 64 Numerical results | 67 Analysis of solutions | 67 Application of Two-dimensional Solutions to Three-dimensional Problems | 73 Integrals over elementary strips | 73 Statement of the diffraction problem | 73

X | Contents

3.1.2 3.1.3 3.1.4 3.1.5 3.2 3.2.1 3.2.2 3.2.3 3.2.4 4 4.1 4.2 4.2.1 4.2.2 4.3 4.3.1 4.3.2 4.4 4.5 4.5.1 4.5.2 5 5.1 5.2 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6 5.3.7

Infinite cylinder | 75 Far zone condition | 76 Fragment of cylindrical surface | 78 Polygonal edge | 79 Application of two-dimensional solutions to three-dimensional problems | 81 Physical optics solution for diffraction by a plane scatterer. Properties of contour integral | 81 Rigorous 3D formulas | 82 Comparison with 2D case | 84 Total current diffraction coefficients | 84 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach) | 85 Statement of the problem | 85 Solution in physical optics approximation | 87 Contour integral with enforced far zone condition | 89 Inputs of edges and vertices | 90 Solution in EECM approximation | 96 Rigorous solution for oblique incident wave | 96 Substitution of polarization components of diffraction coefficients | 98 Modified EECM | 101 Applicability limits of heuristic approaches | 106 Solution algorithm | 106 Applicability limits of heuristic solutions | 108 Propagation of Radio Waves in Urban Environment (Deterministic Approach) | 109 Relevance of the problem | 109 Specifics of radio wave propagation in urban environment | 111 Design formulas | 112 Zone significant for radio wave propagation | 112 Reference solutions | 114 Mutual coupling between two antennas | 115 Energy relationships | 117 Fresnel zone | 119 Derivation of heuristic formulas | 121 Solution algorithm | 121

Contents | XI

6 6.1 6.2 6.3 6.4

Analytical Heuristic Solution for Wave Diffraction by a Plane Polygonal Scatterer | 125 Introduction | 125 Problem formulation for elastic wave diffraction | 126 Approach to derivation of formulas | 130 General form of the solution | 134

7

Conclusion | 135

A A.1 A.2 A.3 A.4 A.5

Application of Stokes Theorem to Diffraction Problems | 139 Stokes theorem. Relationship between the surface and contour integrals | 139 Integral over the surface of a finite-size polygon | 141 Integral over the surface of a plane angular sector | 143 Vertex waves for a finite-size polygon | 144 Phase function and far zone condition | 147

B

Rigorous Two-dimensional Solution for Diffraction by Half-plane | 149

C

Application of Imaginary Edge in Diffraction Problems | 157

D

Summary of Formulas for Diffraction by Plane Angular Sector | 163

E

Fresnel Integral and its Properties | 169

F

Generalized Fresnel Integral and Its Properties | 173

G

Electromagnetic Wave Diffraction by Semi-transparent Plate | 177

H

Generalized Diffraction Coefficient and its Application to Diffraction Problems | 181

Bibliography | 193 Index | 199

Introduction Relevance of the problem Analytical solutions of diffraction theory find applications in numerous scientific and engineering problems. In many cases, though, rigorous analytical solutions cannot be used, either because of their absence, or extreme complexity. Also, in some cases (for example, in the inverse problems), it is desirable to use the simplest and fastest solutions, while rigorous solutions often do not conform these requirements. Another problem is that rigorous solutions for diffraction by large scatterers are not feasible due to limited computing power, even in the cases where it is possible to formulate the problem for numerical analysis, and such a solution, if obtained, would satisfy the researchers. Thus, despite the current rapid development of computational hardware and progress in the numerical methods, obtaining new heuristic solutions remains a topical issue. Results of this book can be applied to studying objects of various shapes, although the main attention is paid to diffraction by polygons and polyhedrons. Diffraction by polygons and polyhedrons has been studied extensively in many works. World renowned monographs (see, e. g., [1]) which played an important role in the research on this topic have been published, as well as many specialized monographs on the topic (see, e. g., [2]). Nevertheless, there exists incessant research interest to obtaining analytical solutions for diffraction by such objects, and the studies are far from being completed. In addition to the above-mentioned practical applications, there are many reasons explaining why these studies are still being continued quite actively. One of these reasons is that studying such objects, one can reveal clearly the characteristic features of diffraction phenomena. Another reason is that analytical formulas for such objects are rather cumbersome and difficult to derive.

Approaches to diffraction problems A number of approaches exist by which diffraction problems can be tackled. Numerical methods. It is known that the highest accuracy of diffraction solutions can be achieved by numerical methods. However, in order to apply such methods, corresponding software must be available. Also, even the modern hardware can prove insufficient for the solution of some important applied problems (for example, calculation of the effective scattering surface of large objects, or electromagnetic wave scattering in urban area). Additionally, numerical simulations are, in fact, a sort of computational experiment: they provide a solution as a whole, without its separation

2 | Introduction

into the inputs of individual geometrical elements of which the scatterer consists, thus making the interpretation and analysis of the results obtained rather difficult. Mathematical formulation of the problem includes [3]: i) wave equation; ii) initial, and iii) boundary conditions. Uniqueness of the solution also requires iv) condition at infinity, and v) condition on the edge. By the numerical methods implemented in corresponding software, solutions can be computed that satisfy all the boundary-value problem conditions. Due to the rapid development of numerical methods and computer hardware, numerical solutions to various problems have been obtained, providing the basis for formulation of reference solutions which can be then used for derivation and verification of heuristic formulas. Rigorous analytical approaches. Analytical solution obtained rigorously are very valuable because they allow one to check approximate analytical formulas, as well as numerical programs, both potentially suffering from inaccuracies or errors. Unfortunately, rigorous analytical formulas exist for a limited number of scatterers with simple enough geometry only. Asymptotic solutions are often represented in a rather cumbersome form, for example, as infinite series of special functions, and clear interpretation of their results is as difficult as that of numerical solutions. In many cases, for example, when it is required to solve an inverse problem more efficiently, availability of a reliable numerical or asymptotic solution does not make development of heuristic formulas unnecessary. Rigorous analytical methods are usually based on the method of separation of variables. The coordinate system in which this method is applied depends on the problem geometry, under which we understand the shape of the scatterer, as well as relative positions of the source and the observation point with respect to the scatterer and to each other. Solution is obtained in quadratures of in explicit form, including infinite series of special functions. The first rigorous analytical solutions were obtained quite a long time ago [4–12], mainly for two-dimensional problems of diffraction by half-planes and wedges. Further progress in obtaining rigorous analytical solutions was achieved by application of the Wiener–Hopf method [13, 14] and studies of objects with geometry chosen to match some special coordinate systems [15]. As was already mentioned above, rigorous analytical solutions are typically very complex due to the complexity of coordinate systems for angular and polyhedral domains, as well as to the complexity of special functions forming an orthogonal system in the given coordinates. Even for a simple key problem (plane angular sector), one has to use the sphero-conal coordinate system. Therefore, the number of rigorous solutions obtained is by far smaller than the number of approximate ones, and derivation of new such solutions takes much longer time. As an illustration, remind how much time passed between the two-dimensional solution for diffraction by a perfectly conducting half-plane, 1894 and 1896 [4, 5], and a solution for diffraction by a perfectly

Introduction

|

3

conducting angular sector, 2003 [16], were obtained. As we see, it took more than 100 (!) years, despite the fact that all this period was featured by incessant interest to the problem, and a number of theses were written (see, e. g., [17–19]. This complication of the problem was caused by adding just one kink on the straight edge in the problem of diffraction by a half-plane. After that, the twodimensional problem turned into a three-dimensional one. Another complication of the problem is possible when a kink on the generatrix of a 2D wedge transforms it into a pair of wedges. In this case, the problem remains 2D, but a size parameter (distance between the wedges) appears. In practical applications, a combination of the two above-mentioned geometry changes is possible (three-dimensionality plus appearance of a size parameter). Addition of boundary conditions further complicates the finding rigorous analytical formulas. Diffraction by a pair of wedges was considered in numerous works [20–28]. When solving some scientific or engineering problem, it would often be advantageous to apply an analytical formula which, however, is unavailable because its derivation encounters significant difficulties. Alternatively, the analytical formula may exist, but cannot be applied to the particular situation. This book offers an approach towards derivation of heuristic analytical formulas of diffraction theory, including the problems where rigorous analytical solutions are absent. Obtaining rigorous solutions takes time (sometimes, decades) and requires significant efforts. And even if these long-time efforts prove successful (albeit, the opposite happens too), the resulting formulas can be very complex, cumbersome, not amenable to necessary analytical transformations, and their physical sense can be rather obscure. That is why analytical formulas are often derived by so-called “heuristic” approaches. Heuristic Methods. These solutions are comprised of a set of analytical formulas and algorithms for their application. They are used when there is no rigorous analytical solution, or it is unreasonable to apply it, for example because of high computational demands or difficulty in the interpretation of the results obtained. As a rule, a heuristic approach is not rigorously justified but its appropriateness is verified by comparison with numerical results, known asymptotic solutions in the limit cases, or by using other techniques (for example, by comparison with numerical or experimental data). On the other hand, comparison with known results makes it possible to refine the heuristic solution. There are a number of reasons for which heuristic solutions have to be applied. 1. Rigorous solutions in the form of slowly convergent series of special functions can be analyzed properly only by experts in the field. Simpler and more reliable formulas are more suitable for engineering applications. 2. Rigorous analytical and numerical methods provide a solution to a model problem for an object as a whole, rather than make it possible to understand how individual elements of the object influence the scattered signal.

4 | Introduction

3.

Rigorous analytical solutions become useless when the object geometry changes (for example, when a half-plane or an angular sector become of some small, but finite thickness, or the wedge is truncated, or diffraction by an object with nonideal boundary conditions is of interest).

Various methods can be applied to construct heuristic solutions to the problem of scattering by complex objects. Numerous papers and monographs are devoted to this issue. Geometrical theory of diffraction (GTD) [41, 42] is applied to scattering problem in many cases but it turned out to be unsuitable for calculation of the radar cross section (RCS) of stealth objects, i. e., objects of a shape formed by polyhedron fragments. It has been found out that physical theory of diffraction (PTD) [1, 44, 45] is suitable for the analysis of such objects. The incremental length diffraction coefficients (ILDC) and the equivalent edge current method (EECM) were developed on the basis of PTD [46–49]. There also exist other heuristic methods for the analysis of diffraction problems [20, 29, 34, 50, 52, 53, 87]. We mentioned only few papers referenced in this book, this list is in no way exhaustive. Comprehensive bibliography might include dozens of books and theses, hundreds of papers and reports. Nevertheless, new works on the topic keep appearing, motivated by the necessity of obtaining simpler and more accurate solutions. Heuristic methods make it possible to solve problems that are ill-posed from the mathematical viewpoint. For example, physicists often employ soft input (i. e., nonreflecting) surfaces. Such a surface cannot be described in terms of the theory of an interface between two media from classic electromagnetic theory. Nevertheless, composite materials applied in engineering can provide for soft-input properties better than any combination of parameters on an interface between two planar media in the classic theory. Such solutions cannot be found with the help of rigorous methods, because there is no rigorous formulation of the problem. Heuristic methods make it possible to find a solution when the formulation of a problem is not rigorous. The variety of heuristic methods enables the researcher to choose the method that is most suitable for the solution of a problem and that is not necessarily the most rigorous from the mathematical viewpoint. It is necessary to say few words on the terminology adopted in the book. For the sake of simplicity, we will use the term “a physical optics object” rather than “an object described in the physical optics approximation”. When considering objects that are used in 3D solutions and that are obtained from 2D solutions, we will use the term “rigorous”, though there is no mathematical rigor, because heuristic solutions are dealt with. The coordinate frame and calculation formulas are borrowed from various works [1, 3, 16, 30, 32, 37]. To facilitate the comparison of new formulas with the results from the above-mentioned original studies, the notation from this paper resembles that from these studies. However, in the aforementioned studies, different types of notation and different ways of measuring the wave incidence angle relative to the inward

Introduction

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5

normal to the contour bounding the scatterer are used. When necessary, matching of notations from different studies is given. Heuristic approaches are based on the physical concept of solution structure [3, 29]. Instead of rigorous mathematical formulation of the problem, heuristic methods use such properties of the field as the principle of field locality in penumbra, reciprocity principle, etc. Sometimes heuristic approaches rely on postulates which are known to decrease the accuracy, but simplify significantly the way in which solution is derived and the resulting formulas. Heuristic solutions (obtained by physics-based view of the problem and intuition) do not have rigorous mathematical validation, but allow for clear interpretation of results, since the solution is built on physical concepts from the very beginning. Comparison with more accurate solutions allows one to evaluate the accuracy of a heuristic approach and its “fine-tuning”. Knowing the solution behavior for an object of one shape, it is possible to make a hypothesis on its behavior for another object of a similar shape, after which to check the hypothesis against a more rigorous solution. Heuristic approaches allow one to derive analytical solutions and deal with formulas which cannot be obtained by rigorous analytical approaches. Heuristic formulas can provide accuracy approaching to that of rigorous solutions. There also exist engineering methods which imitate the rigorous solution by, e. g., phenomenological numerical coefficients and approximating formulas. Unlike the engineering formulas, heuristic formulas not simply imitate the solution, but they correspond to physics of the problem. If the essence of processes is captured adequately, a very successful heuristic approximation can be obtained which will be applicable, with minimum variations, to a wide range of problems of electrodynamic theory, as well as in other fields of physics.

Pros and cons of various approaches The main feature of any solution is its accuracy. Among other important features, we can mention mathematical rigor, simplicity of derivation (including necessity in specialized software), analytical form of the solution, simple formulas, low computational demands, physical clarity. Physically clear solutions are those in which influence of each particular parameter can be established. From this point of view, the most physically clear solutions are those which are expressed by simple formulas, with each input parameter appearing once, or a minimum number of times. Numerical methods allow one to obtain solutions of desired accuracy for any problems. If the problem dimensionality is large, then, instead of solving the complete problem in rigorous formulation, one can obtain a hybrid solution combining numerical and heuristic approaches. The drawbacks of these methods are that computations are relatively slow, software packages must be developed or purchased, also, qualified

6 | Introduction

researchers capable of develop or run corresponding software on required hardware platforms and operating systems; these solutions, as a rule, lack physical clarity. Rigorous analytical approaches. Solutions in explicit form or in quadratures exist just for a limited number of simple geometry objects. As the geometrical complexity grows, obtaining such solutions becomes an increasingly tedious task, rigorous formulas become cumbersome and may involve some rarely encountered and poorly studied special functions. Also, rigorous analytical solutions take a relatively simple form for selected objects with simple geometry, or for objects with geometry chosen in such a way that it matches some special coordinate systems in which the solution can be obtained [15]. Even with slight changes in the shape of the scatterer, derivation of rigorous analytical solution must be performed over again. Heuristic approaches. Their main drawback is the lack of rigor, which makes it necessary to verify (validate) heuristic formulas against a more rigorous and reliable solution. This reference solution can be either rigorous analytical, or numerical. Possessing such practically important advantages as simplicity, low computational costs, physical clarity, heuristic solutions by no means render rigorous (and more accurate) approaches unnecessary. Rigorous solutions (if available, of course) can serve as reference problems, verification solutions, as well as they can provide the basis for derivation of heuristic solutions in the cases where more accurate numerical verification solutions are absent.

Key problems in heuristic approaches Heuristic solutions for diffraction by polygons or polyhedrons can be split clearly into components (key problems) describing the inputs of separate components of the scatterer, i. e., its faces, edges, and vertices. Diffraction by edges. This key problem considers 2D diffraction by an infinite-length edge. If the plane wave is incident on the edge not normally, but at some angle, scattering occurs only in some particular direction with respect to the edge [4–12, 30]. Diffraction by vertices. Here, diffraction by a plane angular sector, pyramid-shaped angle, or a cone is considered [16, 31]. Heuristic approaches also give the “vertex waves”, but such solutions are, as a rule, inaccurate and deviate from the rigorous solution. Non-ideal boundary conditions on the scatterer surface (see, for example, [31, 32]). These can be taken into account for 2D edges and vertices. At the moment, solution for electromagnetic wave diffraction by a plane angular sector has been obtained only

Introduction

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7

for ideal boundary conditions. The simplest solution taking into account the boundary conditions can be obtained for wave interaction with an infinite plane surface [32]. Classification of key solutions by their dimensionality-related complexity. The easiest to obtain are 1D solutions describing plane wave interaction with an infinite plane surface. Diffraction solutions for normal incidence on an edge can be classified as 2D. Upon oblique incidence, the solution becomes more complex, its dimensionality can be classified as “2.5D”. The most complex key solutions are pertinent to diffraction by vertices, their dimensionality can be classified as 3D.

Specifics of heuristic solution derivation Phase function. Solutions to diffraction problems are represented by highly oscillatory integrals with exponential functions in the integrands. The argument of this exponential function contains a geometry-dependent phase function. In turn, behavior of the diffraction solution depends significantly on the phase function. The case where the phase function is constant corresponds to groups of points of stationary phase. The case where the phase function is linear corresponds to the far zone condition. Also, by the method of stationary phase it is possible to obtain an asymptotic estimate of the highly oscillatory integral [33]. Problem geometry, stationary phase point groups, and input of vertices. As was mentioned above, under problem geometry we understand the scatterer shape, as well as relative positions of the source and observation point with respect to the scatterer and to each other. A polygonal or polyhedral scatterer consists of plane sides, therefore, its shape is defined by the vertex coordinates which determine the normals to the sides, as well as inner normals to the edges. On a plane polygon or polyhedron, there exist groups of points which provide significant input into the scattered signal because the phase function of the signal at these points is constant. The first such group includes the edge points corresponding to the problem geometry in which the observation point is located in the mirror reflection direction, or in the “straight through” (i. e., straight forward) direction coinciding with the shadow direction in geometrical optics. This group of points gives maximum scattered signal only in these selected directions. The second such group include the edge points corresponding to the diffraction cones. A diffraction cone arises in such geometry where the direction towards the observation point makes the same angle with respect to the edge as the direction towards the source. The third group of points is comprised of vertices. They provide substantial input when inputs of the first two groups of points are absent. If the scatterer is a polygon or a polyhedron, it is the accuracy of the input of the third group of points that determines the overall solution quality.

8 | Introduction

Far zone condition. An important question arising in diffraction problems is whether the far zone condition is satisfied. If the scatterer is located far enough both from the source and from the observation point, it appears as a point source with tiny angular sizes. Then we can neglect the sphericity of wave fronts and consider them as plane. In this case, the signal phase function on the scatterer surface becomes linear. This approximation simplifies significantly the resulting formulas in a heuristic solution to the diffraction problem. It is not by chance that we discuss here heuristic solutions, because there exist no rigorous diffraction solutions for finite-size polygons and polyhedrons. In the key problems, solutions are obtained for semi-infinite objects for which the far zone condition is not satisfied. Indeed, whatever the distance to a semi-infinite object, its angular sizes will never tend to zero, and the scatterer itself will never look like a point. This circumstance entails a very important consequence. It is well known that the special function describing diffraction by half-plane is the Fresnel integral. Far away from the light–shadow boundary, one can apply a simpler expression, i. e., the asymptotic formula for the Fresnel integral which has a singularity at the light–shadow boundary. This singularity is regarded sometimes as a drawback of the solution, although in reality this is not true. If we consider a signal scattered by a finite-size body in the “mirror reflection” and “straight forward” directions, all signals arriving from individual edges will be singular. However, after the inputs of all edges are summed up, a correct result is obtained. It is interesting to note that the same singular formulas for diffraction coefficients can be obtained if the far zone condition is formally applied not to a finite-size scatterer, but to diffraction by a half-plane in the physical optics approximation, or to the rigorous solution. In exactly the same way, one can simplify the expressions if the far zone condition is applied to diffraction by a plane angular sector. Asymptotic solutions and multiple reflections. Asymptotic solutions obtained for semi-infinite objects, as a rule, do not take into account multiple reflections. Taking into account multiple reflections plays especially important role when one of the reflecting edges is located in shadow and, therefore, is not illuminated by the primary field (for example, in 2D problems). Complexity/accuracy ratio. This philosophical ratio characterizes attractiveness of a solution for researchers in the same way as the “cost-benefit” concept. It is not always that the most accurate approaches are the most efficient for the solution of some particular scientific or engineering problem. Sometimes, solution simplicity outweighs its accuracy. Moreover, in some particular cases simple solutions give not only accurate, but also rigorous results, rendering more complex approaches unreasonable.

Introduction |

9

Known heuristic approaches Method of Geometrical Optics (GO) [34]. In this approximation it is assumed that in the “illuminated” part of space the field is the same as in the absence of the scatterer, while in the shadow part of space the field is absent. This approximation gives correct solution only in two selected directions, the “mirror reflection” and “straight through” ones. Diffraction is not taken into account. Essentially, this method is not suitable for diffraction problems, but it provides basis for other methods, PO and PTD. It is from the light–shadow boundaries given by the geometrical optics solution that the angular distance entering PO, GTD, and PTD solutions is reckoned. Method of Physical Optics (PO) [3, 32, 35–37]. In this approximation the scattered field is expressed via a surface integral. The field being integrated is taken in the GO approximation, i. e., it is assumed that no field perturbation occurs near the boundaries, although, such an assumption never holds true in real problems. The field near the boundaries is always perturbed, however, this perturbation not always gives an input to the surface integral. Therefore, the physical optics approximation sometimes provides accurate enough solutions for some directions of the observation point, which makes it suitable for practical applications. If the surface field was taken rigorously, integration would give us an exact solution. However, the problem is that the rigorous field on the scatterer surface is not known beforehand. PO provides solution in the whole space, but this solution is accurate only in two selected directions (mirror and forward) and in their small neighborhood. PO solution uses the GO solution for an infinite plane surface. The essence of PO method is an integral over the scatterer surface. Field perturbations near edges and vertices are not taken into account. Method of physical optics is attractive because, being relatively simple, it guarantees that a solution corresponding to the physicals sense of the problem will be found. The predicted signal maximums are located properly, namely, on the diffraction cones. Some inaccuracies in the predicted amplitude and polarization of the scattered signal is considered acceptable because it is well known that rigorous solutions for diffraction by three-dimensional objects of complex shape are either not available, or their derivations is very involved for a number of reasons. In the simplest PO approximation one can obtain a coarse solution for any scatterer. In the case of electromagnetic wave diffraction by a half-plane, PO solution satisfies almost all boundary-value problem conditions, except the edge condition. The latter condition matches the solution to the given coordinate system. Simplicity can be the crucial decision factor in favor of some method in application to particular physical problems. That is why PO approximation is so popular. Nevertheless, it will be shown in this book that solutions obtained by physical optics method can be refined without significant complication of the approach and resulting

10 | Introduction

analytical formulas. These refinements are connected with taking into account edge, vertex, and boundary condition effects. Rigorous solutions are the most accurate, but also the most complex ones. One can say that all century-old history of modern diffraction theory was devoted to making a seemingly simple, but, as it turns out, a very tedious step forward, from the accuracy of PO solutions to that of rigorous solutions. Geometrical Theory of Diffraction (GTD) [38–42]. In this approximation, diffraction by a body of finite dimensions occurs exactly in the same way as in the key problem of an infinite scatterer. The method relies of the availability of rigorous solution for edge diffraction. It provides a solution only at the observation points corresponding to stationary phase condition, i. e., on the diffraction cones only. In the absence of points of stationary phase on the edges, classical formulation of this method needs modifications. Physical Theory of Diffraction (PTD), or Method of Edge Waves (MEW) [1, 44, 45]. There exist several versions of MEW, including the method of incremental length diffraction coefficients, equivalent edge current method, and others [46–49]. In the MEW approximation, the field on the edge of a finite-dimension polyhedron is assumed to be equal to that on the edge having the same shape but infinite length. The method relies on availability of a rigorous solution for edge diffraction. It provides a solution in the whole space, but this solution is accurate only on the diffraction cones and in their small neighborhood. The method takes into account field perturbation near the edges, but not near the vertices.

Objectives and research method The objective of this book is the development of new heuristic approaches to diffraction theory. Depending on the problem being considered, different approaches are taken. The method of generalized eikonal is based on the rigorous integral representation generalizing the Sommerfeld integral. This integral representation allows one to obtain solutions for scatterers possessing a size parameter (for example, half-plate thickness). Solutions for a half-plate and truncated wedge are obtained by a heuristic approach to the method of generalized eikonal. Solution for diffraction by a plane angular sector is obtained by the equivalent edge current method. Modification of this method by additional factors allows one to increase the accuracy of EECM solution and make it comparable with that of a rigorous solution. Account for non-ideal boundary conditions on the scatterer surface is performed by the method of generalized diffraction coefficients. This method is based on a heuris-

Introduction

|

11

tic approach in which the reflection and transmission coefficients for an infinite plane surface are multiplied by the elements of diffraction coefficients taken from the diffraction problem for a scatterer with ideal boundary conditions. Substantiation of this approach is carried out by physical interpretation of the solution obtained by the method of generalized eikonal. The target audience of this book includes several reader categories: – Specialists who apply the diffraction theory to various research and engineering problems and may find attractive the simplicity and universality of proposed approaches. – Students who begin diffraction theory studies and may find attractive the simplicity and clarity of the matter presented in the book. – Specialists from various fields of physics, who may be interested in the capabilities of the method of generalized eikonal, new analytical solutions to diffraction problems, and new approaches to the solution of known diffraction problems available in almost any chapter of the book.

Brief contents The book consists of six chapters, eight appendices, introduction and conclusions. In Chapter 1, the method of generalized eikonal is proposed which allows one to derive an integral representation of analytical solution to 2D diffraction problem for a perfectly conducting semi-infinite scatterer. In the auxiliary domain, representation of the generalized geometrical optics function is constructed with the aid of Cauchy residue theorem in the form of an integral over a closed contour. By the method of stationary phase, the integrals involved in this integral representation are calculated in the general form in the case of boundary-value problem for the Helmholtz equation. It is shown that for diffraction by a wedge, the newly obtained integral representation coincides with the known one. In Chapter 2, solutions are obtained by the new method for plane wave diffraction by two particular scatterers: a half-plate of finite thickness, and a truncated wedge. Also, approximate solution for the same types of scatterers is obtained by the method of successive diffractions. Comparison with these results and literature data confirmed good agreement of the results. In Chapter 3, application of two-dimensional solutions to three-dimensional problems is considered for normal and oblique wave incidence on a semi-infinite edge. As a rule, solutions to three-dimensional problems are presented as integrals over the scatterer contour located on the light–shadow boundary. In turn, solutions to two-dimensional problems are written as integrals over semi-infinite elementary integration strips. These strips begin on the light–shadow boundary of a two-dimensional semi-infinite scatterer and span to infinity. In Section 3.1, a method is proposed which allows one to apply two-dimensional solutions to three-dimensional problems with-

12 | Introduction

out performing integration over the elementary strips in the case where an analytical solution on the straight pieces of the generatrix of two-dimensional scatterer is available. In Section 3.2, solution to plane electromagnetic wave diffraction by a plane angular scatterer is analyzed. Elements of this solution play a key role in derivation of heuristic solutions to three-dimensional problems. The “edge” and “vertex” waves are considered. In Section 4.5, features of various heuristic approaches in application to three-dimensional diffraction problems are analyzed, and applicability limits of these approaches are established. In Chapter 4, a heuristic solution for diffraction by a plane perfectly conducting angular sector is offered under the far zone conditions. The solution is obtained in several steps. On the first step, solution to diffraction by a perfectly conducting angular sector is obtained in the physical optics approximation. On the second step, the polarization component of diffraction coefficient of two-dimensional edge is modified, resulting in the solution corresponding to equivalent edge current method (EECM). In doing so, we use the concept of “imaginary edge” which is a specially selected direction on the scatterer. Application of the imaginary edge allowed us to avoid integration over an elementary integration strip. On the third step, EECM solution is refined by the modifying factor chosen from physical considerations. As a result, we obtain a solution comparable in accuracy with the rigorous one, but of the same complexity level as the physical optics solution. In Chapter 5, the methods developed in the previous chapters are applied to the development of a deterministic approach to the problem of radio wave propagation in urban environment. In order to reduce the computational complexity for urban area objects, we limit our consideration to the principal radio wave propagation zone and corresponding first Fresnel zone. Only the object parts falling into this spatial domain are considered, and diffraction by these parts is calculated on the basis of approaches developed in the preceding chapters of the book. In Chapter 6, the methods developed in the previous chapters are applied to the problem of wave field (including elastic wave) diffraction by a plane polygonal scatterer (in particular, by an inhomogeneity in the wave propagation medium). A formula for generalized diffraction coefficient is obtained which takes into account non-ideal boundary conditions on the scatterer surface. To obtain the solution in time domain, Fourier transformation is applied to the diffraction coefficients derived in frequency domain. Comparison with a rigorous solution obtained by numerical methods confirmed efficiency of the proposed approaches. In the Appendices, formulas used in various chapters are summarized. This was done in order to make the text less overcharged by formulas and more readable.

Introduction

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13

Principal results In the book, results are obtained for various components of which heuristic solutions are constructed, namely: – Two-dimensional diffraction by infinite-length edges, – Approach to application of 2D solutions in 3D problems, – Diffraction by a vertex (plane angular sector), and – Non-ideal boundary conditions on the scatterer surface. Also, predictive capabilities of the approach developed in the book are demonstrated on several particular problems: – Diffraction by scatterers with a size parameter: half-plate of finite thickness, truncated wedge, – Radio wave propagation in urban environment, – Diffraction of an elastic wave by a polygonal inhomogeneity in elastic propagation medium. The heuristic approaches developed are suitable for the solution for various diffraction problems, including diffraction of waves of different nature by 3D scatterers of complex shape with non-ideal boundary conditions and edges of complex shape. Results of this book can be used for solving the following applied problems: – Calculation of radar cross-section for aerial, ground and sea radar objects. – Signal scattering by urban area objects. – Indoor propagation of radio signals. – Electromagnetic wave diffraction by crystals. – Diffraction by the open end of rectangular waveguide. – Elementary particle diffraction by traps and other objects. – Diffraction of elastic waves by inhomogeneities in the medium (solids, rocks, etc.). A number of other practically important problems exist where diffraction of waves of various physical nature by polygons and polyhedrons is encountered.

1 Method of Generalized Eikonal 1.1 Integral representation of solution 1.1.1 Statement of the problem Consider an electromagnetic wave P of the form P = A exp (iS)

(1.1)

propagating in a 2D infinite space. In (1.1), P, A, and S depend on the coordinates; A is the wave amplitude; S is the eikonal function; and i is an imaginary unit. Assume that function P satisfies the wave equation Δ z P + k2 P = 0, (or Δ kz P + P = 0,

Δ kz

∂2 ∂2 + 2 2 ∂x ∂y ∂2 ∂2 = + ), 2 ∂ (kx) ∂ (ky)2

Δz =

(1.2)

where k = 2π/λ is the wave number and λ is the wave length. Let a 2D semi-infinite perfectly conducting scatterer bounded by surface S0 be located in the space considered. From the geometrical optics point of view, this scatterer divides the incident field into two components: the incident and reflected ones; accordingly, two light–shadow boundaries arise. It is necessary to find out the scattered field. In Fig. 1.1, a two-dimensional scatterer with a polygonal boundary is shown on which a wave is incident emitted by a linear source parallel to the edge. In the presence of the scatterer in the source field, the original geometrical optics field is subdivided into the light and shadow zones, as well as reflected part.

z

P

⃗ n

S₀

Fig. 1.1: Incidence of a wave from the source at point P on a perfectly conducting scatterer with polygonal boundary. Hatching indicates the shadow zones corresponding to the geometrical optics light– shadow boundaries of the incident and reflected waves.

From the mathematical point of view, the above formulation is reduced to the solution of a boundary-value problem U = u + P. Here, the total field U and scattered field u satisfy the following conditions of the boundary-value problem:

16 | 1 Method of Generalized Eikonal

1. 2. 3.

Wave equation (1.2), Boundary condition for U or dU/dn on surface S0 , Regularity condition u = O(r−1/2 ) and radiation condition ∂u/∂r − iku = o(r−1/2 ) as kr → ∞, and 4. Meixner’s edge condition lim ρ grad u = 0, where ρ is the radius of a small circle ρ→0

enclosing the edge. We consider a two-dimensional semi-infinite scatterer having a polygonal boundary. Polarization type is not important for the construction of integral representation because the final expression allows both kinds of boundary conditions, U = 0 and dU/dn = 0, to be satisfied. Figure 1.2 illustrates the incidence of a plane wave on the scatterer, with P1 and P2 being the light–shadow boundaries for the incident and scattered waves, respectively; z0 is the observation point. The incident and reflected fields are shown in Fig. 1.2 in three domains: (a) the original domain z; (b) domain w obtained from z by conformal mapping, and (c) auxiliary two-sheet domain w12 demonstrating clearly the diffraction process. Domains z and w also consist of two sheets, but since they are located one under another, they cannot be seen clearly. The incident field is shown by solid lines, the reflected by dashed lines. The method of generalized eikonal allows one to construct integral representations that satisfy all conditions of the boundary-value problem [3–7].

1.1.2 Construction of “auxiliary” domain and generalized geometrical optics function Consider a conformal mapping kZ(w) of region w that is the upper half of the complex plane onto the region z that is the exterior of the scatterer (see Fig. 1.2 (a), (b)): kz = kZ(w), r z = r z (w), φ z = φ z (w),

(1.3)

where r z and φ z are the polar coordinates of a point in region z, which are functions of complex variable w. Conformal mapping kZ(w) is chosen such that there is one-to-one correspondence between the points of regions z and w. The real axis of region w corresponds to the boundary of the scatterer in region z. In region w, function P satisfies the wave equation with the variable wave number Δ w P + k2 |dz/dw|2 P = 0,

(1.4)

however, there is a family of curves, k |dz/dw| = |d| = r d = const

(1.5)

1.1 Integral representation of solution

17

w

z P2

|

P P1 + P2

(b) P1 (a) w12 P1

w0

rd0 P2 (c)

Fig. 1.2: Incidence of a plane wave on a perfectly conducting scatterer with polygonal boundary: (a) scheme of incidence in z-domain, (b) scheme of incidence in w-domain, (c) scheme of incidence in w12 -domain.

on which the variable wave number k|dz/dw| is constant. The new variable, d, introduced in (1.5) has the meaning of the derivative d = d(kz)/dw. On the curve r d0 = 1, the variable wave number is equal to the initial wave number. Below, this curve is referred to as “curve r d0 ”. If a solution to boundary-value problem for the Helmholtz equation (1.4) with a constant or variable wave number has been found somehow in the domain w on the curve r d0 , then after conformal mapping (1.3) it will remain the same on this curve in the domain z. For the solutions on curves r d = const ≠ 1, however, the wave equation is changing upon conformal mapping onto domain z. In view of the form in which equations (1.1), (1.2), and (1.4) are written, it is necessary to clarify what is meant by “function P”. This function depends on coordinates (r z , φ z ) of the observation point in domain z. In turn, they depend on the coordinates in domain w, see (1.3). In order to avoid cumbersome notation, we write hereafter P, P (r z , φ z ), or P (w), understanding the same function depending on the observation coordinates in the domain z, regardless of the actual domain (z or w) in which this point is considered. From the geometrical optics point of view, the influence of a perfectly conducting scatterer on the incident field consists in cutting out the part of incident wave coming to the “illuminated” surface and deflecting this radiation according to the specular

18 | 1 Method of Generalized Eikonal

reflection law, creating thus the reflected field. From the mathematical point of view, one can consider the reflected field as a part of incident field which entered the second sheet of physical space. If we find on such a two-sheet domain a continuous field satisfying all conditions of the boundary-value problem, except the boundary conditions, then we can satisfy the boundary conditions by adding in some particular way the field values at two points located symmetrically with respect to the scatterer boundary and belonging to different sheets. Introduce a domain w12 , which is the total complex plane consisting of two sheets, w1 and w2 , of half-plane w, “bonded” together along the real axis. To each point of domain z there correspond now a pair of points belonging to w1 and w2 and located symmetrically with respect to the horizontal axis. In the half-plane w1 we define the incident geometrical-optics field P1 , while on the half-plane w2 we define the reflected field P2 (Fig. 1.2 (c)). Although we can formally substitute into the conformal transformation (1.3) the values of w from both the upper and lower half-planes, we will derive the solution using only the upper half-plane w (more exactly, two sheets of the upper half-plane). The scatterer boundary coincides with the horizontal axis of domain w. Conformal transformations are known to preserve the angles between intersecting lines. Therefore, the boundary normal in domain z will correspond to that in domain w. In exactly the same way, the equality property for the incidence and reflection angles upon wave interaction with a perfectly conducting scatterer will be preserved. If we place the reflected field on the other sheet, the property of angle equality will be equivalent to continuity between the incident geometrical-optics field P1 and the reflected field P2 on the illuminated part of scatterer boundary (Fig. 1.2 (c)). On the shadow part of the scatterer boundary, the geometrical-optics field is equal to zero, i. e., it is continuous as well. Introduce an “auxiliary” domain ŵ 12 , intersecting w12 along the circle r w0 = |w0 |, where w0 is the observation point lying on the curve r d0 (Fig. 1.3). Suppose that there ̂ analytical in the domain ŵ 12 (and, therefore, satisfying the exists a function P c (w) Laplace equation on it) and equal P (w) on the curve r d0 : ̂ = P (w) ŵ 12 = w12 , P c (w)

on r d0 ; Δ ŵ P c = 0 in ŵ 12 .

(1.6)

The auxiliary domain ŵ 12 is sketched in Fig. 1.3. It is shown arbitrarily as a cylindrical surface intersecting the domain w12 along the curve r d0 (thick semi-circle). The observation point w0 is located on the curve r d0 . The closed integration contour encircles the observation point w0 . Then the integration contour is deformed in such a way that it pass through the saddle point in the convergence part (dashed area). Consider a closed contour encircling the illuminated parts of domain ŵ 12 . By the illuminated parts, we understand the domains where the function P c is present (in the geometrical optics sense) on r d0 , while by the shadow ones we mean the domains there this function is absent. Then, applying the Cauchy residue theorem, one can ̂ at the obserobtain on the domain ŵ 12 the integral representation of function P c (w)

1.1 Integral representation of solution

19

|

ŵ 12

w0 w12

Fig. 1.3: Scheme for construction of auxiliary domain ŵ 12

vation point w0 belonging to curve r d0 : ̂ 1 P c (w) d ŵ = K ⋅ P c (w0 ) , ∮ 2πi ŵ − w0

(1.7)

where K = 1 if the point w0 is located in the interior of the contour, otherwise K = 0. In what follows, we will show that domain ŵ 12 can be constructed on the curve r d0 by analytic continuation of the angular variable φ w to the complex domain, while the function P c can be obtained from function P by a certain transformation. ̂ decreases with the Let on the domain ŵ 12 there exist subdomains where P c (w) increase in the wave number k. We refer to these as the “convergence parts” (Fig. 1.4). For a domain with two convergence parts s1 and s2, we obtain: K ⋅ P c (w0 ) = =

̂ 1 P c (w) d ŵ ∮ 2πi ŵ − w0 ̂ ̂ ̂ 1 [ P c (w) P c (w) P c (w) d ŵ − ∫ d ŵ + ∫ d ŵ ] , ∫ 2πi ŵ − w0 ŵ − w0 ŵ − w0 s1 s2 ] [C

(1.8)

where C is the part of closed integration contour remaining after s1 and s2 have been singled out. In Fig. 1.4, the auxiliary domain ŵ 12 is shown. Fragments of the integration contour pass through the saddle points in the shadowed convergence parts s1 and s2 . Denoting in (1.8) the sum of integrals over the convergence parts s1 and s2 by ̂ P c (w) 1 v (w0 ), and the integral over the remaining part C by V(w0 ) = 2πi d w,̂ we ∫C w−w ̂ 0 obtain: V(w0 ) = K ⋅ P c (w0 ) + v(w0 ) = K ⋅ P c (w0 ) +

̂ ̂ P c (w) 1 [ P c (w) d ŵ − ∫ d ŵ ] . ∫ 2πi ŵ − w0 ŵ − w0 s2 ] [s1

(1.9)

20 | 1 Method of Generalized Eikonal

ŵ 12

Fig. 1.4: Auxiliary domain ŵ 12 .

The integrals over s1 and s2 have different signs because these parts are passes in the opposite directions with respect to the coordinate origin.

1.1.3 Boundary conditions Function V(w0 ) from (1.9) is continuous on the whole domain ŵ 12 . In this case, it is possible to satisfy the Dirichlet condition, U = 0, or von Neumann condition, dU/dn = 0, on the scatterer boundary by adding or subtracting the function values at points w0 located symmetrically with respect to the horizontal axis (corresponding to the scatterer boundary): U(k, r z0 , φ z0 , φ0 ) = V(k, r w0 , φ w0 , φ0 ) − V(k, r w0 , −φ w0 , φ0 ) (Dirichlet),

(1.10)

U(k, r z0 , φ z0 , φ0 ) = V(k, r w0 , φ w0 , φ0 ) + V(k, r w0 , −φ w0 , φ0 ) (Neumann).

(1.11)

Now we will show that solutions (1.10) and (1.11) also satisfy other conditions of the boundary-value problem. To prove that this solution satisfies the wave equation, we take the function (1.1) in the form of a plane wave of unit amplitude P = P (r z , φ z ) = exp [−ikr z cos (φ z − φ0 )] ,

(1.12)

1.1 Integral representation of solution

|

21

where (r z , φ z ) are the polar coordinates of a point of the domain z, φ0 is the direction angle of the incident plane wave. In these coordinates, (1.12) takes the form P (w) = P (r w , φ w ) = exp {−ikr z (r w , φ w ) cos [φ z (r w , φ w ) − φ0 ]} .

(1.13)

Let (r z0 , φ z0 ) and (r w0 , φ w0 ) be the coordinates of the observation point in domains z and w, respectively. We choose the function P c in accordance with equation (1.6) in the following way: we fix the radial coordinate r z = r z0 , while considering the angular coordinate as complex-valued φ z → φ cz . Then P c = exp[−ikr z0 cos(φ cz − ̂ it is required to establish relationship beφ0 )]. To obtain the dependence P c (w), tween the variables ŵ and φ cz . Consider a wedge-shaped scatterer with the external angle of the wedge πn. This configuration will be studied in detail in the following chapter; here, we use the principal results, omitting derivations. For a wedge-shaped scatterer, φ z = nφ w . Substitute the variables φ w → φ cw by fixing in the expression w = r w exp (iφ w ) the radial variable r w = r w0 , while considering the angular variable as complex-valued, introducing an additional radial variable r ̂w in the expression for φ w : φ w → φ cw = φ w − i ln (r ̂w /r w0 ). Note that for r ̂w = r w0 we have φ w = φ cw . Therefore, for the observation point w0 located on the circle r w = r w0 we obtain the following formula for the auxiliary variable ŵ depending on the parameter r w0 : ̂ w0 ) = r ̂w exp(iφ w ) = r w0 exp (iφ cw ) =r w0 exp {i [φ w − i ln (r ̂w /r w0 )]}. Taking into w(r ̂ = nφ cw (w) ̂ , we obtain then account that iφ cw = ln ŵ − ln r w0 and φ cz = φ cz (w) ̂ = exp {−ikr z0 cos [nφ cw (w) ̂ − φ0 ]} P c (w) ̂ − φ0 ]} . = exp {−ikr z0 cos [in (ln r w0 − ln w) In Fig. 1.3, the surface of auxiliary variable ŵ (r w0 ) is shown by a cylinder, with height corresponding to quantity ln (r ̂w /r w0 ), while in Fig. 1.4 ŵ is shown in the pô lar coordinates (r ̂w , φ w ). The formula obtained provides explicit dependence P c (w). Since this function corresponds to conditions (1.6), further analysis will be equally applicable to it. In particular, integral representation of V(w0 ) takes the form: V(w0 ) =

̂ 1 P c (w) d w.̂ ∫ 2πi ŵ − w0

(1.14)

C

Two essential points must be emphasized. Firstly, different radii of the observation point r w0 correspond to different auxiliary domains ŵ = ŵ (r w0 ). Secondly, from the mathematical point of view it is not important which of the quantities in the argument of cosine (φ z or φ0 ) becomes complex-valued. This circumstance allows us to admit two interpretations of the integral representation obtained. For the observation point (r z0 , φ z0 ) and the incidence angle of the primary plane wave, equation (1.12) takes the form P (w0 ) = exp [−ikr z0 cos (φ z0 − φ0 )]. According to the approach described above, the integral representation (1.14) takes the form: V(w0 ) =

̂ − φ0 ]} exp {−ikr z0 cos [φ cz (w) 1 d w.̂ ∫ 2πi ŵ − w0 C

(1.15)

22 | 1 Method of Generalized Eikonal Thus, to derive the integral representation (1.15), we made substitution φ z0 → φ cz in the argument of cosine in P(w0 ). However, equally successful would be substitution φ0 → φ0c : V(w0 ) =

̂ exp {−ikr z0 cos [φ z0 − φ0c (w)]} 1 d w.̂ ∫ 2πi ŵ − w0

(1.16)

C

Assuming the arguments of cosine in (1.15) and (1.16) equal φ cz − φ0 = φ z0 − φ0c , we perform some simple transformations in the integral (1.16) taking into account c ̂ = idφ cw = ni dφ cz = ni d (φ cz − φ0 ) = ni d (φ z − φ0c ) = −i relations d (ln w) n dφ 0 : V(w0 ) =

1 d ŵ ŵ exp [−ikr z0 cos (φ z0 − φ0c )] ∫ ̂ 2πi w − w0 ŵ C

1 1 exp [−ikr z0 cos (φ z0 − φ0c )] dφ cz = ∫ 2πi 1 − exp [i (φ z0 − φ cz ) /n]

(1.17)

C

=

1 1 exp [−ikr z0 cos (φ z0 − φ0c )] dφ0c . ∫ 2πi 1 − exp [i (φ0 − φ0c ) /n] C

If we change variables φ0 = α and φ0c = β in (1.17), it will become clear that the integral representation (1.16) is equivalent to the well-known Sommerfeld integral representation [7]: U = ∫ A (β) exp [−ikr cos (φ − β)] dβ, A (β) = C

e iβ/n 1 , iβ/n 2πn e − e iα/n

(1.18)

usually referred to as plane-wave expansion. The meaning of this expression is explained in the following way: the field at a point in space is found in the form of a set of incoming plane waves with various amplitudes A (β). Hence, the solution satisfies the same wave equation which is satisfied by the primary plane waves. The integral representation (1.15) is equivalent to (1.16) and, also, satisfies the original wave equation. However, interpretation of (1.15) is somewhat different: the primary wave field at the observation point is represented by an integral over a closed contour in some complex domain, after which this integral representation is split into parts, some of which can be interpreted as the scattered field. This interpretation allows one to consider the primary excitation fields in more general form, as well as gives more flexibility in dealing with them. What is only necessary is to know the coordinate dependence of the primary field, and to perform such change of coordinates that conditions (1.6) be satisfied. Thus, we have shown that solutions (1.10) or (1.11) satisfy the wave equation (1.2) because function P c satisfies on the domain ŵ 12 the same wave equation as at the observation point w0 , while at the latter point it satisfies the original wave equation (1.2) because the point w0 belongs to the curve r d0 .

1.1 Integral representation of solution

|

23

Solutions (1.10) or (1.11) satisfy the radiation condition because integration in (1.9) is performed over the convergence intervals where P c decreases with increase in the wave number. It remains to prove that solutions (1.10) or (1.11) satisfy the Meixner edge condition. This is not an easy task because we obtain the solution on the curve r d0 which always passes at some distance from the edge. However, in what follows we will be able, analyzing the solution obtained, to prove that this condition is satisfied indeed on the edge. Thus, we have found solutions to the Helmholtz equation by substituting the variable ŵ into the known function P and constructing the integral representation of the resulting function in accordance with the Cauchy theorem on residues. Formulas (1.9), (1.10), and (1.11) give the integral representation of the boundaryvalue problem solution in it general form. To obtain the integral representation of the solution, it is sufficient to find out the conformal mapping, determine the shadow boundaries in the auxiliary domain, and perform integration over the convergence parts of these boundaries according to formula (1.9).

1.1.4 Features of the solution Consider the incident field in the form (1.13). If the scattered field is searched for in the form of plane-wave expansion (1.18), we need to substitute in (1.13) the arrival angle φ0 of the plane wave by the complex variable β. This operation does not affect coordinates. Independently of in which domain (z or w) we search for the solution, the wave equation remains intact, while whether the solution satisfies the boundary condition will depend on how well the plane wave amplitude A (β) in the integrand was chosen or found. On the other hand, if we search for the scattered field in domain w in the form (1.15) by substituting in (1.13) ŵ by w, the factor in the integrand is obtained automatically because it has the form of a simple pole. Also, in domain w ̂ the boundary conditions are satisfied automatically. Consider the function P c (w) obtained in this case. ̂ = exp {−ikr z (r w0 , φ cw ) cos [φ z (r w0 , φ cw ) − φ0 ]} . P c (w)

(1.19)

̂ of general form (1.19) (or of more Analysis of integral (1.14) with function P c (w) complex form) is a mathematical study beyond the scope of this book. In such a study, ̂ satisfies the Laplace equafirst of all the conditions under which the function P c (w) tion in domain ŵ must be established. For wedge scatterers, the Laplace equation is satisfied because r z (r w0 , φ cw ) = const. We could attempt to use the known solution for scattering by a wedge when solving the diffraction problem for a scatterer possessing a size parameter. But direct substitution of the wedge solution to the domain ŵ of a scatterer possessing a size parameter, and subsequent transformation of the solution to domain z by conformal mapping will give incorrect result because this procedure

24 | 1 Method of Generalized Eikonal

will correspond to diffraction by an object in an inhomogeneous medium. The refraction coefficient of this medium will be equal to the ratio of derivatives |dz/dw| for the wedge and a scatterer with a size parameter. However, if we take the wedge solution not in the whole space, but on a specific curve where the ratio of derivatives is equal to one, and then continue it into the remaining space, we can obtain a correct solution of wave equation for the scattered field. This approach agrees with the concept of quasistatic approximation. It is known that if the object sizes are much smaller that the wave length, the field distribution in its neighborhood is the same as in the static case. For a semi-infinite object, its sizes are always larger than the wave length, but the sizes of the curve on which we search for the solution can satisfy the above-mentioned condition. Continuation of solution from the given curve to the remaining space can be performed in different ways. A rigorous approach would be to apply the method of integral equations for a medium with inhomogeneous dielectric permeability [64]. One possible approximate way is to use the method of geometrical optics. Knowing the amplitude, phase and divergence characteristics of the field upon its propagation from the contour into the remaining space, it is possible to find the solution at all points. Integral representations of the solution obtained by the method of generalized eikonal have a number of advantages among which are the generality of approach for scatterers of different shapes, and relative simplicity of the final relationships.

1.2 Asymptotic calculation of contour integrals by method of stationary phase The main steps towards derivation of an analytical solution to boundary-value problem for the Helmholtz equation by the method of generalized eikonal are: 1. Conformal mapping of the upper complex half-plane onto the exterior of the scatterer. 2. Construction of an auxiliary domain and introduction of a complex angular variable. 3. Derivation of the generalized eikonal function by substitution of the complex angular variable into the eikonal function. 4. Derivation of integrals over fragments of closed contours in the auxiliary domain. 5. Application of the integral representations obtained as solutions to the boundaryvalue problem for the Helmholtz equation.

1.2 Asymptotic calculation of contour integrals by method of stationary phase | 25

1.2.1 General solution Consider the generalized function of geometrical optics which on the curve r d0 takes the form (1.1), whereas at the current integration point in the auxiliary domain ŵ is equal to ̂ = A(w) ̂ exp [iS c (w)] ̂ . P c (w) (1.20) We evaluate the integral of the form (1.8) by the method of stationary phase. Let there exist a saddle point ŵ sm of the function S c [ŵ (r w0 )], at which [S c (ŵ sm )]󸀠 = 0, [S c (ŵ sm )]󸀠󸀠 ≠ 0. Then the corresponding integral can be transformed by the approach presented in [1]. The non-singular part of integrand is expanded in a power series in the integration variable, after which the first term of this series is retained and taken out of the integral: ̂ exp [iS c (w0 )] P c (w) 1 d ŵ = ∫ 2πi 2πi ŵ − w0 sm

×∫ sm

≈

A (ŵ sm

) exp [iS c (w 2πi

0 )]

̂ [S c (w) ̂ − iS c (w0 )] ̂ − S c (w0 )] exp [iS c (w) A (w) d ŵ ̂ − S c (w0 ) (1.21) ŵ − w0 S c (w) ̂ − iS c (w0 )] exp [iS c (w) S c (ŵ sm ) − S c (w0 ) d w.̂ ×∫ ̂ − S c (w0 ) ŵ sm − w0 S c (w) sm

In the vicinity of saddle point ŵ sm , the difference of eikonals is obtained by the ̂ where [S c (ŵ sm )]󸀠 = 0, [S c (ŵ sm )]󸀠󸀠 ≠ 0: Taylor expansion of function S c (w), 2

(ŵ − w sm ) +... . (1.22) 2 Taking the first two terms in the expansion (1.22), we can evaluate the integral on the right-hand side of equation (1.21) by the method of stationary phase: ̂ − S c (w0 ) ≈ S c (ŵ sm ) − S c (w0 ) + [S c (ŵ sm )]󸀠󸀠 S c (w)

∞

1

−∞

∞

̂ − iS c (w0 )] exp [iS c (w) ̂ t − iS c (w0 ) t] dt d ŵ = ∫ d wî ∫ exp [iS c (w) ∫ ̂ − S c (w0 ) S c (w)

sm

∞

1

̂ t] d ŵ = i ∫ exp [−iS c (w0 ) t] dt ∫ exp [iS c (w) ∞

−∞ ∞

1

i 󸀠󸀠 2 = i ∫ exp {i [S (ŵ sm ) − S c (w0 )] t} dt ∫ exp [ [S c (ŵ sm )] (ŵ − w sm ) t] d ŵ 2 c

∞

−∞

1

󸀠󸀠 −1/2

= 2i ∫ exp {i [S c (ŵ sm ) − S c (w0 )] t} √2πi {[S c (ŵ sm )] } ∞

=

2i√2πi √[S c (ŵ sm )]󸀠󸀠 [S c (ŵ sm ) − S c (w0 )]

dt 2√t

√S c (ŵ sm )−S c (w0 )

∫ ∞√S c (ŵ sm )−S c (w0 )

exp (iq2 ) dq.

26 | 1 Method of Generalized Eikonal

In the derivation, we used 1

exp iz2 = i ∫ exp (iz2 t) dt, z2 ∞

as well as ∞

i 󸀠󸀠 2 󸀠󸀠 −1/2 . ∫ exp [ [S c (ŵ sm )] (ŵ − ŵ sm ) t] d ŵ = √2πi {[S c (ŵ sm )] t} 2

−∞

Thus, we proved that ∫ sm

̂ − iS c (w0 )] exp [iS c (w) 2i√2πi d ŵ ≈ c c ̂ − S (w0 ) S (w) √[S c (ŵ sm )]󸀠󸀠 [S c (ŵ sm ) − S c (w0 )] √S c (ŵ sm )−S c (w0 )

×

∫

(1.23) exp (iq2 ) dq.

∞√S c (ŵ sm )−S c (w0 )

Substituting (1.23) into (1.21), we obtain 2iA (ŵ sm ) exp [iS c (w0 )] √ S c (ŵ sm ) − S c (w0 ) ̂ P c (w) 1 ̂ dw ≈ ∫ 2πi 󸀠󸀠 ŵ − w0 (ŵ sm − w0 ) √2πi [S c (ŵ sm )] sm (1.24)

√S c (ŵ sm )−S c (w0 )

×

exp (iq2 ) dq,

∫ ∞√S c (ŵ sm )−S c (w0 )

where the sign of the integral is taken according to the direction of passing the contour in the vicinity of ŵ sm . With reference to the known asymptotic formula X

∫ exp (iq2 ) dq ≈ ∞⋅X

2

e ikX for |X| ≫ 1, 2iX

(1.25)

one can show that if we set X = √ S c (ŵ sm ) − S c (w0 ), the integral (1.24) over the vicinity of the saddle point takes its asymptotic form ̂ P c (ŵ sm ) P c (w) 1 1 d ŵ ≈ √ , ∫ 󸀠󸀠 2πi ŵ − w0 2πi [S c (ŵ sm )] ŵ sm − w0

(1.26)

sm

󵄨 󵄨 provided that 󵄨󵄨󵄨√ S c (ŵ sm ) − S c (w0 )󵄨󵄨󵄨 ≫ 1. The sign of the root is taken negative in the light domain, and positive in the shadow domain. Equation (1.26), which is a particular case of (1.24), gives the result which is known quite well in the theory of the method of stationary phase (see, e. g.,[33]).

1.2 Asymptotic calculation of contour integrals by method of stationary phase

| 27

Integral (1.24) can be recast in a more explicit form: ̂ 1 P c (w) d ŵ ≅ ∫ 2πi ŵ − w0 sm

2i√ S c (ŵ sm ) − S c (w0 )

P (ŵ sm )

c c ̂ 󸀠󸀠 (ŵ sm − w0 ) √2πi [S c (ŵ sm )] exp [iS (w sm ) − iS (w0 )]

(1.27)

√S c (ŵ sm )−S c (w0 )

×

exp (iq2 ) dq.

∫ ∞√S c (ŵ sm )−S c (w0 )

In the case of two saddle points, we obtain v (w0 ) ≅ ∑ m=1,2

P (ŵ sm )

2i√ S c (ŵ sm ) − S c (w0 )

c c ̂ 󸀠󸀠 (ŵ sm − w0 ) √2πi [S c (ŵ sm )] exp [iS (w sm ) − iS (w0 )]

(1.28)

√S c (ŵ sm )−S c (w0 )

×

exp (iq2 ) dq.

∫ ∞√S c (ŵ sm )−S c (w0 )

To obtain the final result, it remains to substitute into equation (1.28) the values ̂ 󸀠󸀠 ̂ at the observation point w0 and its derivative [S c (w)] of generalized eikonal S c (w) at the saddle points ŵ sm . In the solution to boundary-value problem for the Helmholtz equation, formula (1.28) gives the general form of scattered component obtained by the method of generalized eikonal. It will be shown later that application of this formula to scattering by half-plane and wedge gives correct results. It follows from (1.28) that scattered field is represented in the form of a product of the incident field value at the saddle point and a factor discontinuous at the shadow boundary. Far away from the shadow boundary we have, according to (1.25): v (w0 ) ≈ ∑ m=1,2

P (ŵ sm ) 󸀠󸀠 (ŵ sm − w0 ) √2πi [S c (ŵ sm )]

.

(1.29)

Each of the two terms in equation (1.29), as expected, coincides with equation (1.26). The final expressions for the scattered field can be obtained after the scatterer shape is specified, by substitution into (1.28) and (1.29) the expressions ̂ and S c (w). ̂ for P c (w)

1.2.2 Solution of diffraction problem for plane and cylindrical waves by the method of generalized eikonal We choose the incident wave function (1.1) in the form P (r z , φ z ) = A (r z , φ z ) exp [iS (r z , φ z )] ,

(1.30)

28 | 1 Method of Generalized Eikonal

where for a plane wave A (r z , φ z ) = 1,

S (r z , φ z ) = −kr z cos (φ z − φ0 ) ,

(1.31)

while for a cylindrical wave A (r z , φ z ) = f (φ z ) /√ kρ, S (r z , φ z ) = kρ = k√ r2z + r20 − 2r z r0 cos (φ z − φ0 ).

(1.32)

Here, ρ is the distance between the source and observation point, r0 and r are the distances from the source and observation point to the edge, respectively; k = 2π/λ, f (φ z ) is the dependence of the amplitude on the angle. The amplitude in equation (1.32) is an asymptotic at kρ → ∞ of the rigorous relation which involves the (1) (1) Hankel function of the first kind H0 (kρ), with H0 (kρ) ≈ √2/ (πkρ) exp (ikρ − iπ/4) for kρ ≫ 1. More rigorous expressions for the amplitude and incident wave function in the case of cylindrical wave are A (r z , φ z ) = f (φ z ) √

πi (1) H (kρ) exp (−ikρ) , 2 0

(1.33)

πi (1) P (r z , φ z ) = f (φ z ) √ H0 (kρ) . 2 Expressions (1.33) are reduced to (1.32) for kρ ≫ 1. Consider first diffraction of cylindrical wave (1.32) by a wedge scatterer for which the conformal mapping (1.3) and derivative (1.5) take the form kz (w) = A ⋅ w n ,

d [kz(w)] = A ⋅ n ⋅ w n−1 = d, dw

(1.34)

where A is the proportionality coefficient which is constant at all points w and, therefore, is independent of this variable. In the general case, |d| ≠ 1. According to the method of generalized eikonal, the solution is sought at the observation point w0 belonging to the curve r d0 at which the condition |d0 | = r d0 = 1 is satisfied. Substitute into (1.34) an arbitrary value of w0 : n ⋅ kz(w0 ) d [kz(w0 )] , = d0 = A ⋅ n ⋅ w0n−1 = dw w0 󵄨󵄨 1−n 󵄨󵄨 󵄨󵄨w0 󵄨󵄨 󵄨 |w0 | 󵄨 󵄨. , A= 󵄨 |kz(w0 )| = 󵄨󵄨󵄨A ⋅ w0n 󵄨󵄨󵄨 = n n

(1.35)

It follows from (1.35) that for |d0 | = 1 the value of |kz(w)| does not depend on the coefficient A which, in turn, depends on |w0 | and, thus, changes the conformal mapping. The condition that the coefficient A is independent of w can be satisfied by fixing |w0 | at some particular value, for example, by setting |w0 | = 1.

1.2 Asymptotic calculation of contour integrals by method of stationary phase | 29

On the other hand, we can normalize the quantity w N = w/|w0 |, so that |w N | = 1. Then kz (w) =

w n |w0 | |w0 | ( ) = (w N )n , n n |w0 |

d [kz(w)] = d N = |w0 | (w N )n−1 . dw N

(1.36)

It also follows from (1.36) that the condition |d N | = 1 is satisfied only for |w0 | = 1. If we proceed now to normalized variable w N , the conformal mapping has to be taken in the form kz (w N ) = w nN /n, in which case for the normalized variable the following holds true: |d N | = 1. We obtain the same expression if we eliminate in (1.35) the dependence of coefficient A on the variable w by setting |w0 | = 1. Therefore, we have A = 1/n. Thus, the normalized expressions for the conformal mapping and its derivative in the case of cylindrical wave diffraction by wedge scatterer take the following form: kz (w) =

1 n w , n

d [kz(w)] = w n−1 = d. dw

(1.37)

In the case of wedge scatterer, the curve r d0 is represented by a fragment of circle |w| = 1. Since the observation point w0 is located on the circle r d0 , we have r w0 = |w0 | = 1. In the domain z on this circle the following is true: kr z0 = 1/n. Change the variable w → ŵ (r w0 ) (or φ w → φ cw ): w = r w exp(iφ w ),

ŵ (r w0 ) = r w0 exp(iφ cw ) = r ̂w exp(iφ w ),

φ cw = i ln r w0 − i ln ŵ (r w0 ) = i ln r w0 − i ln r ̂w + φ w = −i ln

r ̂w + φw , r w0

(1.38)

where r w0 = |w0 |, and w0 is the observation point always located on the curve r d0 . Transformation (1.38) introduces an additional radial variable r ̂w , while the primary radial variable is fixed as r w = r w0 , and φ w remains intact. In the polar coordinates (r ̂w , φ w ), there appears an auxiliary domain ŵ (r w0 ) in which the complex-valued variable φ cw takes real values and is equal to φ w on the circle r ̂w = r w0 . ̂ = P (w) at all points of the circles r w0 = const The generalized eikonal function P c (w) because at these points r w = r ̂w = r w0 , φ w = φ cw , and w = ŵ (r w0 ). Thus, to show that transformation (1.38) satisfies conditions (1.6), it remains to prove that Δ ŵ P c = 0 in ŵ 12 . The latter follows from the fact that P c is an analytic function of ŵ (r w0 ), in contrast to P which is not an analytic function of w or z. Function P depends on the variables (r z , φ z ), with r z = |z| and φ z = arg (z). Transformations of this kind are not analytic, therefore function P satisfies the wave Helmholtz equation (1.2), rather than the Laplace equation. However, if we fix, according to (1.37), r z = r z0 = r nw0 / (kn) by setting r w = r w0 , than we obtain on this circle, with reference to (1.38), φ cz = −i ln (z) + i ln (r z0 ) = −i ln (

rn wn ) + i ln ( w0 ) = nφ cw . kn kn

(1.39)

30 | 1 Method of Generalized Eikonal ̂ satisfies the Laplace equation (1.6). This function is analytic, and function P c (w) Therefore, all further discussions concerning integral representations are valid for the ̂ function P c (w). The auxiliary domain ŵ (r w0 ) depends on r w0 , however, it is independent of the scatterer type. Therefore, the total field U (w0 ) found by integral transformations on the domain ŵ (r w0 ) is also independent of the curve r d0 type. On the other hand, formula for the derivative (1.5) and the curve r d0 are dependent on the scatterer shape. Therefore, the same value of U (w0 ) at the point w0 of domain w is mapped onto domain z with different values z0 and |d (kz) /dw| which depend on the scatterer type. As was mentioned before, when the solution is mapped onto domain z, it remains intact only at the points |d (kz) /dw| = 1. The complex-valued angular variable φ cw takes a real value at the observation point w0 which always belongs to the curve r d0 . Therefore, on r d0 we have ŵ = w0 and r ̂w = r w0 . If we fix r w0 = const, it is also true that r z0 = const on the whole domain w.̂ Also, equation (1.37) implies that relationship between the angular variables is linear, φ z = nφ w . Therefore, on r d0 (where, according to (1.38), r ̂w = r w = r w0 ) substitution of ŵ for w into the eikonal formula is reduced to the change of real-valued angular variables φ w and φ z by complex-valued φ cw and φ cz , respectively, with φ cz = nφ cw . Expression for ŵ can be obtained by formal substitution in w of old coordinates (r w , φ w ) by new ones: (r w0 , φ cw ), where r w0 is a constant, while φ cw is a complex variable expressed via φ w and r ̂w . One can take that the auxiliary orthogonal coordinate r ̂w is perpendicular both to φ w and r w (i. e., normal to the plane (r w , φ w )). Thus, instead of a pair of real coordinates (r w , φ w ) we obtain a triplet (r w , r ̂w , φ w ). By fixing (r w , r ̂w = r w0 , φ w ), we obtain the domain w, by fixing (r w = r w0 , r ̂w , φ w ), we obtain the domain w.̂ Substituting φ cw for φ w in P(r z , φ z ) from (1.30), we obtain the generalized geometrical optics function P g (r w , r ̂w , φ w ) depending on three real-valued arguments: P [r z (r w , φ cw ), φ z (r w , φ cw )] = exp {−ikr z (r w , φ cw ) cos [φ z (r w , φ cw ) − φ0 ]} = P g (r w , r ̂w , φ cw ).

(1.40)

By similarity with the generalized function of geometrical optics, one can write, according to (1.30), an expression for generalized eikonal: S [r z (r w , φ cw ), φ z (r w , φ cw )] = −kr z (r w , φ cw ) cos [φ z (r w , φ cw ) − φ0 ] = S g (r w , r ̂w , φ cw ).

(1.41)

It will be demonstrated below that properties of function (1.41) determine the features of the new method, therefore we refer to it as the “method of generalized eikonal”. By fixing in the coordinate triplet (r w , r ̂w , φ w ) one of the radial variables (setting r ̂w = r w0 or r w = r w0 ), it is possible to enforce desired properties for the geometrical optics function (1.40). In the case r ̂w = r w0 , we obtain φ cw = φ w and P g (r w , r w0 , φ w ) =

1.2 Asymptotic calculation of contour integrals by method of stationary phase

| 31

P [r z (r w , φ w ), φ z (r w , φ w )], while for r w = r w0 we obtain a new function P c (φ cw ) of complex argument φ cw : P [r z (r w0 , φ cd ), φ z (r w0 , φ cw )] = exp {−ikr z (r w0 , φ cw ) cos [φ z (r w0 , φ cw ) − φ0 ]} = P g (r w0 , r ̂w , φ w ) = P c (φ cw ).

(1.42)

The old function P(r z , φ z ) satisfies the Helmholtz equation in domain z, while on w it satisfies the Helmholtz equation with variable wave number. At the same time, the new plane wave function P c (φ cw ), according to the analytic function properties, satisfies on ŵ the Laplace equation: Δ ŵ P c (φ cw ) = 0. This occurs because P(r z , φ z ) depends not on the complex variable z as a whole (or, on d or w), but on separate coordinates r z and φ z (which can be obtained from z only by the operation of complex conjugation). On the contrary, P c (φ cw ) depends on a single complex variable φ cw as a whole. Thus, we established properties of the generalized geometrical optics function P g (r w , r ̂w , φ w ) for a fixed value of one of radial coordinates. By fixing the values of both radial coordinates, r w = r ̂w = const = r w0 , we obtain the curve r w0 common to both domains w and w;̂ on this curve P = P c and S = S c . For example, for a plane wave (1.31), we obtain, with reference to (1.37): S c [ŵ (r w0 )] = −kr z0 cos (φ cz − φ0 ) = −n−1 r nw0 cos (nφ cw − φ0 ) .

(1.43)

The convergence domains of the function (on which the function decreases with increase in the wave number) P c = exp (iS c ) are determined by the condition Im S c > 0, which, taking into account − cos (φ cz − φ0 ) = − cos (Re φ cz − φ0 ) cosh (Im φ cz ) + i sin (Re φ cz − φ0 ) sinh (Im φ cz ) ,

(1.44)

as well as (1.38), results in the inequality sin (Re φ cz − φ0 ) sinh (Im φ cz ) > 0, or

sin (φ z − φ0 ) [(

r w0 n r ̂w n ) −( ) ] > 0. r w0 r ̂w

(1.45)

The convergence domains are shown in Fig. 1.4 by hatching. For a cylindrical wave (1.32), we obtain: S c [ŵ (r w0 )] = k√ r2z0 + r20 − 2r z0 r0 cos (nφ cw − φ0 ).

(1.46)

A similar equation is also obtained for a plane wave (1.31). In the curve r d0 , the function S c (w0 ) is equal to a real-valued number because we have in it φ cw = φ w0 . On the other hand, it follows from (1.38) that d dφ cw d 1 d = = . c dφ w d ŵ dφ cw i ŵ d ŵ

(1.47)

32 | 1 Method of Generalized Eikonal

Taking into account relations (1.46) and (1.47), it is easy to obtain an expression for the first-order derivative dS c [ŵ (r w0 )] kr z0 r0 sin (nφ cw − φ0 ) n = . d ŵ i c 2 2 √ r z0 + r0 − 2r z0 r0 cos (nφ w − φ0 ) ŵ

(1.48)

The contours of integration involved in (1.8) contain parts passing along the shadow boundary. On these parts, the first-order derivative is zero at the points where the variable φ cw is real-valued and satisfies the relationship nφ cw − φ0 = ±π. In this case, the equalities sin (nφ cw − φ0 ) = 0 and cos (nφ cw − φ0 ) = −1 are valid. Calculate now the second-order derivative of function S c [ŵ (r w0 )] with respect to ŵ at the above-mentioned points: d2 S c [ŵ (r w0 )] kr z0 r0 n 2 = ( ) for nφ cw − φ0 = ±π. 2 ̂ r dw z0 + r 0 ŵ

(1.49)

Since, in contrast to the first-order derivative, the second-order derivative is not equal to zero, the points nφ cw − φ0 = ±π are, by definition, the saddle points of function S c [ŵ (r w0 )] in the domain w.̂ These points are denoted by ŵ sm , where m is the saddle point number. We then rewrite (1.49) in the form 󸀠󸀠

[S c (ŵ sm )] =

kr z0 r0 n 2 ( ) . r z0 + r0 ŵ sm

(1.50)

Substituting ŵ sm into function S c [ŵ (r w0 )], we obtain S c (ŵ sm ) = k (r z0 + r0 ) .

(1.51)

We will need expressions (1.50) and (1.51) later in order to evaluate the integral in the neighborhood of the saddle point by the method of stationary phase. To obtain the solution of boundary-value problem, it remains to find out the coordinates of observation points w0 located on the curve r d0 of the scatterer. Note that in the cases r z0 ≫ r0 or r z0 ≪ r0 , the expression r z0 r0 / (r z0 + r0 ) in (1.50) becomes approximately equal to the smaller of r0 and r z0 . The case kr0 ≫ 1 corresponds to plane wave (1.31), while for r z0 ≫ r0 (cylindrical wave (1.32)) the observation point is located in the far zone with respect to the source. Substituting into (1.27) the values from (1.46), (1.50), and (1.51), we obtain 2i√ S c (ŵ sm ) − S c (w0 ) ̂ P (ŵ sm ) 1 P c (w) ŵ sm /n d ŵ ≅ ∫ c c 2πi ŵ − w0 √2πi rkr z0+rr0 ŵ sm − w0 exp [iS (ŵ sm ) − iS (w0 )] sm z0 0 (1.52)

√S c (ŵ sm )−S c (w0 )

×

∫ ∞√S c (ŵ sm )−S c (w0 )

exp (iq2 ) dq.

1.2 Asymptotic calculation of contour integrals by method of stationary phase | 33

󵄨 󵄨 Under the condition (r z0 + r0 )2 ≫ 󵄨󵄨󵄨2r z0 r0 [1 + cos (φ − φ0 )]󵄨󵄨󵄨 (namely, in the vicinity of shadow boundary (φ − φ0 = ±π), where cos (φ − φ0 ) ≅ −1) or r2z0 + r20 ≫ 1, including r z0 ≫ 1 or r0 ≫ 1, the binomial expansion formula can be applied to eikonal 1.32 ρ = √(r z0 + r0 )2 − 2r z0 r0 [1 + cos (φ − φ0 )] 1/2

φ − φ0 2r z0 r0 2 cos2 ] 2 (r + r0 )2 φ − φ0 r z0 r0 ≅ (r z0 + r0 ) [1 − 2 cos2 ] 2 (r z0 + r0 )2 r z0 r0 φ − φ0 2 cos2 . = r z0 + r0 − r z0 + r0 2 = (r z0 + r0 ) [1 −

(1.53)

With reference to equation (1.50), we obtain: φ − φ0 kr z0 r0 2 cos2 r z0 + r0 2 2 φ − φ0 󸀠󸀠 ŵ sm = [S c (ŵ sm )] ( . ) cos2 n 2

S c (ŵ sm ) − S c (w0 ) =

(1.54)

󸀠󸀠

Substituting into equation (1.28) [S c (ŵ sm )] from (1.54) and S c (w0 ) from (1.31), we obtain ŵ s2 2i ŵ s1 ψ e−ikr z0 cos ψ − v (r, ψ) = ( ) cos n ŵ s1 − w0 ŵ s2 − w0 2 √πi √2kr z0 cos(ψ/2)

×

(1.55) exp (iq ) dq.

∫

2

∞ cos(ψ/2)

General expression (1.55) for the scattered field excited by the incident plane wave (1.31) agrees completely with the known formulas of diffraction theory. As was mentioned before, in the case of diffraction by a wedge, the curve r d0 becomes a circle with r wsm = r w0 . If we substitute into (1.55) the coordinates of points ŵ sm = r wsm exp [i (φ0 ± π) /n] and w0 = r w0 exp (iφ z /n), φ +π

φ −π

exp (i 0n ) exp (i 0n ) ŵ s1 ŵ s2 − − = ŵ s1 − w0 ŵ s2 − w0 exp (i φ0 +π ) − exp (i φ z ) exp (i φ0 −π ) − exp (i φ z ) n n n n = =

exp (i πn )

0 exp (i πn ) − exp (i φ z −φ n )

i sin cos

π n

π n

− cos

φ z −φ0 n

,

−

exp (−i πn )

0 exp (−i πn ) − exp (i φ z −φ n )

(1.56)

34 | 1 Method of Generalized Eikonal we obtain from equations (1.28) and (1.52) with r z0 ≪ r0 a well-known solution to the problem of plane wave diffraction by a wedge [1]: sin πn −2 v (r z , ψ) = n cos π − cos n

ψ n

ψ e−ikr z0 cos ψ cos 2 √πi

√2kr z0 cos(ψ/2)

∫

exp (iq2 ) dq,

(1.57)

∞ cos(ψ/2)

where ψ = φ z − φ0 , S c (ŵ sm ) − S c (w0 ) = kr z0 (1 + cos ψ) = 2kr z0 cos2 (ψ/2). The integral representations (1.21) and (1.24) derived by the method of generalized eikonal and evaluated approximately by the method of stationary phase contain the difference between the generalized eikonal function at the saddle point and observation point S c (ŵ sm ) − S c (w0 ). In the diffraction theory, this expression is interpreted as the difference between eikonals of the edge and incident (or reflected) waves. It follows from (1.51), (1.53), and (1.57) that in the case of plane wave, as well as in the case of cylindrical wave scattered by a wedge, this expression agrees with know formulas (see, e. g., [3, 30, 43, 63]). S c (ŵ sm ) − S c (w0 ) = Sedg − Sinc = k (r z0 + r0 − ρ) ,

(1.58)

ref

where ρ = √ r2z0 + r20 − 2r z0 r0 cos (φ − φ0 ) corresponds to the eikonal (1.32). Expression for diffraction of a cylindrical wave by a wedge can be obtained by substituting into the general equation (1.28) the difference of eikonals (1.58) together with the second-order derivative (1.50). Thus, substitution of expressions for the incident wave (1.31) and (1.46) into the general solution for scattered field (1.28) obtained by the method of generalized eikonal gives correct analytic solutions for diffraction of plane and cylindrical waves by the wedge. Application of the method of generalized eikonal to scatterers of different shapes allows one to obtain analytic solutions to new diffraction problems.

2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal 2.1 Introduction In Chapter 1, the basic principles of method of generalized eikonal (MGE) were presented. This method allows one to obtain analytical solutions to two-dimensional diffraction problems. The method is based on integral representation of solution in the auxiliary domain obtained by conformal mapping. In Chapter 1, asymptotic formulas for integral representations of diffraction coefficients for perfectly conducting semiinfinite scatterers, including those possessing a size parameter, are derived in the general form. Application of these formulas to diffraction by half-plane and wedge reproduces the known results, which confirms the validity of proposed method. In this chapter, analytical solution is obtained for a half-plate of finite thickness. Results are obtained with the asymptotic solution to the same problem in which double reflection of the scattered signal on the half-plate vertices is considered, as well as with literature data. Good agreement of the results is demonstrated. For wedge scatterers having no size parameter, integral representation (1.9) is applicable at all point of the domain w, including those not belonging to the curve r d0 (which for the wedge scatterer is a circle r w0 = const). This is confirmed by the fact that for wedge scatterers the integral representation of the solution coincides completely with the known integral representation derived by different means. On the contrary, for scatterers possessing a size parameter, mapping of solution for the scattered field from the domain ŵ (r w0 ) onto domain w gives correct results not at any point of domain w, but only at points w0 belonging to curve r d0 at which the condition |d (kz) /dw| = 1 (or k = |dw/dz|) is satisfied. At the points w0 of domain w, the wave equation (1.4) for the field U (w0 ) coincides with the original wave equation (1.2). If, however, at some points of domain w we have |d (kz) dw| ≠ 1 (or k ≠ |dw/dz|), the wave equation (1.4) at these points does not coincide with the original wave equation. Therefore, the solution mapped from ŵ (r w0 ) onto w will match the original problem with a different wave number k1 satisfying the condition k1 = |dw/dz| (or |d (k1 z) /dw| = 1). Thus, for scatterers possessing a size parameter, correct mapping of solution for the scattered field from the domain ŵ (r w0 ) onto domain w can only be performed at the points w0 belonging to the curve r d0 . According to (1.38), the form of domain ŵ (r w0 ) and dependence of the generalized eikonal function on the complex-valued angular variable in the integral representation of the solution will be the same as the corresponding dependencies for a wedge scatterer. Therefore, solution for a scatterer possessing a size parameter can be found by placing its curve r d0 in the domain w where solution for wedge scatterer satisfying the original wave equation in the whole domain z is already available. However, it

36 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

P

ρ z0 rz0

kh

kh = 2; π/2; 1,2; 0.8; π/8

Fig. 2.1: Family of curves r d0 upon incidence of a wave from point source on a perfectly conducting half-plate. For kh = π/2 and above, the curve r d0 consists of two separate parts. The smaller kh, the closer the shape of curve r d0 to a circle.

will be shown below that placement of the curve r d0 must be accompanied by its prenormalization.

2.2 Diffraction by half-plate Consider a scatterer of a particular type, namely, a half-plate of finite non-dimensional thickness kh (k = 2π/λ is the wave number, h is the dimensional thickness of the halfplate) for which the conformal mapping w(kz) takes the form πkz(w) = (w + 1) √ w (w + 2) − ln [w + 1 + √ w (w + 2)] , kh

(2.1)

with the derivative d πkz(w) [ ] = 2√ w (w + 2) = 2D(w) = 2d = 2r d exp (iφ d ) . dq kh

(2.2)

Specifying different values of |d| = r d , we obtain a family of curves on w and z domains shown in Fig. 2.1. Variable w is non-dimensional. Whatever the value of kh, the upper vertex of the half-plate corresponds to w = 0, at the middle of the end surface we have w = −1, while at the lower vertex w = −2. The curve r d = const can be simply or doubly connected. Large values r d ≫ 1 correspond to simply connected curves r d = const for which |w| ≫ 1, while small values r d ≪ 1 correspond to doubly connected curves r d = const for which |w| ≪ 1. In the latter case, there exist points w located on the part of doubly connected curve r d = const which envelops the coordinate origin w = 0. For the other part of doubly connected curve r d = const enveloping point w = −2, for r d ≪ 1 we will have |w + 2| ≪ 1. According to the principle of uniqueness of solution it is evident that any value kh must correspond to a distinct curve r d = const which in the general theory of the method of generalized eikonal was referred to as “curve r d0 ”. In Fig. 2.1, the family

2.2 Diffraction by half-plate | 37

of curves r d0 is shown for a perfectly conducting half-plate (shown by the thick line). The observation point z0 is located on the curve r d0 . For large values of kh (π/2 and larger), the curve r d0 is split into two parts. The smaller the value of kh, the closer the shape of curve r d0 approaches a circle. To establish one-to-one correspondence kh ↔ r d , consider a half-plate with variable value of kh as an intermediate case between a wedge and half-plane. It is known that wedge scatterers possess no size parameter, and for them the curves r d0 at any value of the external angle of the wedge πn are circles kr z0 = 1/n in the z-domain, or |w0 | = r w0 = 1 in the w-domain. In the limiting cases of diffraction by half-plane (kh ≪ 1, r d ≫ 1, |w| ≫ 1) or wedge (kh ≫ 1, r d ≪ 1, |w| ≪ 1), the curve r d0 in the w-domain for a thick half-plate must also become stabilized in the vicinity of circle r w0 = 1, with the dependence on kh vanishing. Thus, we normalize the wdomain as: w N = C (kh) ⋅ w, (2.3) where C (kh) is a function depending on the size parameter, such that in the limiting cases kh ≪ 1 and kh ≫ 1 condition |w N | ≅ 1 holds true. Since we consider scattering at a fixed frequency, normalization (2.3) is linear, i. e., is independent of w. To obtain function C (kh) in the explicit form, we write in the domain w N a conformal mapping for the wedge scatterer kz (w N ) corresponding to an infinitely thin half-plane at n = 2 and a to wedge at n = 3/2, as well as its derivative: kz (w N ) =

1 n w , n N

d = dN . [kz(w N )] = w n−1 N dw N

(2.4)

On the curve r d0 of wedge scatterer we have |w N | = 1. Since |d N | = 1, it follows that d N is a normalized derivative. Evidently, under the normalization (2.3), the conformal mapping (2.1) and derivative (2.2) are reduced in the cases r d ≫ 1 and r d ≪ 1 to (2.4) for n = 2 (kh ≪ 1) and n = 3/2 (kh ≫ 1), respectively. Equate now the conformal mappings (2.1) and (2.4) to each other: kz(w) = kz(w N ),

(2.5)

and differentiate both sides of the equation with respect to w N : d [kz(w)] kh dw =2 d = dN . dw N π dw N

(2.6)

Equalizing the two conformal mappings (2.5), we introduce an additional condition formulated from the physical standpoint. As was mentioned before, if the half-plate with variable kh is considered as an intermediate case between a wedge and half-plate, the curves r d0 of the variable-kh half-plate must be reduced in the limiting cases kh ≪ 1 and kh ≫ 1 to the circle |w N | = 1, which represents the curve r d0 for wedge scatterer at arbitrary values of the external apex angle of the wedge πn. Since the curve r d0 for a scatterer with a size parameter and that for a wedge have different

38 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

shapes, equality (2.5) will hold true not at all point of the curve r d0 , but at the point of their intersection; in the limiting cases kh ≪ 1 and kh ≫ 1, though, the curves r d0 for the two scatterer types will be practically coinciding. Since the relation (2.3) between w and w N is linear, we have in (2.6) dw/dw N = w/w N , therefore d⋅w π π 󵄨 󵄨 = , 󵄨󵄨󵄨w√ w(w + 2)󵄨󵄨󵄨󵄨 = . (2.7) d N ⋅ w N 2kh 󵄨 2kh Since for all factors containing the variable w linear normalization is performed according to the power exponents in the left-hand side of equation (2.7), we can relate to each other the normalized and non-normalized quantities: 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2kh 2kh 2kh 2kh 2kh 󵄨󵄨w√ √ √ √ √ w +2 = wN . (w )󵄨󵄨󵄨󵄨 = 1 = |d N | , w√ 󵄨󵄨 π π π π 󵄨󵄨 π 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨

(2.8)

Thus, having established the normalizing function from (2.3), C (kh) = √2kh/π, we obtain the relationship between w and w N . Also, using (2.8) one can obtain the intersection point of the curve r d = const and circle |w N | = 1. The intersection point is located at the midpoint of the curve r d = const, i. e., on the normal to the plate end face, because only at this point the distances to the vertices of half-plate turn out to 󵄨 󵄨 be equal: 󵄨󵄨󵄨󵄨w N + 2√2kh/π 󵄨󵄨󵄨󵄨 = |w N | = 1 = r w0 . Domains w and w N are shown together with curves r d0 in Figures 2.2 and 2.3, respectively. w

Fig. 2.2: Family of curves r d0 in domain w.

wN

Fig. 2.3: Family of curves r d0 in domain w N .

2.2 Diffraction by half-plate | 39

In Fig. 2.2, the family of curves r d0 is shown in domain w. This is the same family of curves as is shown in Fig. 2.1; the shape of the curves is changed due to conformal transformation. In Fig. 2.3, the family of curves r d0 is shown in domain w N . Normalization of the curve sizes is performed according to the normalization function C(kh). The central point of each curve r d0 is located on the circle |w N | = 1 (shown by the thick line). In the case of multiply connected curve r d = const, field interaction with the scatterer is localized in the neighborhood of the edges, therefore, they must be considered separately. When multiply connected curve becomes simply connected, the pieces of curve r d = const merge, and the curve envelopes both edges. In this case, diffraction by both curves can be considered as a single process. The shape of curve r d = const depends on the value of kh. It follows from (2.8) that for a simply connected curve r d = const with kh ≪ 1 and |w| ≫ 1, we have |w| ≅ |d| ≅ [π/ (2kh)]1/2 = r d . Evidently, after changing to the normalized variables, condition (1.5), r d = 1, must be revised in the following way: r dN = 1, or |d (kz) /dw N | = |d N | = 1, d N = d√2kh/π.

(2.9)

In this case, the condition that the derivative is equal to unity by absolute value holds true on the curves satisfying the condition r d = √ π/ (2kh). In what follows, we refer to these curves as “curves r d0 ”. The latter formula establishes the relationship between the non-dimensional size parameter kh and the value r d which determines the shape of curve r d0 . Normalization of variable w allows us to place the curve r d0 for a scatterer with a size parameter on the domain w12 correctly; at all points of this domain a solution to the diffraction problem for a wedge scatterer already exists (formulas (1.28), (1.30), (1.32), and (1.52), see also Fig. 2.4). In Fig. 2.4, incidence of a wave emitted by a point source on a wedge is shown. The thick circle corresponds to the curve r d0 for the wedge.

P

ρ z0

r0 rz0

Fig. 2.4: Incidence of a wave from a point source on a wedge.

40 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

2.2.1 Solution on the given curve r d0 Suppose that we have found somehow (for example, numerically) a solution u(r z , φ z ) to the problem of unit-amplitude plane wave diffraction by a half-plate of thickness kh P(r z , φ z ) = 1 ⋅ exp [−ikr z cos(φ z − φ0 )] .

(2.10)

The wave is incident in such a way that the end face of the half-plate is shadowed. In this case, on the surface of half-plate there exists just a single point (illuminated vertex) which belongs to the shadow boundary. We place the total field u + P on the domain w12 . Also, place on this domain the curve r d0 corresponding to the given scatterer type. Further on, find out another solution u1(r z , φ z ), corresponding to plane wave diffraction by a wedge with the external apex angle πn e P1(r z , φ z ) = A e ⋅ exp [−ikr z cos(φ z − φ0e )] ,

(2.11)

and place the total field u1 + P1 in domain w12 (Fig. 2.5). The choice of parameters n e , A e , and φ0e will be explained later.

w12

rd0 rwN

Fig. 2.5: Domain w12 , curves r d0 and r wN , ray structure of the incident and scattered fields.

In Fig. 2.5, the ray structure of scattered field is shown for diffraction by an “equivalent” wedge (to which the circle r wN corresponds). taking this solution on the curve r d0 for a half-plate, we obtain a heuristic solution. Geometrical parameters of the “equivalent” wedge are chosen in such a way that the light–shadow boundaries for the equivalent wedge and half-plate coincide. Taking the solution u1 + P1 at the points of curve r d0 , we obtain the total wave field u2 + P in the domain corresponding to half-plate and compare it with the initial solution. The scattered field u2 satisfies all condition of the boundary-value problem for half-plate, as is the case with the field u. Let the parameters n e , A e , and φ0e be

2.2 Diffraction by half-plate | 41

chosen in such a way that far away from the edge the shadow boundaries of u2 coincide with those of u, and let the total field on these boundaries be continuous. Then, if the solution for scattered field u2 deviates from the solution for scattered field u, uniqueness of the solution will be violated. Thus, we have shown that solution for a half-plate can be chosen from the family of solutions for wedge scatterers. By the “family of solutions”, we understand the set of solutions corresponding to different wedge apex angles πn and different angles of incidence φ0 . The search for wave field by its values on the curve r d0 can be formulated in the form of an integral equation. However, as the first approximation, the solution can be evaluated by the method of geometrical optics. Let us assume that in the medium with constant refraction index, a wave originating on the curve r d0 propagates inside a constant angular sector, with the amplitude and phase of the scattered field u(r z , φ z ) varying as u(r z , φ z ) = A(r z , φ z )

√ kr2z exp [ik (ρ − r2z )]

= f (φ z )

√ kr2z /kρ , exp [ik (ρ − r2z )]

(2.12)

where A(r z , φ z ) is the complex amplitude of the field on the interval of curve r d0 , 1/r2z is the curvature of this interval, 1/ρ is the curvature of wave front at the observation point, f (φ z ) is a function proportional to the field amplitude but independent of distance, i. e., remaining constant in the angular sector. We refer to the function f (φ z ) as the wave field pattern. Our task now is to find the field scattered by half-plate via the field scattered by the equivalent wedge, the latter depending on the wedge shape (i. e., parameter n e ) and parameters of the incident wave (2.11) (A e and φ0e ). If we determine the wave field by the method of geometrical optics, it is evident that after conformal mapping the exit directions of the shadow boundaries of equivalent wedge and curve r d0 must coincide with those for half-plate: φ0e + π = φ w (φ0 + π) , ne

φ0e − π = φ w (φ0 − π) , ne

(2.13)

where φ w (φ0 + π) and φ w (φ0 − π) are the exit directions of shadow boundaries on the curve r d0 in domain w12 (Fig. 2.5). Since, upon conformal mapping (2.1) all fragments of domain w are rotated by the angle φ d , condition (2.13) is reduced to finding out such point on the curve r d0 at which the following condition is satisfied: φ w + φ d = φ0 + π,

φ w + φ d = − (φ0 − π) .

(2.14)

Requiring that condition (2.14) is satisfied, one can obtain parameters n e and φ0e of the equivalent wedge. Parameter A e will be determined later.

42 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

2.2.2 Power normalization Upon the conformal mapping which transforms the solution from domain w to z, it may well occur that the length and curvature of the fragment Δl of the curve r d0 be different in the domains z and w. In this case, the field amplitude is changed on the interval Δl, which has to be taken into account. Consider wave propagation in the domain z inside a ray tube with the wave front fragments Δl1 and Δl2 orthogonal to the wave propagation direction. Then for fields A1 and A2 the following relationship holds true on the corresponding fragments: A22 Δl2 = A21 Δl1 .

(2.15)

Making use of relations (1.32) for a cylindrical wave, we write A2 kr = f (φ) ,

(2.16)

where A is the complex amplitude of the wave field, f (φ) is the wave field pattern. Pose now the condition of power conservation in the angular sector, analogous to the power conservation law in the ray tube: f2 Δφ2 = f1 Δφ1 ,

(2.17)

where f1 and f2 are the wave patterns in the domains 1 and 2, Δφ1 and Δφ2 are corresponding angular sectors. Note that relations (2.15) and (2.16) are only valid for domains where the wave number is constant, while relation (2.17) is valid as well for domains with variable k. Consider the variation of field amplitudes when the solution is transformed from domain z0 for equivalent wedge to domain w with curves w1 and w2, and then to domain z2 of half-plane. The scheme of solution transformation between different domains is shown in Fig. 2.6. The circle kz = 1/n e in domain z0 corresponds to the condition |d| = 1 for conformal mapping kz = w n e /n e . In the domain w, this curve is matched by the circle w1 : r w = 1. The curve w2 is, in fact, curve r d0 for conformal mapping (2.1) in the domains w and z. We first find the solution for the equivalent wedge on the curve w1 in domain z0 , then in the domain w we map the solution from curve w1 onto curve w2 , after which we map the domain w onto domain z2 corresponding to the half-plate. The field values on the curve w2 on the domain z are taken to be a heuristic solution for diffraction by a half-plate. Then, in the case of half-plate we obtain f z2 Δφ z2 = f w2 Δφ w2 = f w1 Δφ w1 = f z0 Δφ z0 = f z0 Δφ w1 n e , or

f z2

Δl z2 Δl w1 = f z0 n e , kr z2 kr w1

(2.18)

where f z1 is the field pattern in the domain z in the angular sector Δφ z1 for equivalent wedge, f z2 is the field patters in domain z in the angular sector Δφ z2 for half-plate

2.2 Diffraction by half-plate | 43

z2 z0

w1

w

w1

w2

w1

w2 (a)

(b)

(c)

Fig. 2.6: Correspondence between solutions in z0, w, and z2-domains.

of finite thickness, f w1 is the field patters in domain z in the angular sector Δφ w1 for half-plate of finite thickness, Δφ w1 = Δφ z1 /n e is the corresponding angular sector in domain w. As a result, we obtain for the field patterns f z2 and f z0 , with reference to Δl z2 = Δl w1 : n e r z2 f z2 = f z0 , (2.19) r w1 where 1/r w1 and 1/r z2 are the local curvatures of corresponding parts of curve r d0 . In order to determine r z2 , consider the element Δl of the curve r d0 in domains d, w, and z, denoting it by Δl d , Δl w , and Δl z . In the domain d, we have Δl d = r d0 Δφ d , while Δl w and |Δl z | are: 󵄨󵄨 dw 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨 Δl d 󵄨󵄨󵄨 = r d0 󵄨󵄨󵄨 Δl w = 󵄨󵄨󵄨 󵄨 Δφ d , 󵄨󵄨 dd 󵄨󵄨 󵄨󵄨 dd 󵄨󵄨󵄨 (2.20) 󵄨󵄨󵄨 dz 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Δl w = r d0 󵄨󵄨󵄨 dz 󵄨󵄨󵄨 󵄨󵄨󵄨 dw 󵄨󵄨󵄨 Δφ d . Δl z = 󵄨󵄨󵄨 󵄨󵄨 dw 󵄨󵄨󵄨 󵄨󵄨󵄨 dw 󵄨󵄨󵄨 󵄨󵄨󵄨 dd 󵄨󵄨󵄨 Elements Δl w and Δl z have equal lengths because on the curve r d0 we have |dz/dw| = 1. Consider now an element Δl wN of the circle w N (phase front of the signal scattered by the equivalent wedge) intersecting Δl w at its mid-point at an angle (φ wN − φ nw ), where φ wN and φ nw are the angles of normals to these elements (Fig. 2.7). In Fig. 2.7, the sector between the rays emitted from the center at angles φ wN2 and φ wN1 are shown. The difference of these angles is the divergence angle of the rays. The field amplitude corresponding to these angles, corresponds to the diffraction problem solution for the equivalent wedge (Fig. 2.5), while the phase front corresponds to the circle r wN . We take these amplitudes on the curve r d0 for the half-plate and assume that the phase front also corresponds to the curve r d0 . Therefore, we assume that the rays now leave the curve r d0 at angles φ n2 and φ n1 . According to the energy conservation law, changes in thee divergence angle cause changes in the field amplitude. The lengths of elements Δl wN and Δl w in the domain w are related by Δl wN = Δl w cos (φ wN − φ nw ). Find out the change of element Δl wN curvature upon conformal mapping Z (w). The mapping Z (w) rotates all vectors by the angle φ d , therefore,

44 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

w12

φn1 φwN1

φn2 φwN2

Fig. 2.7: Variation of the divergence angle and scattered field ray direction on the curve r d0 .

the normal to r d0 is rotated by φ z = φ w + φ d , while the difference between vector directions at the ends of element Δl is equal to Δφ z = Δφ w + Δφ d . Accordingly, Δφ nz = Δφ nw + Δφ d . Because conformal mapping is angle-preserving, the vectors at the ends of element Δl wN are rotated by the angle Δφ zN = Δφ wN + Δφ d

Δl wN = Δφ wN + Δφ d cos (φ wN − φ nw ) , Δl w

(2.21)

since the angle difference between the normals to curve r d0 at the ends of element Δl wN is proportional to the projection of element Δl wN on the curve r d0 . In turn, projection of element Δl wN on the curve r d0 , as well as projection Δl wN of the element Δl w on the circle w N , is proportional to the cosine of the angle between the normals cos (φ wN − φ nw ). The curvature radius of element Δl zN of the circle w N on the curve r d0 in domain z is r zN = Δl zN /Δφ zN = Δl wN /Δφ zN . The angular element Δφ wN , corresponding to the element Δl wN , is related to the radius r wN by Δφ wN = Δl wN /r wN . Therefore: Δl wN Δφ zN = + Δφ d cos (φ wN − φ nw ) r wN (2.22) 󵄨󵄨 dw 󵄨󵄨 r 󵄨󵄨 󵄨󵄨 d0 = (󵄨󵄨 + 1) Δφ d cos (φ wN − φ nw ) . 󵄨 󵄨󵄨 dd 󵄨󵄨󵄨 r wN Upon the conformal mapping, the difference between the angles (φ wN − φ nw ) remains constant, therefore we obtain 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Δl zN = Δl z cos (φ wN − φ nw ) = 󵄨󵄨󵄨 󵄨󵄨 󵄨 r Δφ cos (φ wN − φ nw ) . 󵄨󵄨 dw 󵄨󵄨󵄨 󵄨󵄨󵄨 dd 󵄨󵄨󵄨 d0 d Hence,

󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨 󵄨󵄨 dd 󵄨󵄨 r d0 cos (φ wN − φ nw ) Δl zN = 󵄨󵄨 dw󵄨󵄨󵄨 r 󵄨 r zN = Δφ zN (󵄨󵄨󵄨 󵄨󵄨󵄨 d0 + 1) cos (φ wN − φ nw ) 󵄨 dd 󵄨 r wN (2.23) 󵄨󵄨 dz 󵄨󵄨 1 1 Δl z 󵄨󵄨󵄨 dw 󵄨󵄨󵄨 , = + . = Δφ nz r zN r wN 󵄨󵄨󵄨󵄨 dw 󵄨󵄨󵄨󵄨 r d0 dd 󵄨 󵄨 Substituting r zN for r z2 in (2.19), we obtain the required coefficient for the field pattern when the equivalent edge solution is transformed into the solution for finitethickness half-plate.

2.2 Diffraction by half-plate | 45

Now we have to transform the expressions for field patters into those for the fields. It follows from (2.16) that A2 Δl = f (φ) Δφ. Upon the integration of total energy propagating through the given contour, the following must hold true: L

Φ

󵄨 󵄨 ∫ |A (l)| dl = ∫ 󵄨󵄨󵄨f (φ)󵄨󵄨󵄨 dφ, 2

0

(2.24)

0

where L is the total length of the contour, Φ is the total angle of field pattern. Let A (l) and f (φ) be related by A 2 (l) = A n f (φ) ,

(2.25)

where A n is a normalization constant. The same relation must as well be true for the average values: 󵄨 󵄨 (2.26) |A (l)|2 = A n 󵄨󵄨󵄨f (φ)󵄨󵄨󵄨. Write down formula (2.24) for average values the amplitude squared and field pattern: L

Φ

󵄨 󵄨 L ⋅ |A (l)|2 = ∫ |A (l)|2 dl = ∫ 󵄨󵄨󵄨f (φ)󵄨󵄨󵄨dφ = Φ ⋅ f (φ), 0

(2.27)

0

With reference to equations (2.25) and (2.26), we obtain from (2.27): A2 (l) = A n f (φ) =

Φ f (φ) . L

(2.28)

Consider equation (2.28) in the case where the curve r d0 of wedge scatterer is transformed from domain z into domain w. In this case, on the circle kr z = 1/n e with reference to dl z = kr z dφ z , where dφ z = n e dφ w is obtained independently of n e : dl z = dφ w ,

L z = πn e kr z = π = Φ w ,

A2z = f w (φ w ) .

(2.29)

As was mentioned above, for wedge scatterers the solution can be taken at all points of domain w, rather than only on the curve r d0 . In this case, we always have Φ w = π, while f w (φ w ) remains constant (at least far away from the shadow boundaries), while the amplitude A z decreases with the increase in the length of curve L z , according to equation (2.28). For other scatterer types, equation (2.28) also gives correct relationship between the amplitude and field pattern. When a solution to the problem of scattering by half-plate is sought for by the method of generalized eikonal, the field values are transformed between domains w and z twice. Firstly, the solution for equivalent wedge is transformed to domain w. Secondly, the field values on the curve r d0 are transformed from w to z. Therefore, two normalization factors of the type (2.28) appear, each related to its own field transformation. Each transformation involves a curve, the length L of which must be evaluated.

46 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

The length of circle w1 in domain z0 is equal to π. Evaluate now the length of curve w1 in domain z upon conformal mapping corresponding to the half-plate, for some particular valued of kh. It is equal to the integral z2 π π 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 L = ∫ dz (φ w ) = ∫ 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 dφ w = ∫ 󵄨󵄨󵄨 󵄨 dφ w 󵄨󵄨 dφ w 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨󵄨 󵄨󵄨󵄨 dφ w 󵄨󵄨󵄨 z 1

0 π

0 π

0

0

(2.30)

󵄨 󵄨 = ∫ |d ⋅ w| dφ w = ∫ 󵄨󵄨󵄨󵄨2w√ w (w + 2)󵄨󵄨󵄨󵄨 dφ w . Calculate now the length of curve r d0 . Integrating with respect to parameter φ d in the domain d, we obtain: z2

π

1

0 π

π 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dz 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨 󵄨󵄨 dd 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 Ld = ∫ dz (φ d ) = ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 dφ d = ∫ 󵄨󵄨󵄨 󵄨 dφ d 󵄨󵄨 dφ d 󵄨󵄨 󵄨󵄨 dw 󵄨󵄨󵄨 󵄨󵄨󵄨 dd 󵄨󵄨󵄨 󵄨󵄨󵄨 dφ d 󵄨󵄨󵄨 z 0

π󵄨 󵄨󵄨 dw 󵄨󵄨 󵄨󵄨 d3 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 dφ . = ∫ 󵄨󵄨󵄨d2 󵄨󵄨 dφ d = ∫ 󵄨󵄨󵄨󵄨 d 󵄨 󵄨󵄨 dd 󵄨󵄨 󵄨󵄨 √1 + d2 󵄨󵄨󵄨 0

(2.31)

0

Integrals (2.30) and (2.31) are evaluated numerically. Taking into account equations (2.19) and (2.28), we obtain: A2z2

Ld n e r z2 L n e r z2 = A2z0 . = f z2 = f z0 π r w1 π r w1

(2.32)

As a result, we obtain from (2.32) that the complex amplitudes of the field are related on the curve r d0 by A z2 = A z0 √

L n e r z2 √ . Ld r w2

(2.33)

Since the length of curve L in the general case is different from π, in order to obtain the given field values at these points it is necessary to alter the amplitude of the excitation field for the equivalent edge. In turn, this means that we obtained the parameter A e from (2.11): (2.34) A e = √ L/π. This factor was already taken into account in equation (2.33), however, in equation (2.34) we calculated it separately because later on it will be necessary to substitute the parameters A e and φ0e into equation (2.11).

2.2.3 Solution by method of successive diffractions (MSD) The above method was applied to calculate scattering by a finite-thickness half-plate of a plane wave incident at an angle of 40∘ . The solution obtained by MGE was compared with the approximate solution obtained by the method of successive diffractions

2.2 Diffraction by half-plate | 47

(MSD) which takes into account secondary reflection from the end face of the wave scattered by the illuminated edge. First, a cylindrical wave A1 resulting from diffraction of the primary plane wave by the wedge with n = 3/2 and apex coinciding with the illuminated vertex was found. Then, a cylindrical wave A2 resulting from diffraction of the cylindrical wave A1 by the wedge with n = 3/2 and apex coinciding with the shadow vertex was calculated. And finally, a cylindrical wave A3 was calculated which results from diffraction of the cylindrical wave A2 by the wedge with n = 3/2 and apex coinciding with the illuminated vertex. The cylindrical waves A1, A2, and A3 were calculated by formula (1.52). In order to calculate A1, substitute r z and r0 for a plane wave into (1.52). This substitution results in the well-known formula (1.57) derived in the method of edge waves [1]. To obtain A2 and A3, substitute into (1.52) r z and r0 for a cylindrical wave. Also, apply equations (1.46) and (1.51) to determine S c (ŵ sm ) and S c (w0 ). In this case S c (ŵ sm ) − S c (w0 ) = k [r z + r0 − √ r2z + r20 − 2r z r0 cos (φ z − φ0 )] .

(2.35)

If the source is located at some vertex, and diffraction occurs at some other vertex, we have to substitute into equation (2.35) r0 = h, φ0 = ±π/2. For a plane wave, the difference of eikonals (2.35) is reduced to expression (1.54). The saddle points ŵ sm belong to the curves r d0 of illuminated and shadow vertices (wedges with n = 3/2); these curves are circles of radius kr z = 1/n = 2/3 centered at the corresponding vertices. Coordinates of the saddle points were obtained from equation (1.49). Thus, in accordance with (1.9), to obtain v(w0 ) in the form (1.28) we need to evaluate two integrals of the type (1.52). For the excitation field in the form of unit plane wave (1.30) and (1.31), we obtain: A1 (r z , φ w ) =

sin πn −2 n cos π − cos n

ψ n

cos

ψ e−ikr z cos ψ 2 √πi (2.36)

√2kr z cos(ψ/2)

×

∫

exp (iq ) dq, 2

∞ cos(ψ/2)

where ψ = 3φ w /2 − φ0 . Evidently, the argument ψ depends on φ w in the domain w12 . This angle varies in the range −π < φ w < π. A1 (r z , φ w ) = A1 (ŵ sm ) is the total field after scattering of the primary plane wave by the illuminated edge. It exists only beyond the wedge connected with the illuminated edge.

48 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal Then, we find out diffraction of field A1 (ŵ sm ) by the shadow edge: 2i√ S c (ŵ sm ) − S c (w0 ) ̂ P (ŵ sm ) ŵ sm /n P c (w) 1 d ŵ ≅ ∫ c c 2πi ŵ − w0 z h ŵ sm − w 0 exp [iS ( ŵ sm ) − iS (w 0 )] √ 2πikr sm r +h z

(2.37)

√S c (ŵ sm )−S c (w0 )

×

∫

exp (iq2 ) dq.

∞√S c (ŵ sm )−S c (w0 )

Substituting into this formula the excitation field in the form P (ŵ sm ) = A1 (ŵ sm ) twice (for the upper and lower half-planes in the domain w12 ), and then substituting the results into (1.28) and (1.9), we obtain the total field A2 (ŵ sm ) after scattering by the shadow edge: A2 (ŵ sm ) = U (ŵ sm ). This field exists only in the domain external to the wedge related to the shadow edge. And, finally, substituting P (ŵ sm ) = A2 (ŵ sm ) twice into (2.37) and then substituting the results into (1.28) and (1.9), we obtain A3 (ŵ sm ) = U (ŵ sm ). This result is also applicable only in the domain external to the wedge related 󵄨 󵄨 to the illuminated edge; we, however, take it in an even narrower angle range 󵄨󵄨󵄨φ z 󵄨󵄨󵄨 < π/2. In this case, solution becomes continuous on the shadow boundary related to the illuminated edge. The saddle points ŵ sm are located at the intersection of the shadow boundary related to the shadowed vertex and of the circle centered at this vertex and passing through the observation point. For example, if the observation point is located at a distance r z from the illuminated vertex, and the distance from it to the shadow vertex is r2z , then upon diffraction by the shadow vertex of the wave from the source located at the illuminated vertex, the saddle point is located at the distance r2z + h from the shadow vertex, i. e., its angular coordinates are (r2z + h, 3π/2). For diffraction by illuminated vertex of the wave from the source at the shadow vertex, the angular coordinates of the saddle point are (r z , π/2). When applying formulas (2.36) and (2.37), we took into account whether the observation point is located in the upper or lower half-plane w12 (or, equivalently, on which of the two sheets of domain z).

2.2.4 Results of calculations In Figures 2.8a–e, the total field amplitudes on curve r d0 are compared for the solutions obtained by method of generalized eikonal (MGE) and method of successive diffractions (MSD) for different values of kh. The solutions are compared for the case where the wave is incident on a thick halfplate at an angle φ0 = 40∘ with respect to the horizontal axis. The left panels of the figures show the amplitudes of the solutions to the problem of diffraction by half-plate obtained by the method of generalized eikonal (MGE, thin solid lines) and method of successive diffractions (MSD, dashed lines); also shown are solutions to the problem of

2.2 Diffraction by half-plate | 49

1.2

kh = 1.571

2 1.8

1

1.6

MSD

0.8

1.4

MGE

1.2

0.6

HP

1 0.8

0.4 0.2

dU/dn = 0 MGE MSD

0.6 HP

U=0

0.4 0.2

0 (a)

–360 –180

1.2

0 ϕj

180

360

kh = 1.2

0

0

90

180 ϕj

270

360

90

180 ϕj

270

360

90

180 ϕj

270

360

2 1.8

1

1.6 1.4

0.8

1.2 1

0.6

0.8 0.4

0.6 0.4

0.2

0.2 0

–360 –180

(b) 1.2

0 ϕj

180

360

kh = 0.8

0

0

2 1.8

1

1.6 1.4

0.8

1.2 0.6

1 0.8

0.4

0.6 0.4

0.2

0.2 0 (c)

–360 –180

0 ϕj

180

360

0

0

Fig. 2.8: Comparison of total field amplitudes for various kh values (see legend in the upper row).

50 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

1.2

kh = 0.4

2 1.8

1

1.6 1.4

0.8

1.2 0.6

1 0.8

0.4

0.6 0.4

0.2

0.2 0 (d)

–360 –180

1.2

0 ϕj

180

360

kh = 0.04

0

0

90

180 ϕj

270

360

90

180 ϕj

270

360

90

180 ϕj

270

360

2 1.8

1

1.6 1.4

0.8

1.2 1

0.6

0.8 0.4

0.6 0.4

0.2

0.2 0 –360 –180 (e) 1.2

0 0 ϕj

180

360

kh = 4 × 10‒3

0

2 1.8

1

1.6 1.4

0.8

1.2 0.6

1 0.8

0.4

0.6 0.4

0.2

0.2 0

–360 –180

(f)

Fig. 2.8 (continued)

0 ϕj

180

360

0

0

2.2 Diffraction by half-plate | 51

scattering by half-plane (HP, dotted lines). In the region z, the observation angle φ z is varied from −360° to 360° (i. e., this angle is considered in the double-sheeted region). The right panels of the figures show the corresponding solutions for two polarizations: U = 0 (the lower group of curves) and dU/dn = 0 (the upper group of curves). The size parameter kh takes the values π/2 (h = λ/4), 1.2, 0.8, 0.4, 0.04, and 0.004. It is seen that initially the MGE and MSD solutions coincide as the size parameter decreases. Then, as the size parameter decreases, the MSD solution stabilizes to form a pattern having the shape different from the regular one, whereas the MGE solution approaches the solution for a half-plane. In Fig. 2.9, results obtained by our heuristic formulas are compared with the literature data. Decimal logarithms of the reflection coefficients at the shadowed end face of the plate are plotted for plane TE-wave (having electric field vector normal to the edge) incident at an angle φ0 = 0° on a perfectly conducting half-plane. The results obtained by MGE and MSD methods (denoted by LRMGE, thick solid line, and LRMSD, thin solid line) are compared with those presented in [65]. In this work, the decimal logarithms of reflection coefficients were calculated by solution expansion in direct (denoted by LRR, dashed line) and reciprocal (denoted by LRA, dash-anddot line) powers of quantity q = h/λ (kh = 2πq). These expansions are overlapped near q ≅ 0.25. The results were compared in the far zone, i. e., away from the curve r d0 ; MSD formulas were applied directly, while MGE solution away from the edge was found by formula (2.12). Comparison of the solutions has shown that the asymptotic formula derived by MSD method in the form of expansions following from MGE is valid for any values of q, though it gives correct field amplitudes on the curve r d0 only for kh > 0.8. This, however, corresponds to q = 0.127, which is much smaller than 0.25. Therefore, –1.1

–1.2 LRMSD –1.3

LRA LRMGE

–1.4 –1.45 0

LRR 0.25

0.5 q

0.75

1

Fig. 2.9: Decimal logarithms of reflection coefficients for plane wave reflection from the shadowed end face of a half-plate.

52 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

MGE allows one not only to obtain solutions to new diffraction problems, but also to construct asymptotic solutions to known problems valid in a wider parameter range. At the same time, the solution obtained directly by MGE agrees satisfactorily with the series expansion containing powers of q when q < 0.25 (kh < π/2). For the size parameter kh > π/2, the solution obtained by MGE becomes invalid because the curve r d0 becomes doubly connected. On the other hand, splitting of curve r d0 into two parts plays a positive role as well because it gives a clear physical interpretation to diffraction phenomena. Diffraction is incipient on the light–shadow boundary line located on the scatterer surface. The energy of corresponding linear source is equal to the energy scattered by the equivalent wedge. As long as the curve r d0 encircling the linear source is simply connected, diffraction process can be viewed as “energy flux redistribution” over the curve r d0 which encircles the linear source generating this energy. Within the curve r d0 , the ray tubes along which energy propagates can be curvilinear, while outside this curve they become rectilinear. When the curve r d0 becomes multiply connected, diffraction must be considered separately for each part. Considering the iso-lines of conformal transformation derivatives in the neighborhood of various scatterers, one can determine the optimal approach to studying diffraction by these structures. Thus, for kh < π/2 one can rely on the MGE solution, while for kh > π/2 the MSD solution works satisfactorily. These solutions overlap for 0.8 < kh < π/2. For any value of the size parameter, one of these solutions coincide with known data. Therefore, solutions obtained by MGE are also applicable to other diffraction problems. In Fig. 2.10 (a–d), the total fields calculated in single-sheet and two-sheet domains are presented. Decimal logarithms of scattered field amplitude are plotted for MGE solution continued from the curve r d0 to the far zone by formulas analogous to (2.12), kh = 1.2

krz0 = 200 –0

–0

–0.5

–0.5 LHGEj

LEGEj LESDj

–1

LHSDj

–1

LHHPj –1.5

LEHPj –1.5

∙∙∙∙

∙∙∙∙

LEHKj –2 .....

LHHKj –2 ..... –2.5

–2.5 –3

(a)

0

60

120 180 240 300 360 ϕj

–3

0

60

120 180 240 300 360 ϕj

Fig. 2.10: Scattered field amplitudes in the far zone for different values of kh.

2.2 Diffraction by half-plate | 53

kh = 0.8

krz0 = 200 –0

–0

–0.5

–0.5 LHGEj

LEGEj LESDj

–1

LHSDj

–1

LHHPj –1.5

LEHPj –1.5

∙∙∙∙

∙∙∙∙

LEHKj –2 .....

LHHKj –2 ..... –2.5

–2.5 –3

0

60

(b)

–3

120 180 240 300 360 ϕj

kh = 0.4

120 180 240 300 360 ϕj

60

120 180 240 300 360 ϕj

60

120 180 240 300 360 ϕj

–0 –0.5

–0.5 LHGEj

LEGEj –1

LHSDj

–1

LHHPj –1.5

LEHPj –1.5

∙∙∙∙

∙∙∙∙

LEHKj –2 .....

LHHKj –2 ..... –2.5

–2.5 –3

0

60

(c)

–3

120 180 240 300 360 ϕj

kh = 0.04

0

krz0 = 200 –0

–0

–0.5

–0.5 LHGEj

LEGEj LESDj

60

krz0 = 200

–0

LESDj

0

–1

LHSDj

–1

LHHPj –1.5

LEHPj –1.5

∙∙∙∙

∙∙∙∙

LEHKj –2 .....

LHHKj –2 ..... –2.5

–2.5 –3

0

60

(d) Fig. 2.10 (continued)

120 180 240 300 360 ϕj

–3

0

54 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

with the curvature of fragment 1/r2z variation with distance from the shadow boundary taken into account (diagrams for E and H-polarizations denoted by LEGE and LHGE and shown by solid lines on the left and right plates, respectively). Also, similar plots are shown for the solution obtained by method of successive diffractions (LESD and LHSD, dashed lines), solution obtained for half-plane (LEHP and LHHP, big dots), as well as solution for half-plane obtained by Kirchhoff’s method (LEHK and LHHK, small dots). Polarizations E z and H z correspond to the vector components directed, upon normal wave incidence, along the edge (oriented along the z-axis), they are sometimes referred to as TH and TE-polarizations (because in this case the H⃗ and E⃗ vectors of the incident wave are perpendicular to the edge). Diagrams were obtained at a distance kr z = 200 from the edge, with kh equal to 1.2, 0.8, 0.4, and 0.04 (see legends on the plots). From the analysis of field patterns, we can conclude that MGE describes scattering process correctly. Discrepancy between the results near the shadow boundary can be eliminated applying special approaches (for example, analogous to method of edge waves) in the course of the solution. A common feature of MGE and MSD methods is that in both methods solution is obtained by processing the first approximation which is the solution describing diffraction by a wedge having its apex at the illuminated edge. However, the first approximations used and the ways in which they are processed are different between the methods. In MGE, the first approximation is the solution for an equivalent wedge with 3/2 < n e < 2 defined at all points of the curve r d0 . Its processing in MGE is performed by field integration in the far zone. This operation can be carried out approximately by continuing the field into the far zone according to the same law as that describing field decay with distance in the far zone in the case of diffraction by a wedge, with the curves r d0 being circles (equation (2.12)). In MSD, the first approximation is the solution for a wedge with external apex angle πn = 3π/2. The signal scattered by this wedge is diffracted successively by the shadow and illuminated vertices. To describe diffraction by wedge, in both MGE and MSD methods the first term if asymptotic formula obtained by the method of stationary phase is used. Subsequent terms decrease proportionally to inverse powers of kR, where R is the distance from the edge to observation point in domain z. The curve r d0 is located close to the edge, therefore, the field on it will be described accurately neither in MGE or in MSD. However, in most practically important problems, of interest is the field behavior at large distances from the scatterer. As the distance to the edge increases, the field will more and more correspond to the first term of asymptotic formula, and the accuracy will be increasing for both MGE and MSD methods. It can be shown that the integral representation used in MGE is a generalization of the classical integral representation which was used by Sommerfeld for construction of diffraction problem solutions for half-plane and wedge. Consider now the similar-

2.2 Diffraction by half-plate | 55

ities and differences between two solutions for diffraction by semi-infinite scatterers: the Sommerfeld solution, and the solution obtained by method of generalized eikonal. These solutions can be regarded similar because they both were obtained as integral representations on a multiple-sheet surface. For wedge scatterer, these representations coincide. Sommerfeld’s solution is heuristic, it is valid for a wedge scatterer with the external apex angle of the wedge πn on multiple-sheet Riemann surface, with n being a rational number n = l/m, where l is the number of sheets of the surface, and m is an integer number. For example, for the ambient space of a rectilinear wedge l = 3 and m = 4. The heuristic factor of the integrand is a 2πl-periodic function. Further development of Sommerfeld’s solution allowed it to be extended to arbitrary numbers l (including non-integer). Substantiation of Sommerfeld’s solution was achieved by separation of variables. Nevertheless, it is applicable to wedge-shaped scatterers only. MGE solution consists of a sum of integrals over the contour sides on the complex plane. Its integral representation is obtained by very simple mathematical and physical principles. It follows naturally from the simplest form of Cauchy residue theorem (first-order pole), the closed contours about the illuminated part is physically clear. Integral representation of MGE for semi-infinite scatterers of any shape is derived in twosheet auxiliary domain (full complex plane) connected with the shape of the scatterer via conformal mapping function. The boundary conditions for semi-infinite scatterers are satisfied rigorously on the whole boundary (horizontal axis), and only on it. Thus, this solution is much more general that Sommerfeld’s one, but has equally simple form. The price we pay for the generality and simplicity is that the MGE solution is only valid on some particular curve, whereas Sommerfeld’s solution is valid in the whole space. Nevertheless, by applying simple mathematical transformations, we can use MGE solution in the whole space as well (albeit, approximately). Additionally, MGE solution is substantiated better than Sommerfeld’s one. MGE solution can be used, together with other approaches, to substantiate the latter solution. This substantiation applies not to the final result, but to the aspects which were proposed by Sommerfeld from heuristic considerations. In particular, the form of integrand function can be substantiated. MGE takes a new view of the diffraction process physics by introducing the curve r d0 which separates the domains of “curved” and “rectilinear” eikonals. The fact that the critical circle r d0 exists in reality and has small radius in the domain z (for half-plane kz = 1/2, z = λ/ (4π)), is confirmed indirectly by the results obtained in [66]. The main bending of energy flux density lines near the edge occurs inside the circle having the radius approximately equal to the value given above. Of interest is to establish the class of scatterers for which the scattering problem can be solved by the proposed method, as well as to substantiate mathematically how rigorous are the solutions obtained. Such studies, however, are beyond the scope of the current work.

56 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

2.3 Diffraction by a truncated wedge In this section, we apply several heuristic approaches in order to derive an analytical solution for diffraction by a canonical 2D structure, truncated wedge, which is a semiinfinite wedge with its tip cut off (Fig. 2.11). Such a structure can be considered as an intersection of two semi-infinite wedges that are located at the vertices, the sides parallel to the sides of the truncated wedge forming the corresponding vertex. In this case, the common fragment corresponds to the edge surface of the truncated wedge. Alternative names of the structure (thick screen and a pair of combined wedges) can be found in literature. Similar problems are solved when diffraction by a pair of half planes [20] or a pair of wedges [21–23] is considered. For the analysis of diffraction by two vertices, a special function (generalized Fresnel integral considered in detail in [55]) is used in [20–23]. Note that the generalized Fresnel integral is studied in [55] in application to the theory of scattering of radio waves above the point of a stepwise variation in the ground electric properties, rather than as an element of the problem of diffraction by two wedges. In Fig. 2.11, truncated wedge geometry with the external angles πα and πβ at the vertices is shown. The length of the end face is equal to kh. The light–shadow boundaries of the incident and reflected fields are shown by hatching. The additional light–shadow boundaries are directed along the continuations of the end face of the plate.

P P2

πα P1

kh πβ

Fig. 2.11: Problem formulation for diffraction by a truncated wedge.

A half plate, which is also known as a half-plane with finite thickness or thick semiinfinite plate, is a particular case of the truncated wedge. First studies of such a structure were performed by different authors and research groups [14, 24, 65]. Note that complexity of the problem depends on complexity of the configuration being studied. With respect to increasing complexity, the problems can be ranked in the following way: diffraction by a single half plane, diffraction by a pair of half planes,

2.3 Diffraction by a truncated wedge

|

57

diffraction by a pair of isolated wedges, diffraction by a pair of non-parallel wedges, and diffraction by half plate. Diffraction by a 2D truncated wedge is an important canonical problem. Such structures are often used as fragments of objects with complicated configurations. Studies of truncated wedge also have quite a long history [25–28]. Below, we demonstrate that the problem of diffraction by a truncated wedge is more complicated than the problem of diffraction by half-plate. The previous discussion on the advantages and drawbacks of various approaches to the solution of diffraction problems fully applies to the case of truncated wedge. Despite substantial successes achieved, derivation of an analytical solution for the diffraction by a truncated wedge is a topical problem, since the corresponding results are necessary for researchers and engineers. An additional advantage of heuristic approaches lies in the fact that the physical principles can be used to interpret and illustrate terms of mathematical formulas and to perform individual analysis of such terms. Note that the solution can be studied in the range where the size parameter (distance between the vertices) uniformly tends to zero. Such an advantage is provided by the recently developed method of generalized eikonal (MGE) [50–53, 72, 75, 79, 80]. In this section, the basics of this method are presented, mainly according to the lines of work [54].

Basics of the method of generalized eikonal For the truncated wedge, the solution is constructed by the method of generalized eikonal in the same way as for half-plate. In accordance with the MGE method, the integral representation of the field that is uniform in domain w12 is constructed in the domain of complex variable ŵ that represents the analytical continuation from curve r d0 to the complex domain. Substitution of coordinates of variable ŵ for coordinates of variable w in functions P and S ̂ and S(w) → changes the coordinate dependencies of these functions P(w) → P c (w) ̂ as well as their properties. The eikonal S(w) is transformed to the generalized S c (w), ̂ which gives the name to the method. eikonal S c (w), ̂ satisfy Change of variables w → ŵ must be such that functions P(w) and P c (w) the following conditions: ̂ = P(w) must be satisfied; a) on the curve r d0 conditions ŵ = w and P c (w) b) function P(w) must satisfy the wave equation (1.4) on the domain w; c) function

̂ P c (w)

(2.38)

must be analytical on the domain w.̂

Using the Cauchy theorem on residues, we construct the integral representation of solution on the domain ŵ for the scattered field at the observation point w0 (1.7). In the presence of two shadow boundaries (for the incident and reflected waves) and

58 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

two convergence parts s1 and s2 , we obtain the general formula (1.9). Evaluating the integrals (1.9) by the method of stationary phase, we obtain (1.28). Provided that the change of variables w → ŵ satisfying conditions (2.38) is available, one obtains the scattered field in the form (1.28) which describes the diffraction problem solution for a wide class of scatterers.

Diffraction by a wedge Applying the method of generalized eikonal, one can obtain the known solution for diffraction by a wedge with the external apex angle πn. The conformal transformation and its derivative (1.5) are equal (by analogy with (1.34)): kz = w n /n,

d (kz) /dw = w n−1 ,

n−1 󵄨 󵄨 n . r d0 = 󵄨󵄨󵄨󵄨w n−1 󵄨󵄨󵄨󵄨 = r n−1 w = (nkr z )

(2.39)

The family of curves r d0 is transformed into the family of circles r w0 = const. Variable φ w is the angular coordinate of a point on the circle r w0 = const in the domain w12 . For the integral representation, we employ a complex variable φ cw , but the final formulas contain only coordinates of points on the curve r d0 , which is common for domains w and w.̂ Coordinates of these points are real-valued numbers. For the wedge with conformal mapping (2.39), it is possible to find the relationship between variables ŵ and w satisfying conditions (2.38) and to obtain the scattered field. Transforming the general expression (1.28), we find the scattered field on the curve r d0 in the case of two saddle points w s1 and w s2 (corresponding to the shadow boundaries of the incident and reflected waves), see (1.28): P (w sm ) √ 2πikr z0 r 0 / (r z0 + r 0 ) m=1,2

v (w0 ) ≅ ∑

2i√ S (w sm ) − S (w0 ) ŵ sm /n × ŵ sm − w0 exp [iS (w sm ) − iS (w0 )]

√S(w sm )−S(w0 )

∫

(2.40) exp (iq2 ) dq.

∞√S(w sm )−S(w0 )

In this expression, the saddle points w sm = r w0 exp[i(φ0 ± π)/n] and observation point w0 = r w0 exp(iφ w0 ) = (nkr z0 )1/n exp(iφ z /n) are located on curve r d0 . The source polar coordinates in z-domain are (r0 , φ0 ). The observation point polar coordinates in z-domain are z0 = (r z0 , φ z0 ), while in w-domain they are w0 = (r w0 , φ w0 ). Dimensionless (i. e., multiplied by k) eikonals at the saddle points S (w sm ) and observation point S (w0 ) are written as S (w sm ) = k (r z0 + r0 ) , S (w0 ) = kρ = k√(r z0 + r0 )2 − 2r z0 r0 [1 + cos (φ z0 − φ0 )], the values entering these formulas are shown in Fig. 2.4.

(2.41)

2.3 Diffraction by a truncated wedge |

59

Symmetrization of solution We consider equation (2.40) for the field scattered by a wedge on an infinite twosheeted domain w12 with variable wave number. This equation is valid on almost the whole domain z (and almost at any point of w12 ) except for a small neighborhood of the illuminated vertex. This neighborhood is located inside the curve r d0 . If no special precautions are taken, solution (2.40) has discontinuities. In the twosheeted domain w12 , the angular coordinate of observation point φ w0 ranges from 0 to 2π. By setting certain boundaries of this interval, we can make solution (2.40) continuous. To this end, we reckon the angular coordinate of observation point φ w0 from the direction of the incident field, so that the angle φ w0 ranges from φ0 /n − π to φ0 /n + π. The corresponding angle φ z is expressed as angle φ z = nφ w . Substitute it in formula (2.41); solution (2.40) will become continuous in domain w12 , only its derivative will exhibit discontinuity in directions φ0 /n + π and φ0 /n − π. Both directions correspond to the same spatial ray in the lower half-plane w2 of domain w12 . We refer to the above procedure for determination of the starting and ending points φ w0 as “symmetrization” of solution.

2.3.1 Schwarz–Christoffel integral In the theory of complex variable, the Schwarz–Christoffel integral provides the conformal mapping z(w) of upper half plane w in domain z (external domain of a 2D semi-infinite scatterer with vertex angles πα and πβ (Fig. 2.11)). The conformal mapping and its derivative are given by w

kz(w) = C ∫ w α−1 (w + 2)β−1 dw + C + 1, w0

dkz(w) = Cw α−1 (w + 2)β−1 . dw

(2.42)

Here, C and C1 are constants that depend on the system configuration. By applying expression (2.42), one can obtain formula (2.1) for a half-plate. Consider a structure with two vertices fixed in domain w12 at points 0 and −1, and assume that a size parameter (distance between the vertices) in the domain z is kh. Then, the condition kh = |kz(−1)| must be satisfied. For a half-plate, we have α = β = 3/2, in which case the integral in (2.42) can be evaluated to yield an explicit representation of conformal mapping and its derivative. For the half-plate with thickness kh, the conformal mapping and derivative are given by equations (2.1) and (2.2). Curves r d0 for half-plate in the domain z are presented in Fig. 2.1. In order to put the curves r w0 for the wedge and curves r d0 for the half-plate on the same domain w12 , we introduce linear normalization (similar to that in (2.3)): w√2kh/π = w N ,

r d = √ π/ (2kh).

(2.43)

60 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal Figure (2.3) demonstrates domain w N with the circle r wN = 1 and normalized curves r d0 . As was mentioned before, the circle r wN intersects curves r d0 at the middle point. Figure 2.5 shows the curve r d0 and system of circles r w0 in domain w. The thick line shows the circle r d0 = r wN = 1. We perform the symmetrization of solution along the dashed straight line. Curves r d0 are concentrated in the vicinity of vertices in domain z, while in domain w they are concentrated in the vicinity of points 0 and –1 on the horizontal axis. We construct the solution for diffraction by half plate using the known solution for the wedge. We assign the total field values from the solution to the boundary-value problem for the equivalent wedge on the curve r d0 in the normalized domain w12 (Fig. 2.5). The total field amplitude is taken with the correcting coefficient sN which compensates for variation in the curvature of corresponding fragment of curve r d0 upon the conformal mapping. The correcting coefficient is given by V z2 = V ze sN,

sN = √ n e r z2 /r w2 ,

(2.44)

where V z2 and V ze are amplitudes of functions V(1.9) for the half-plate and equivalent wedge with external apex angle πn e , respectively. Function sN characterizes the effect of the curvature of curve r d0 on the field pattern (1/r w1 and 1/r z2 are the local curvatures of the corresponding fragments of curve r d0 upon the conformal mapping of the curve from domain w to domain z).

2.3.2 Features of solution for half-plate Each term on the right-hand side of equation (2.40) contains four multiplicands. The first multiplicand is equal to the product of the field at the saddle point (at the shadow boundary) and the factor which describes the dependence of solution on the distances between the source and the point of observation. The second multiplicand is the half of diffraction coefficient with a period of 2π in domain w12 . The total diffraction coefficient for a particular type of polarization can be obtained by adding or subtracting the values of this factor at points w0 symmetric about the horizontal axis in domain w12 . The third and fourth multiplicands are the products of the Fresnel integral and its asymptotic expressions. They characterize the dependence of the field on the angular distance from the shadow boundary. For the field scattered by half-plate (in contrast to that scattered by the wedge), equation (2.44) contains the fifth multiplicand, sN, which characterizes the variation of field amplitude due to the variation in curvature of curve r d0 upon the conformal mapping. Comparing the solutions for two wedges, we see that they are different only by the second multiplicand. All other multiplicands remain unchanged in domain z for all types of wedges and do not depend on the conformal mapping. Contrary to this, the second multiplicand remains unchanged, accurate to a constant factor 1/n for all types of wedges in domain w12 . We call this multiplicand the “amplitude” factor, while

2.3 Diffraction by a truncated wedge |

61

the multiplicands that remain unchanged in domain z will be referred to as “ray” factors. These factors are related to the spatial congruences of rectilinear rays (diverging from the vertices), and the amplitude factor provides the additional amplitude. Thus, we obtain a solution for the half plate in domain z by taking the solution for the equivalent wedge on curve r d0 in domain w12 , multiplying it by function sN, and performing conformal mapping of domain w12 . We apply formulas (1.10) and (1.11) in order to satisfy the boundary conditions. If we take the solution (2.44) for half-plate and trace the rays that correspond to the equivalent wedge, we observe that these rays are “bending” in domain z. On the one hand, the rays must be rectilinear in the domain with constant wave number. On the other hand, the diffracted power flux is bent around the scatterer due to the field superposition. The ray bending also takes place in the domain with variable wave numbers. The rays are deflected to (from) the regions with greater (smaller) wave numbers. Using the heuristic solution, the conformal mapping, and the non-uniformity of wave number, which follows from the mapping, we simulate the power streamlines in the vicinity of the scatterer. The power flow around the scatterer also exists in the rigorous solution due to superposition of the scattered field and a part of the incident geometrical optics field (in particular, a part of the field that passes by or is reflected by the scatterer) on the light–shadow boundary. Note that the fields interfere in domain z with constant wave number. Simulation of a physical process with the aid of another physical process is not mathematically rigorous and does not guarantee accuracy of the solution. However, we demonstrate that such an approach (i) provides sufficient accuracy and (ii) makes it possible to describe the feature of solution that cannot be analyzed using alternative approaches. In particular, we can analyze behavior of the solution as the size parameter kh tends to zero. Expression (2.40) contains two terms that correspond to two saddle points, s1 and s2. These terms differ only by amplitude factors. We denote the sum of these terms over the two saddle points by pN (with constant factor i omitted). The explicit expression for pN is (see also (1.56)): pN =

w s1 w s2 − = w s1 − w0 w s2 − w0 cos

i sin π n

π n

− cos

φ z0 −φ0 n

.

(2.45)

This expression is 2π and 2πn-periodic in domains w12 and z, respectively. Here, 2πn is the external apex angle of the truncated wedge. The singularity at φ z0 = φ0 ± π (i. e., at the shadow boundaries) in expression (2.40) for the field scattered by the wedge is compensated for by the zero difference of eikonals S(w sm ) − S(w0 ). The difference of eikonals is zero only at the shadow boundary and nowhere else in domain z. We define function pN for the equivalent wedge. For other scatterer types, formal substitution of pN from (2.45) may lead to uncompensated singularities in domain z. To eliminate such singularities, we must apply symmetrization to function pN and fix the value of this function on curve r d0 in domain w12 . Then, upon conformal mapping

62 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

transforming the curve r d0 with fixed values of pN from domain w12 to domain z, no uncompensated singularities will arise. Function sN given by (2.44) characterizes the effect of “bending” rays on the field pattern. In turn, curvature of these rays results from the variation in curvature of curve r d0 due to conformal mapping. It is the variation of curvature of curve r d0 describes the diffraction by additional shadow boundaries, which are absent for the equivalent wedge and are observed for the truncated wedge (the lines passing to both sides along the end surface in Fig. 2.11). We do not introduce additional shadow boundaries for the solution using the equivalent wedge. However, the above analysis shows that diffraction by the additional boundaries is automatically taken into account. The above solution with curved rays obtained using MGE and the Fresnel integral (involved in the solution for the equivalent wedge), is referred to as the MGE-FI solution

2.3.3 Solution by method of successive diffractions We use the method of successive diffractions (MSD) for the verification of heuristic solution for diffraction by half plate. The MSD solution represents the result of double reflection of the scattered field (2.40) from the vertices of scatterer. First, we describe the incident field diffraction by the illuminated vertex. Then, we consider the diffraction by the shadow vertex of the secondary source located on the illuminated vertex. Finally, we describe the diffraction of the secondary source located on the shadow vertex by the illuminated vertex. We can consider MSD an alternative procedure for filling the domain w12 with a uniform field. The MSD solution has additional shadow boundaries on the lines that extend the end surface to both sides. Note the absence of curved rays in the MSD solution. Comparing the MGE-FI and MSD solutions, we see that both approaches possess advantages and disadvantages. The advantages of MGE-FI solution are: (i) it employs the solution for the equivalent wedge with only two shadow boundaries, (ii) it works well when the size parameter tends to zero and makes it possible to obtain the solution when two vertices merge into a single one. The disadvantage of the MGE-FI solution lies in the fact that it employs the solution with curved rays, which indicates violation of the wave equation and absence of mathematical rigor. This circumstance may lead to inaccuracy of the heuristic solution. To eliminate the disadvantage, we can compare the solution with a more rigorous solution. The MSD solution is more mathematically rigorous. The disadvantage of the MSD solution lies in its inadequacy when the size parameter tends to zero. The MGE-FI and MSD solutions are in good agreement with each other for diffraction by half plate.

2.3 Diffraction by a truncated wedge

|

63

Diffraction by a truncated wedge The solution for diffraction by truncated wedge is more complicated in comparison with diffraction by half plate. Note that the Schwarz-Christoffel integral (2.42) is not expressed explicitly as was the case for half plate (expressions (2.1) and (2.2)). When the solution for the truncated wedge is obtained using the method similar to the one used to construct the solution for the half plate, we do not obtain good agreement with the test MSD solution for all values of α and β. Therefore, for truncated wedge diffraction by the shadow vertex is not described adequately. To resolve this problem, we introduce two phenomenological coefficients ksn and ka that modify the function sN. As a result, expression (2.44) takes the form V z2 = V ze sN ksn ka.

(2.46)

One of the phenomenological coefficients (ksn) is a constant power exponent, the other one (ka) is a constant factor. The phenomenological coefficients ksn and ka are tuned so that the MSD and MGE-FI solutions coincide at relatively large values of the size parameter kh (kh = π/2). It is interesting that for a half-plate it is not necessary to change the solution by phenomenological coefficients. As became clear upon the analysis of phenomenological coefficients ksn and ka for a truncated wedge, these coefficients remain constant when the size parameter decreases from kh = π/2 to kh = π/4. After that, MSD solution becomes invalid. Maintaining the same values of phenomenological coefficients, we obtain the MGE-FI solution for the size parameter kh taking smaller values, down to zero.

2.3.4 Principles for the construction of heuristic solutions for diffraction by truncated wedge In this example, we demonstrate the procedure for the construction of heuristic solutions. The main principles of this approach are: generalization of the solution valid for one type of scatterers to a wider class of scatterers, comparison of the solution with the rigorous solution, and refinement of the solution. Then, the steps of the procedure are repeated. We employ the solution for the wedge to obtain the solution for the half plate. Then, we compare the solution with the MSD solution and specify the solution for the half plate using function sN. Then, we use the solution in the problem of the diffraction by the truncated wedge. We compare the new solution with the MSD solution and tuned it by using phenomenological coefficients ksn and ka. Despite the fact that the phenomenological coefficients are not mathematically substantiated, only the MGE–FI approach makes it possible to describe the solution behavior when the vertices are approaching each other and merging. The reason for the application of the phenomenological coefficients lies in the fact that the above heuristic approach does not allow the correct solution for the diffraction by the

64 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

shadow vertex. The methodical substantiation for introduction of the phenomenological coefficients is that, within this heuristic approach, it provides the fastest and sufficiently accurate procedure which enables good agreement with the test solution to be achieved. It may well be that the phenomenological coefficients can be eliminated if a more appropriate function r d0 (kh) than (2.3) and (2.43) is found. We hope that the analysis of the behavior of phenomenological coefficients for the diffraction by the truncated wedge under various conditions will make it possible to reveal some unknown diffraction features which could be used for the construction of new heuristic models. Thus, the MGE–FI approach allows us to obtain the solution for the diffraction by truncated wedge that is uniformly valid when the size parameter kh tends to zero.

2.3.5 Solution with generalized Fresnel integral An alternative approach to the problem of diffraction by truncated wedge is possible in which, in addition to the Fresnel integral, another special function is used to construct the solution, namely, the generalized Fresnel integral [55]. As distinct from the Fresnel integral, the generalized Fresnel integral depends on two (rather than one) arguments. The problem of diffraction by two half planes is considered in [20]. The field that results from the diffraction of the primary field by the first half plane serves as the source that illuminates the second half plane. It is known that such field is written in terms the integral representation. In [20], the integral representation is substituted as a source in the second integral representation, so that the final expression represents a double integral over complex plane. The mathematical transformations allow this integral to be reduced to a single integral that depends on two arguments, the generalized Fresnel integral. When only one plane is illuminated by a wave with unit amplitude, the solution is [20] V(w 0 ) = F [SMS0 (φ0 , r z , φ z )] − G [SMNSM (r z , φ z ) , SMS0 (φ0 , r z , φ z )] , y

∞

−∞

x

exp [i (ξ 2 + y2 )] dξ y 1 , F (y) = ∫ exp (iξ 2 ) dξ, G (x, y) = ∫ 2π ξ 2 + y2 √πi

(2.47)

where SMS0 (φ0 , r z , φ z ) = S M (P) − S0 (P), SMNSM (r z , φ z ) = S MN (P) − S M (P), the points for the determination of quantities S0 (P) = QP, S M (P) = QM + MP, S MN (P) = QM + MN + NP are shown in Fig. 2.12; these points are used for determination of eikonals involved in the generalized Fresnel integral entering in a possible expression for the field scattered by the truncated wedge. The solution for diffraction by two half planes can be used for the construction of a heuristic solution for diffraction by a truncated wedge. We construct the solution by

2.3 Diffraction by a truncated wedge

|

65

Q

M

N

P

Fig. 2.12: Points Q, M, N, P for determination of eikonals involved in the generalized Fresnel integral

the same approach as was used to derive the MGE-FI solution for the truncated wedge, the difference being that the rays in domain z are now not “bending”. Therefore, the solution does not contain function sN. Function pN, however, is still present in the solution affecting the field scattered by the illuminated vertex. Comparing the diffraction by half plane and wedge, we can interpret the multiplication by function pN as the transformation of the diffraction problem solution for a thin structure to the corresponding solution for a finite-thickness structure. In this respect, we can assume that multiplication by function pN in the diffraction problem for two half-planes transforms this solution to the one for a “thick” structure having the surface which envelopes the vertices located at the same points as the ends of the half-planes. Note a significant difference of the above approach from the MGE-FI approach used in the problem of diffraction by half-plane. In the MGE-FI approach, we used all of the solution for the equivalent wedge on curve r d0 , including both its “ray” and amplitude components. The changes to the solution correspond to bending of rays, which is taken into account via function sN. In the MGE-GFI approach, we use only the amplitude component of the solution for the equivalent wedge, i. e., function pN. The ray component of the solution is taken from solution (2.47) of the diffraction problem for two half planes. Subtracting the primary field in the illuminated region from the total field, we obtain the field that is scattered by the illuminated vertex. Then, we multiply this field by pN and add again the primary field. Adding the GFI, we obtain the desired solution: V(w0 ) = [F (SMS0) − P01] pN1 + P01 − G (SMNSM, SMS0) .

(2.48)

Here, P01 is the primary field (see equation (1.1)) in the illuminated region, pN1 = pN cos[(φ z0 − φ0 )/2]. Multiplication of function pN by the cosine of half difference of angles, which follows from the difference of eikonals (see, for example, asymptotic expression (1.54)) results in the transformation of function pN from equation (2.45), which has singularity at the shadow boundary, into function pN1, which is equal to unity at the shadow boundary. Evidently, if we use function pN1 expression (2.40)

66 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

and include the cosine of half difference of angles into the “amplitude” factor pN, obtaining thus the function pN1, this factor will no longer enter the “ray” component. Solution (2.48) has been obtained using a simplified procedure. In contrast to the conventional Fresnel integral, the generalized Fresnel integral describes diffraction by the shadow vertex. Instead of pN1, one can construct a function that is equal to unity on all shadow boundaries, i. e., on the shadow boundaries of the incident and reflected waves related to diffraction of the primary wave by the illuminated vertex, as well as on the shadow boundaries related to diffraction of the secondary wave by the vertices. However, such a solution must be “tuned” against a more accurate numerical solution, rather than against the MSD solution. Thus, function pN plays different roles in the MGE-FI and MGE-GFI solutions. The common feature of the approaches is that we use function pN on curve r d0 , and the values of this function in domain w12 are taken equal to those of function pN for the equivalent wedge. The difference, however, is that the directions of the shadow boundaries in domain z are defined differently. In the MGE-FI approach, we take the shadow boundaries at such a point of curve r d0 where the directions of ray emission are parallel to the shadow boundaries determined by geometrical optics, but do not coincide with them. We do not single out the shadow boundaries directed along the end surface, but obtain them automatically due to conformal mapping and corresponding effect of function sN on the solution. In the MGE-GFI approach, the ray pattern automatically results from the solution for two half-planes, and the shadow boundaries coincide with those predicted from geometrical optics. In solution (2.48), we also perform symmetrization of the angular argument φ w with respect to the propagation direction of the incident wave. We do not present here the corresponding trivial expressions. The generalized Fresnel integral becomes indeterminate when both arguments simultaneously tend to zero [22, 55]. In this case, the asymptotic expression for the GFI is equal to arc tangent of the ratio of arguments. Depending on the ratio of small (tending to zero) arguments, the arc tangent can change abruptly between 0 and π. This feature of GFI must be taken into account in the calculations. For example, in [22], the data are tabulated and the linear interpolation is performed in the vicinity of the point where both arguments are zeros. This behavior of GFI can be attributed to the ill-posedness of the physical formulation of the problem. In particular, when the distance between the half planes decreases to zero, they can either merge to a single half plane, or remain separate. If we decide which scenario is realized, we can fix the ratio of arguments as they both tend to zero and, therefore, avoid the abrupt changes of GFI.

2.3 Diffraction by a truncated wedge |

67

2.3.6 Numerical results Numerical calculations were performed using the analytical formulas based on the MSD, MGE-FI, and MGE-GFI approaches. We solve the problem of plane wave diffraction by a truncated wedge using the three methods for various angles α and β, with the size parameter kh decreasing from π to π/20. Figure 2.13 presents the results obtained for the truncated wedge with the size parameter kh = π/2 and several angles α and β. The angle of incidence of the plane wave in domain z is φ0 = 40°. The dotted lines correspond to the MSD solution (equation (2.40)), solid lines are MGE-FI solutions (equation (2.46)), while dashed lines are MGE-GFI solutions (equation (2.48)), respectively. One can see that solutions obtained by different methods are in reasonable agreement.

2.3.7 Analysis of solutions 1. MSD describes primary diffraction by the illuminated vertex, secondary diffraction by the shadow vertex, and subsequent diffraction by the illuminated vertex. This verified method works well at relatively large values of size parameter kh but fails to describe the diffraction when the size parameter tends to zero. Such behavior is inherent in the MSD procedure. In addition, the intermediate solutions for diffraction by auxiliary wedges related to the illuminated and shadow vertices exhibit singularities for the values of angles inside the wedges. This circumstance is insignificant in the analysis of diffraction by each wedge but may lead to problems in the heuristic approaches to diffraction by a half-plate or by a truncated wedge, where the regions with singularities are working regions. 2. The MGE-FI method involves the special function which describes diffraction by a single (illuminated) vertex. As distinct from the remaining two methods, no problems are encountered in this method when the small parameter tends to zero, two vertices merge to one vertex, and additional shadow boundaries disappear. The disadvantage of the MGE-FI approach is that solution behavior at the shadow vertex is described inaccurately. This disadvantage can be eliminated using the phenomenological approach in which the solution is corrected by two phenomenological coefficients ksn and ka, the exponent of function sN and a constant factor, respectively. The coefficients are calculated for the given values of vertex angles (illuminated and shadow) and remain unchanged as the size parameter is varied. 3. The MGE-GFI method, which takes into account diffraction by both vertices simultaneously (primary diffraction by the illuminated vertex and single secondary diffraction by the shadow vertex), is the most mathematically substantiated method, since this special function is obtained from the problem in a rather rigorous formulation.

68 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 (a) –360

–180

0

180

360

0 (b) –360

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 (c) –360

–180

0

180

360

0 (d) –360

–180

0

180

360

–180

0

180

360

Fig. 2.13: Example numerical solutions for truncated wedge with size parameter kh = π/2. Total field V on curve r d0 vs. angle φ zn of the rays issuing from curve r d0 in the MGE-FI solution in the two-sheet domain z for (a) α = 1.35, β = 1.25, ksn = 0, and ka = 1.05; (b) α = 1.35, β = 1.65, ksn = 1, and ka = 1.15; (c) α = 1.65, β = 1.35, ksn = 1, and ka = 1; and (d) α = 1.5, β = 1.5, ksn = 1, and ka = 1.1.

A disadvantage of the MGE-GFI solution lies in the complexity of the special function and its uncertainty when both arguments simultaneously tend to zero. This uncertainty, though, is inherent in the formulation of the problem, it seems to correspond to the uncertainty of whether the gap between two half-planes remains existent or disappears as the distance between them tends to zero. Another disadvantage of the MGE-GFI method lies in the fact that both GFI and the corresponding coefficient pN describe only single diffraction [20]. Analysis of

2.3 Diffraction by a truncated wedge |

69

secondary reflections requires additional study, which only makes sense when a numerical solution is available. The MGE-GFI method allows correct analysis of the main features of solution and does not require phenomenological coefficients. However, the special function (GFI) exhibits uncertainty when both arguments tend to zero. This circumstance leads to uncertainty in the solution when the size parameter tends to zero.

Conclusions to Chapter 2 The essence of the approach can be formulated as follows. According to MGE, diffraction problem solution is mapped from the initial domain z external to the scatterer, onto the domain w12 occupying the complete plane (Fig. 1.2 (c)). The upper halfplane w1 of domain w12 can be mapped onto the domain z by conformal transformation Z(w). We write complex numbers in polar coordinates: z = r z exp(iφ z ), w = r w exp(iφ w ). It should be noted that all coordinates r z , φ z , r w , and φ w of domains z and w12 are real-valued. At this stage, we apply the conformal mapping theory only to perform orthogonal coordinate transformation. The lower half-plane w2 of domain w12 can also be mapped onto the domain z by the same conformal transformation Z(w); this mapping law is a mirror image with respect to the horizontal axis of the mapping law for half-plane w1 . The horizontal axis corresponds to the scatterer boundary z. The purpose of all these geometrical transformations is to ensure that boundary conditions are satisfied, similar to what is done in the mirror image method. Solution to the boundary-value problem for wave equation with constant wave number on the domain z with complex boundary is equivalent to finding a continuous field on twosheet infinite domain w12 with variable wave number. This continuous field must satisfy three conditions of the original boundary-value problem: initial conditions, condition at infinity, and the Meixner condition. Also, it must satisfy the wave equation with variable wave number. Boundary conditions of the first or second kind on the scatterer boundary are satisfied automatically after we perform two steps. Firstly, we add or subtract in the upper half-plane w1 the field values from w1 and w2 at the points symmetric with respect to the horizontal axis (both point correspond to the same point in the real space, i. e., in domain z). Then we make inverse transformation from domain w1 to z. At the first glance, transformation to infinite domain w12 with variable wave number can only complicate solution of the boundary-value problem. However, things are simplified substantially if we search for the solution not on the whole domain, but on one of the curves r d0 . In contrast to the solution for a wedge, in the case of a scatterer possessing a size parameter, it may become very difficult to find the corresponding change of variables w → ŵ satisfying the conditions (2.38). Therefore, we apply heuristic ap-

70 | 2 Solution of Two-dimensional Problems by the Method of Generalized Eikonal

proach. Instead of trying to find the rigorous integral representation (1.28), we take on the domain w12 the integral representation for so-called “equivalent wedge”, but consider this solution on the curve r d0 corresponding to the scatterer with size parameter. Equivalent wedge is a scatterer of wedge shape, with the illuminated edge coinciding with that of the actual scatterer. The incidence angle φ z0e and external wedge angle πn e are chosen in such a way that after inverse mapping of the field from domain w12 onto z the angles at which the rays leave the curve r d0 coincide with the angles of geometrical optics directions of shadow boundaries; the latter ones are determined by the problem initial conditions. The field amplitude on the curve r d0 is taken equal to the field amplitude of the equivalent wedge on the domain w12 , taking into account the change of curvature of r d0 upon the inverse field mapping from the domain w12 onto z. The amplitude is corrected in such a way that the energy propagating in a given ray tube remain constant. In the rigorous solution satisfying the conditions (2.38), one can take curves r d0 not only with r d0 = 1, but with other values too. Upon the inverse mapping onto domain z, the wave number will be equal to the original one for any value of r d0 . In the heuristic solution, this is no longer true. We find the solution for equivalent wedge on its circle r w0 , after which transfer it to the curve r d0 for half-plate. In the heuristic approach which uses the equivalent edge concept, solution behavior as a function of size parameter kh is determined solely by the choice of a particular shape of the curve r d0 . In turn, this shape is determined by the choice of a particular value of r d0 , and the latter is determined only by the size parameter kh. When applying this approach, one can take for half-plate, as the first approximation, such a working curve r d0 that its points were not too distant from the circle r w0 for the equivalent wedge. For example, one can require that the unit circle r w0 intersect any of the curves r d0 at the middle point (Fig. 2.3). In this chapter, we presented heuristic solutions to the problem of diffraction by truncated wedge obtained by three methods: MSD, MGE-FI, and MGE-GFI procedures. The MSD solution is the simplest one. It can be constructed using the existing solutions for the wedges whose sides coincide with vertices without application of curve r d0 for the truncated wedge. Note that application of this allows the MSD solution to be improved. The solution becomes ineffective when the size parameter is relatively small. To improve the procedure, we factor out of the original integral a special function that describes correctly the field in the vicinity of the scattering edge, rather than FI. As a result, we obtain the most accurate MSD solution, possibly the most accurate solution of all considered in this Section. For the MGE-FI solution, we must employ function pN on the curve r d0 and function sN (in general, with correcting phenomenological coefficients ksn and ka, see (2.46)). This solution is not strictly substantiated from the mathematical point of view, but it works better than others when the size parameter kh tends to zero.

2.3 Diffraction by a truncated wedge |

71

The MGE-GFI solution describes single secondary diffraction by the shadow vertex, and function sN is unnecessary. However, the generalized Fresnel integral becomes singular when both arguments tend to zero. Further analysis of the proposed heuristic approaches requires a reliable numerical solution for verification. Then, all three approaches can be further developed in order to increase their accuracy. Prospects for the application of the above approaches to 3D problems are related to the application of the results presented in the following chapters of this book.

3 Application of Two-dimensional Solutions to Three-dimensional Problems 3.1 Integrals over elementary strips In this section, electromagnetic wave scattering by infinite and finite cylindrical edges is considered. Respective formulas for the vector potentials are compared. It is shown that in the case where the directrix of the integration contour encircling the cylindrical edge is a polygonal line, the analytical formulas obtained previously for electromagnetic wave scattering by infinite cylindrical edges can be applied in order to describe scattering by fragments of similar finite-length edges. It is concluded that this approach can be applied to edges of arbitrary shape and to arbitrary boundary conditions. In a number of problems where approximate solutions are obtained by the method of edge waves or equivalent edge current method [1, 62], it is required to describe electromagnetic wave scattering by a fragment of cylindrical edge which has finite length in the coordinate directed along its generatrix. There available many analytical and numerical solutions for cylindrical bodies of various shapes and with different boundary conditions; however, these solutions imply that the bodies are infinite in the above coordinate. Due to specific geometry of the object, in such problems waves are scattered only along the diffraction cones, i. e., the direction of wave scattering is oriented with respect to the scattering edge generatrix at the same angle as the direction vector of the incident wave. At the same time, wave scattered by a finite-size fragment of cylindrical edge in all directions; in the directions of diffraction cones one can apply the solution for an infinite cylindrical edge, whereas in all other directions this solution is not applicable. However, it is in these directions signals are emitted, which takes place, for example, in the problems where scattering by finite-size objects with large number of linear edges is studied. In this book, it is shown in which cases and by what means the “two-dimensional” solutions obtained for infinite cylindrical edges can be applied to finding out “three-dimensional” solutions for finite cylindrical edges.

3.1.1 Statement of the diffraction problem According to the well-known approach [32, 62], the diffraction problem is formulated as follows. Consider an electromagnetic field E⃗ 0 , H⃗ 0 , incident on the surface of a scat⃗ . terer. This field induces electric and magnetic surface currents of densities j e⃗ and j m When an electromagnetic wave is scattered by a body of finite dimensions, the complex-valued vector amplitudes of the electric and magnetic fields (having the time dependence exp(−iωt)) can be written as

74 | 3 Application of Two-dimensional Solutions to Three-dimensional Problems

1 E⃗ = − (grad div A⃗ e + k2 A⃗ e ) − rot A⃗ m , ik (3.1) 1 H⃗ = − (grad div A⃗ m + k2 A⃗ m ) + rot A⃗ e , ik where the vector potentials A⃗ e and A⃗ m can be expressed in terms of the surface currents:

⃗ = 1 ∬ exp (ikr) j e⃗ ds, A⃗ e (R) c r S

⃗ ds. ⃗ = 1 ∬ exp (ikr) j m A⃗ m (R) c r

(3.2)

S

⃗ are the and are the electric and magnetic vector potentials; j e⃗ and j m Here, 󵄨 󵄨 electric and magnetic surface currents; S is the surface of the scatterer; r = 󵄨󵄨󵄨R⃗ − ρ⃗ 󵄨󵄨󵄨 is the distance between the scattering surface point with position vector ρ⃗ = (ξ, η, ζ ) 󸀠󸀠 󸀠󸀠 󸀠󸀠 and observation point with position vector R⃗ = n⃗ 󸀠󸀠 R = (n󸀠󸀠 x R, n y R, n z R); n⃗ is the unit vector pointing to the receiver; R is the distance from the coordinate origin to the receiver; k = 2π/λ is the wave number; c is the speed of light; i is imaginary unit. For the field on an arbitrary surface enveloping the scatterer, the electric and mag⃗ are expressed via the fields E⃗ 0 and H⃗ 0 on the scatterer netic surface currents j e⃗ and j m surface as: c ⃗ = − c [n⃗ × E⃗ 0 ] , j e⃗ = (3.3) [n⃗ × H⃗ 0 ] , j m 4π 4π where n⃗ is the normal to surface S pointing to the domain occupied by the field. In the far zone (kr → ∞), formulas (3.1) written in the spherical coordinates are reduced to well-known relationships [32] A⃗ e

A⃗ m

E ϑ = H φ = ik (A eϑ + A m φ), E φ = −H ϑ = ik (A eφ − A m ϑ ),

(3.4)

E r = H r = 0, where A r , A ϑ , and A φ are related to A x , A y , and A z via linear transformations of physical coordinates of the vector (see equation (G.20) from Appendix G). Components A x , A y , and A z , in turn, are obtained by integrating the corresponding components of the currents along the scattering surface. If we substitute E⃗ 0 and H⃗ 0 in equation (3.3) by the exact values E⃗ and H,⃗ equations (3.4) will give the exact fields in the far zone. Unfortunately, exact fields E⃗ and H⃗ on the scattering surface are unknown, and we have to use E⃗ 0 and H⃗ 0 , introducing the systematic error which is characteristic for the physical optics approximation. Obtaining the exact currents j ⃗ on the surface S, or direct search for the exact solution is a complicated problem; its solution by rigorous methods is hindered by substantial computational demands, increasing as the scatterer dimensions are increased. For very large scatterers, finding a rigorous solution is not feasible. At the

3.1 Integrals over elementary strips

|

75

same time, accuracy of asymptotic methods for bodies with edges [1, 81] increases with the increase in the scatterer sizes, with the computational demands remaining equally low. In this section, an approach to using available two-dimensional solutions is proposed, enabling three-dimensional asymptotic solutions to be derived quite simply, as well as efficiency of numerical algorithms to be increased substantially.

3.1.2 Infinite cylinder Consider the integrals (3.2) for an infinite cylindrical edge of arbitrary shape, the generatrix of which coincides with the axis X. Let the excitation be performed by a planar electromagnetic wave ⃗ . E⃗ (i) = E⃗ 0 exp {ik (n⃗ 󸀠 , r)} (3.5) Here, E⃗ 0 is the complex amplitude, n⃗ 󸀠 is the direction vector of the incident wave, r ⃗ = (x, y, z) is the position vector of an arbitrary point in space. Expressions (3.2) can be recast in the scalar form 2 2 2 󸀠󸀠 󸀠󸀠 exp {ik [√(ξ − n󸀠󸀠 x R) + (η − n y R) + (ζ − n z R) ]} 1 A󸀠󸀠 = ∬ j ⃗ (ρ)⃗ ds. 2 2 2 c 󸀠󸀠 󸀠󸀠 √(ξ − n󸀠󸀠 x R) + (η − n y R) + (ζ − n z R) S

(3.6)

We decompose ρ⃗ into two parts: ρ⃗ = e⃗ x ξ + ρ⃗ 0 ,

(3.7)

where ρ⃗ 0 are components of vector ρ⃗ on the plane YOZ of the directrix of cylindrical surface C; e⃗ x is the unit coordinate vector. It is known that, on a cylindrical surface with infinite span along the x-axis, the currents vary along this axis according to the same law as the incident field (3.5): j ⃗ (ρ)⃗ = j ⃗ (ρ⃗ 0 ) exp (ikξn󸀠x ) .

(3.8)

2 2 󸀠󸀠 Denote by r yz = √(η − n󸀠󸀠 y R) + (ζ − n z R) the distance between the integration and observation points on the YOZ plane and rewrite equation (3.6), with reference to (3.8), in the form 2 2 󸀠 exp {ik [√(ξ − n󸀠󸀠 x R) + r yz + ξn x ]} 1 A = ∬ j ⃗ (ρ⃗ 0 ) ds 2 c 󸀠󸀠 2 √ R) + r − n (ξ x yz S 󸀠󸀠

=

∞

󸀠 exp {ik [√ ξ 2 + r2yz + (ξ + n󸀠󸀠 x R) n x ]}

−∞

√ ξ 2 + r2yz

1 ∫ j ⃗ (ρ⃗ 0 ) ∫ c C

(3.9) dξ dl.

For convenience, we applied the following variable substitution: instead of the combination (ξ − n󸀠󸀠 x R) in the integral over S, only ξ is present in the integral over dξ . The

76 | 3 Application of Two-dimensional Solutions to Three-dimensional Problems

integration contour C on the plane YOZ is the directrix of the cylindrical surface; dl is an element of C. It is also convenient to make use of the relationship [15] (see also [1]): ∞

∫

exp {ik [√ ξ 2 + r2yz + ξn󸀠x ]} √ξ 2

−∞

+

r2yz

(1)

dξ = iπH0 (kr yz √1 − (n󸀠x ) ) . 2

(3.10)

2 Introduce the notation k1 = k√1 − (n󸀠x ) and apply the asymptotic formula for the Hankel function for large values of its argument: (1)

H0 (k1 r yz ) ≅ √

2 exp (ik1 r yz ) . iπk1 r yz

(3.11)

Taking into account equations (3.10) and (3.11), rewrite (3.9) in the form ∞

A󸀠󸀠 =

󸀠 exp (ik1 r yz ) exp (ikn󸀠󸀠 x Rn x ) √ 2πi dl. ∫ j ⃗ (ρ⃗ 0 ) ∫ c k1 √r yz C

(3.12)

−∞

3.1.3 Far zone condition For a body of finite dimensions, provided that the far zone condition (A.35) from Appendix A is satisfied, r ≥ 2D2 /λ, (3.13) where D is the characteristic size of the body (for example, its maximum transverse size), the following equation holds true: ⃗ , r ≅ R − (n⃗ 󸀠󸀠 , ρ)

(3.14)

and integrals (3.2) for the vector potential A⃗ at the observation point can be reduced to a set of scalar relationships of the form A≅

1 exp (ikR) ⃗ ds. ∬ j exp {−ik (n⃗ 󸀠󸀠 , ρ)} c R

(3.15)

S

However, relations (3.14) and (3.15) are not valid for formula (3.12) because the cylindrical surface to which this formula applies is of infinite length along the x-axis, and condition (3.13) is not satisfied at any r. In the general case, the surface can be semi-infinite with respect to contour C, however, this circumstance does not prevent application of relations (3.14) and (3.15). According to the method of edge waves [1], we subdivide the currents j into the “uniform” (described by physical optics) and “non-uniform” (provided by deviation

3.1 Integrals over elementary strips |

77

of the surface shape from infinite plane) parts. The uniform part of the current is distributed over the whole scatterer surface, while the non-uniform part is localized near the edges. It can be proven rigorously [70, 83] that it is allowable to substitute the integration of physical optics component of plane wave over a flat plate in the far field by integration over the contour of this plate. In equation (3.12), this will correspond to taking into account only one point of contour C located at the intersection of the half-plane edge with plane YOZ. Besides (as can be demonstrated, for example, for half-plane [1]), both the uniform and non-uniform components of the current give after integration the inputs into A󸀠󸀠 at large distances from the edge of the order of 1/√r typical for cylindrical waves diverging from a linear edge. The above reasoning allows us to state that the far zone condition (3.13) can be applied in YOZ-plane even for semi-infinite contours C, with D being taken equal to the size of the zone in which non-uniform part of current propagates. In this case, relation (3.14) takes the form 󵄨󵄨 󸀠󸀠 󵄨󵄨 r yz ≅ R yz − (n⃗ 󸀠󸀠 (3.16) yz / 󵄨󵄨󵄨 n⃗ yz 󵄨󵄨󵄨 , ρ⃗ 0 ) . As before, subscript yz denotes projection of an object (without subscript) on YOZ󵄨󵄨 󸀠󸀠 󵄨󵄨 plane. Clearly, ρ⃗ 0 = ρ⃗ yz . Vector n⃗ 󸀠󸀠 yz /󵄨󵄨 n⃗ yz 󵄨󵄨 is a unit vector in YOZ-plane, aligned with 󸀠󸀠 n⃗ yz . Taking into account equation (3.16), represent (3.12) in the form A󸀠󸀠 =

󸀠 exp (ikn󸀠󸀠 2πi x Rn x + ik 1 R yz ) √ c k1 R yz 󵄨󵄨 󸀠󸀠 󵄨󵄨 × ∫ j ⃗ (ρ⃗ 0 ) exp {−ik1 (n⃗ 󸀠󸀠 yz / 󵄨󵄨󵄨 n⃗ yz 󵄨󵄨󵄨 , ρ⃗ 0 )} dl

(3.17)

C

=

󸀠 exp (ikn󸀠󸀠 2πi 󸀠󸀠 x Rn x + ik 1 R yz ) √ I , c k1 R yz

where the notation is introduced 󵄨󵄨 󸀠󸀠 󵄨󵄨 I 󸀠󸀠 = ∫ j ⃗ (ρ⃗ 0 ) exp {−ik1 (n⃗ 󸀠󸀠 yz / 󵄨󵄨󵄨 n⃗ yz 󵄨󵄨󵄨 , ρ⃗ 0 )} dl.

(3.18)

C

Formula (3.17) is valid for oblique (n󸀠x ≠ 0), as well as for normal (n󸀠x = 0) incidence on an infinite edge. The latter case is a purely two-dimensional problem, whereas oblique incidence on the edge at various angles is known to be reduced to twodimensional problems with different k1 . Note the following feature of the problem: waves scattered by an infinite cylindrical edge are propagating only in the directions of the diffraction cones, while the waves scattered by a finite fragment of the cylindrical surface are propagating in all directions. This fact is reflected in the solution (3.18) 󸀠󸀠 because the projection n⃗ 󸀠󸀠 yz of vector n⃗ on plane YOZ is used in such a way that only its direction, but not the absolute value is taken into account. Therefore, we can generally 󸀠 assume that the vector n⃗ 󸀠󸀠 is located on the diffraction cone. Then, n󸀠󸀠 x = n x and, 󵄨󵄨 ⃗ 󸀠󸀠 󵄨󵄨 󵄨󵄨 ⃗ 󸀠 󵄨󵄨 √ 󵄨󵄨 ⃗ 󸀠󸀠 󵄨󵄨 󸀠 2 consequently, 󵄨󵄨n yz 󵄨󵄨 = 󵄨󵄨n yz 󵄨󵄨 = 1 − (n x ) . In this case, k1 /󵄨󵄨n yz 󵄨󵄨 = k, and the direction of wave scattering will coincide with vector n⃗ 󸀠󸀠 .

78 | 3 Application of Two-dimensional Solutions to Three-dimensional Problems

3.1.4 Fragment of cylindrical surface Consider the vector potential of a cylindrical surface fragment which is finite along x-axis; let the fragment be illuminated by the wave (3.5), and assume that relationship (3.14) is valid for it. According to the method of edge waves [1], we assume that the currents on the fragment are the same as on a corresponding infinite in x cylinder and, therefore, they are the same as in formula (3.17). With reference to equations (3.8) and (3.15), we write the vector potential for a finite fragment of cylindrical surface in the form Aa ≅

2πi 1 exp (ikR) √ c R k1 R yz b

× ∫ exp {ikx (n󸀠x − n ax )} dx ∫ j ⃗ (ρ⃗ 0 ) exp {−ik (n⃗ a , ρ⃗ 0 )} dl a

(3.19)

C

1 exp (ikR) exp {ikb (n󸀠x − n ax )} − exp {ika (n󸀠x − n ax )} a I , = c R ik (n󸀠x − n ax ) where

I a = ∫ j ⃗ (ρ⃗ 0 ) exp {−ik (n⃗ a , ρ⃗ 0 )} dl.

(3.20)

C

Here, a and b are the edge boundaries along x-axis; n⃗ a is an arbitrary direction to the receiver, in the general case not located on the diffraction cone (in contrast to n⃗ 󸀠󸀠 ). Therefore, solutions to integral (3.20) are, generally speaking, not contained in the class of values of integral (3.18). Let a = −b. Then, equation (3.19) can be written as Aa =

1 exp (ikR) 2b sin [kb (n󸀠x − n ax )] a I . c R kb (n󸀠x − n ax )

(3.21)

It is clear from equation (3.21) that if n ax = n󸀠x , i. e., the direction vector of the receiver lies on the diffraction cone of the scattered signal, then 1 exp (ikR) 2bI a , c R

(3.22)

1 exp (ikR) 2I a . c R k (n󸀠x − n ax )

(3.23)

Aa ≅ whereas for n ax ≠ n󸀠x Aa ≅

Since beyond a narrow neighborhood of diffraction cones kb (n󸀠x − n ax ) is large enough, it follows from (3.22) and (3.23) that the scattered signal is determined primarily by the components located on the diffraction cones. This fact is widely used for obtaining asymptotic solutions by the geometrical theory of diffraction (GTD) [81]. To find the diffraction coefficients of the signals on the diffraction cones, two-dimensional

3.1 Integrals over elementary strips |

79

solutions are used, whereas other signal components are considered small and, therefore, neglected. This technique is only applicable when at any spatial point there exists a signal located on some diffraction cone. However, there exist such classes of scatterers for which this condition is not satisfied, for example, polyhedrons or plane polygons. The signal scattered by such bodies is primarily located beyond the diffraction cones, and other asymptotic methods become more advantageous, for example, the method of edge waves, or equivalent edge current method [1, 81]. Let us return to solution (3.19). The main difficulty lies in the calculation of integral (3.20). If the currents j are known, it remains just to substitute them into (3.20) and perform integration, e. g., numerically. An alternative approach is to use available analytical or numerical solutions which are known to be much simpler than threedimensional ones. Suppose that we have in hand analytical formulas from which E⃗ and H⃗ can be obtained; this also means, with reference to (3.2) and (3.4), that A󸀠󸀠 on the left-hand side of equation (3.17) can be evaluated. It does not matter if we have a ready-to-use analytical solution (available, for example, for a perfectly conducting wedge), or the analytical formula was obtained empirically or as an approximation or interpolation of numerical solutions obtained by rigorous methods. Then, taking into account (3.17), one can obtain easily an analytical expression for I 󸀠󸀠 , without resorting to direct integration according to (3.18). Moreover, if the contour C possesses some particular properties (namely, is formed by a polygonal line), we can also evaluate the integral (3.20), solutions of which are not generally contained in the class of solutions of integral (3.18).

3.1.5 Polygonal edge Let the contour C consist of N rectilinear segments C κ . Then, due to linearity of transformations for A󸀠󸀠 and A a , the following relation is valid N

A = ∑ Aκ .

(3.24)

κ=1

Introduce the notation ρ⃗ 0 = ρ⃗ κ1 + ρ⃗ κτ ,

(3.25)

where ρ⃗ κ1 is the initial value of ρ⃗ 0 on the interval C κ ; ρ⃗ κτ is the vector aligned with the rectilinear segment C κ . Taking into account equations (3.24) and (3.25), represent formulas (3.17) and (3.18) in the form A󸀠󸀠 κ =

󸀠 exp (ikn󸀠󸀠 2πi 󸀠󸀠 x Rn x + ik 1 R yz ) √ I , c k1 R yz κ

(3.26)

80 | 3 Application of Two-dimensional Solutions to Three-dimensional Problems 󵄨󵄨 󸀠󸀠 󵄨󵄨 I κ󸀠󸀠 = exp {−ik1 (n⃗ 󸀠󸀠 yz / 󵄨󵄨󵄨 n⃗ yz 󵄨󵄨󵄨 , ρ⃗ κτ )}

󵄨󵄨 󸀠󸀠 󵄨󵄨 × ∫ j ⃗ (ρ⃗ 0 ) exp {−ik1 (n⃗ 󸀠󸀠 yz / 󵄨󵄨󵄨 n⃗ yz 󵄨󵄨󵄨 , ρ⃗ κτ )} dl.

(3.27)

Cκ

while formulas (3.19) and (3.20) as A aκ =

1 exp (ikR) exp {ikb (n󸀠x − n ax )} − exp {ika (n󸀠x − n ax )} a Iκ , c R ik (n󸀠x − n ax )

I κa = exp {−ik (n⃗ a , ρ⃗ κ1 )} ∫ j ⃗ (ρ⃗ 0 ) exp {−ik (n⃗ a , ρ⃗ κτ )} dl.

(3.28) (3.29)

C

Suppose that analytical formulas are available for all A󸀠󸀠 κ , and, taking into account (3.26), they are also available for each I κ󸀠󸀠 . As was noted above, this means that an analytical formula for the integral in (3.27) is available. Write down the scalar products involved in the exponential function arguments in the integrands (3.27): 󵄨󵄨 󸀠󸀠 󵄨󵄨 󸀠󸀠 󸀠 2 √ k1 (n⃗ 󸀠󸀠 yz / 󵄨󵄨󵄨 n⃗ yz 󵄨󵄨󵄨 , ρ⃗ κτ ) = k 1 − (n x ) ρ κτ cos φ κ

(3.30)

k1 (n⃗ a , ρ⃗ κτ ) = k√1 − (n ax ) ρ κτ cos φ aκ .

(3.31)

and (3.29): 2

a 󸀠󸀠 a Here, φ󸀠󸀠 κ and φ κ are the angles between the vector ρ⃗ κτ and vectors n⃗ yz and n⃗ yz , re󸀠󸀠 spectively. If the angle φ κ is involved as a parameter in the analytical formulas for I κ󸀠󸀠 on the left-hand side of (3.27), then formal substitution into them of such φ󸀠󸀠 κ for which

cos φ󸀠󸀠 κ =

(n⃗ a , ρ⃗ κτ ) ρ κτ √1 −

2 (n󸀠x )

=

√1 − (n ax )2 √1 −

2 (n󸀠x )

= cos φ aκ ,

(3.32)

as can be seen from comparison of (3.27) and (3.29), as well as (3.30) and (3.31), results in the following exponential function arguments in (3.27) and (3.29) 󸀠󸀠 a k1 (n⃗ 󸀠󸀠 yz /| n⃗ yz |, ρ⃗ κτ ) = k ( n⃗ , ρ⃗ κτ ) ,

(3.33)

on the whole rectilinear segment C κ of contour C. Because the currents j ⃗ (ρ⃗ 0 ) in (3.27) and (3.29) are the same, we have 󵄨󵄨 󸀠󸀠 󵄨󵄨 a a I κ󸀠󸀠 exp {ik1 (n⃗ 󸀠󸀠 yz / 󵄨󵄨󵄨 n⃗ yz 󵄨󵄨󵄨 , ρ⃗ κ1 )} = I κ exp {ik ( n⃗ , ρ⃗ κ1 )} .

(3.34)

With I κ󸀠󸀠 known from (3.26), one can obtain an analytical expression for I κa by using (3.34), without the need of performing all calculations (3.29). This is advantageous because this analytical expression can be entered into more complicated equations, as well as used to evaluate the derivatives. Since the vector n⃗ a is chosen arbitrarily, the angle φ󸀠󸀠 κ determined by relation (3.32) can be complex-valued. Substitution of complex parameters into the analytical expression for I κ󸀠󸀠 in the left-hand side of formula (3.27) gives analytical continuation of

3.2 Application of two-dimensional solutions to three-dimensional problems |

81

real-valued function to the complex plane; this, however, is known not to affect the validity of all relationships obtained. Formulas for scattering in an arbitrary direction by a perfectly conducting wedge were obtained in [47, 84] by direct analytical integration of the currents from rigorous solution on both sides of the wedge. An integral similar to (3.29) was obtained; the solution involved complex-valued angles, as in (3.32). Incremental diffraction coefficients for perfectly conducting three-dimensional scatterers were analyzed in detail in [49] where a problem for further research was formulated: to clarify if the “direct substitution method” developed for perfectly conducting scatterers can be extended to other boundary conditions and curved surfaces. This question is answered automatically when formula (3.2) from this book is applied, because in the Huygens–Kirchhoff principle [32] from which these formulas follow, any surface S enveloping the scatterer can be used, including surfaces not coinciding with its boundary. No requirements on the type of boundary conditions are posed. In order to apply the technique described in this section, it is only necessary that the surface S was formed by a set of fragments of cylindrical contours, each having its directrix as a combination of closed or open polygonal lines. By eliminating the repeated integration over the linear parts of directrices, one can speed up corresponding numerical algorithms. In the case of diffraction by half-plane, when the contour C becomes a straight line, the situation becomes extremely simple because there appear only two angles φ aκ , equal to each other up to π. Having in hand the two-dimensional solution for half-plane with arbitrary boundary conditions, one can obtain a solution to three-dimensional problem by simple substitution of corresponding φ󸀠󸀠 κ.

3.2 Application of two-dimensional solutions to three-dimensional problems 3.2.1 Physical optics solution for diffraction by a plane scatterer. Properties of contour integral In this section, rigorous relationships for 3D diffraction of plane wave by a plane scatterer in the physical optics approximation are compared with corresponding 2D diffraction coefficients. It is shown that physical optics diffraction coefficients (PODC) are tending to infinity along the diffraction cones, however, this behavior is compensated for by vanishing of the integral over the closed “shadow contour” in the same directions. Currently, diffraction methods are being developed quite actively which allow electromagnetic wave scattering by complex object to be studied. Among these methods are: method of edge waves (MEW) in the physical theory of diffraction (PTD) (new formulation) [84, 85], equivalent edge current method (EECM), improved EECM [47], representation of scattered field in terms of incremental length diffraction coefficients

82 | 3 Application of Two-dimensional Solutions to Three-dimensional Problems

(ILDC) [46, 49]. All these methods are based on the PTD concept [1]. One of the problems of EECM and ILDC methods is that the diffraction coefficients for the components of PO current become infinite in the most important directions, such as diffraction cones, and, especially, in the forward and mirror directions. In a number of works, attempts are undertaken to overcome this difficulty (see, e. g., [48]). We show in this section that tending to infinity of the diffraction coefficients for the components of PO currents is a very important property of this problem. Moreover, attempts to circumvent this behavior can lead to incorrect results. It is shown that the tending to infinity of individual “quasi-2D” diffraction coefficients for components of the physical optics currents is compensated by vanishing of the integral over a closed contour for a 3D scatterer [36].

3.2.2 Rigorous 3D formulas Consider a perfectly conducting plane scatterer (with normal n)⃗ excited by a plane ⃗ wave with temporal function exp {ik (n⃗ 󸀠 , r)}): ⃗ = (H0x , H0y , H0z ) exp {ik (n⃗ 󸀠 , R)} ⃗ , H⃗ (i) = H⃗ 0 exp {ik (n⃗ 󸀠 , R)}

(3.35)

where H⃗ (i) and H⃗ 0 are the distance-dependent and distance-independent magnetic field vectors, n⃗ 󸀠 is the direction vector of the incident wave, R⃗ = n⃗ 󸀠󸀠 R is the position vector of the observation point, n⃗ 󸀠󸀠 is the direction vector of the observation point, i is imaginary unit, k = 2π/λ is the wave number, λ is the wave length. In the PO approximation, the vector potential is similar to (3.2) and is equal to ⃗ = 1 ∬ exp {ikr} [n⃗ × H⃗ (i) ] ds, A⃗ (R) 2π S r

(3.36)

󵄨 󵄨 where r = 󵄨󵄨󵄨n⃗ 󸀠󸀠 R − ρ⃗ 󵄨󵄨󵄨 is the distance between a point on the scatterer and the observation point, ρ⃗ is the position vector of the point on the scatterer surface, R is the distance from the coordinate origin to the observation point, S is the surface area of the scatterer. We assume, without loss of generality, that the plane scatterer is located on the XOY-plane. Also, suppose that the far zone condition r ≅ R−(n⃗ 󸀠󸀠 , ρ)⃗ is satisfied, in which case 1 exp {ikr} ⃗ ds, A⃗ ≅ (−H0y , H0x , 0) I, I = ∬ exp {ik (Δ,⃗ ρ)} 2π R S

(3.37)

where Δ⃗ = (n⃗ 󸀠 − n⃗ 󸀠󸀠 ) − n⃗ [(n⃗ 󸀠 − n⃗ 󸀠󸀠 ) , n]⃗ is the projection on the scattering surface (in our case, on XOY-plane) of the difference (n⃗ 󸀠 − n⃗ 󸀠󸀠 ) between the direction vectors of the incident wave and observation point. Applying the Stokes theorem, one obtains (see Appendix A, formula (A.16)): ⃗ ds = I = ∬ exp {ik (Δ,⃗ ρ)} S

i ⃗ ⃗j ⃗ ⃗ 󵄨󵄨 ⃗ 󵄨󵄨2 ∮ (Δ, n ) exp {ik (Δ, ρ)} dt, k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 C

(3.38)

3.2 Application of two-dimensional solutions to three-dimensional problems |

83

where n⃗ j is the inner unit normal to contour C encircling the scatterer, ρ ⃗ 󸀠 is the unit vector tangent to the contour, t is the coordinate measured along the contour. Denote the phase of the integration point in the exponent by ⃗ . Φ = k (Δ,⃗ ρ)

(3.39)

If the contour is a polygon with N vertices, then aj

Ij =

=

i

⃗ ⃗j ⃗ ⃗󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 ∫ (Δ, n ) exp {ik (Δ, t ρ j )} dt k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 0 ia j (Δ,⃗ n⃗ j ) exp {ik (Δ,⃗ ρ⃗ j )} − exp {ik (Δ,⃗ ρ⃗ j−1 )} , 󵄨 󵄨2 ik (Δ,⃗ ρ⃗ j − ρ⃗ j−1 ) k 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

or N

I = ∑ Ij , Ij = j=1

ia j (Δ,⃗ n⃗ j ) sin [(Φ j − Φ j−1 ) /2] exp {i (Φ j + Φ j−1 ) /2} , 󵄨 󵄨2 (Φ j − Φ j−1 ) /2 k 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

(3.40)

where Φ j = k (Δ,⃗ ρ⃗ j ) is the phase of signal from j-th vertex with direction vector ρ⃗ j , a j is the length of j-th side of the polygon (bounded by j − 1-th and j-th vertices). If the observation point is located on the diffraction cone, i. e., (Δ,⃗ ρ ⃗ 󸀠 ) = 0, then Δ⃗ ‖ n⃗ j ,

󵄨 󵄨 |Δ|⃗ = 󵄨󵄨󵄨󵄨(Δ,⃗ n⃗ j )󵄨󵄨󵄨󵄨 ,

(Δ,⃗ n⃗ j ) = (n⃗ 󸀠 , n⃗ j ) − (n⃗ 󸀠󸀠 , n⃗ j ) = sin β (− cos φ0 − cos φ) ,

(3.41)

where (φ0 + π) and φ are the angles between the projections of direction vectors n⃗ 󸀠 and n⃗ 󸀠󸀠 on the plane perpendicular to ρ ⃗ 󸀠 , and the inner normal n⃗ j to contour C, β is the angle between n⃗ 󸀠 (or n⃗ 󸀠󸀠 ) and ρ ⃗ 󸀠 (these vectors have the same angle to the edge because on the diffraction cone not only n⃗ 󸀠 , but also n⃗ 󸀠󸀠 is directed along the generatrix of the cone determined by vector n⃗ 󸀠 ). Finally, we obtain Ij =

−ia j exp {iΦ j } ia j exp {iΦ j } = , k sin β (cos φ0 + cos φ) k (Δ,⃗ n⃗ j )

(3.42)

with Φ j−1 = Φ j . Now, by factoring the 3D PODC out of vector potential A j for the j-th side of a plane polygonal scatterer, it is easy to show that the rigorous expression for the 3D case coincides exactly with the approximate 2D PODC [1] for an obliquely incident wave. The angles φ and φ0 (same as in [1]) are reckoned from the inner normal to the plate contour, β1 is the angle between directions n⃗ 󸀠 and n⃗ 󸀠󸀠 and the plane normal to the edge.

84 | 3 Application of Two-dimensional Solutions to Three-dimensional Problems

3.2.3 Comparison with 2D case 2D PODC [1] are valid far away from the light–shadow boundary only. Near these boundaries, correct 2D PODC will involve the Fresnel integrals (see, e. g., [48]). However, rigorous 2D PODC are valid only for half-plane, whereas our 2D PODC are valid for finite-dimension scatterers which (and only which) are of interest in practical applications. For |Δ|⃗ = 0 (“forward” and “mirror” directions) the PODCs become singular, which becomes a problem in practical applications. In order to eliminate this singularity, in [48] a “uniform asymptotic solution” involving the Fresnel integral was derived. However, it follows from the discussion in the current section that rigorous PODC must have a singularity, otherwise the integral over the closed contour C would vanish, as was shown in [36]. The rigorous integral over contour C is equal to i ⃗ ⃗j ⃗ ⃗ 󵄨󵄨 ⃗ 󵄨󵄨2 ∮ (Δ, n ) exp {ik (Δ, ρ)} dt k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 C ⃗ ds = S, = ∬ exp {ik (Δ,⃗ ρ)}

I=

S

(3.43)

⃗ |Δ|=0

which is a rigorous result that can be obtained either from the contour, or from the surface integral.

3.2.4 Total current diffraction coefficients Results relevant to PODC can be extended to the total current diffraction coefficients (TCDC). Asymptotic formulas for 2D TCDC [1], as well as PODC, are valid only far away from the light–shadow boundary. Singularities of 2D TCDC are the same as those of 2D PODC and, as was just shown, the same as those of 3D PODC. In the directions of light–shadow boundaries, TCDC and PODC must coincide. Therefore, if we apply 2D TCDC in three-dimensional case, we obtain correct result not only far away from the light–shadow boundary, but in all directions as well. Apropos, it follows from the above discussion that corrections [48] to earlier work [47] eliminating singularities in 2D TCDC are meaningless in three-dimensional case because the equivalent edge currents from [47], referred to as TCDC in this book, are valid in 3D case for arbitrary observation directions, including the neighborhood of light–shadow boundary. Properties of integrals I are considered in more detail in Appendix A.

4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach) Electromagnetic wave diffraction by a plane angular sector is a long-standing problem, however, an analytical solution for diffraction by quarter-plane was found only few years ago [16]. In this section, it is show that analytical solutions for electromagnetic wave diffraction by a plane angular sector can be refined by a modified heuristic equivalent edge current method (EECM). In the modified EECM, instead of integration over elementary strip in the course of the solution, we apply available two-dimensional solution for diffraction by an imaginary edge which is placed at some angle to the real edge. EECM solution was further refined by taking into account heuristically the effect of edge end on the scattered signal. Calculated results are presented which confirm good agreement with known solutions. Importance of this problem is related to the recent progress in the stealth technology which allows objects invisible for radars to be designed. This technology involves the development of absorbing coatings and designing objects of a special shape formed from polyhedrons and polygons. Results of this study can additionally be used for calculation of diffraction by an open waveguide end, development of a theory of radio wave propagation in urban environment, calculation of scattering by a crystal material surface, and for solution of other problems. The topicality of the problem is confirmed by the fact that the URSI Commission B established a prize in 2003 and, in 2004, awarded this prize for analytical solution of the problem of scattering by a quarter–plane. The solution was obtained in the form of series of special functions summed with the help of a special technique [16]. Since validation has become possible, availability of such a solution stimulates derivation of heuristic formulas for calculation of the aforementioned objects.

4.1 Statement of the problem The problem geometry is presented in Fig. 4.1, it is taken from [16] where rigorous solution was obtained for electromagnetic wave diffraction by a plane angular sector. In Fig. 4.2, rigorous results for electromagnetic wave diffraction by a plane angular sector are shown according to [16]. Coefficients D describe the relationship between polarizations of the incident and scattered waves, they were calculated according to the following expressions from [16]: (

exp (ikR) D ϑϑ E∞ ϑ )= ( ∞ kR Eφ D φϑ

E inc D ϑφ ) ( ϑinc ) . Eφ D φφ

(4.1)

86 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

x

x φ

φ

E⃗ inc

⃗

E inc ϑ

ϑ

45˚

45˚ y

y

z

z (a)

(b)

Fig. 4.1: Problem geometry from [16].

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

2 1.5 1 0.5 0 0 (a)

45

90 φ→

135

180 (b)

0

45

90 φ→

135

180

Fig. 4.2: Calculated coefficients D from [16].

Here, E are components of the incident and scattered fields. The subscripts denote the unit vectors for the corresponding angles, and the superscripts denote the incident (inc) and scattered (∞) fields at the observation point located at a large distance R from the scatterer. Our task is to obtain the same dependencies by means of a heuristic method. We construct a heuristic solution to the problem of diffraction by a plane angular sector as follows. 1. In the far-zone approximation, we obtain a rigorous physical optics solution for scattering by a plane polygonal plate. 2. In the obtained physical optics solution, we separate the contribution of the scalar integral over the surface and the vector component that is constant over the entire surface of the plate.

4.2 Solution in physical optics approximation

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87

3.

In the scalar integral over the surface, we separate the integral along the edge and the integral over an elementary strip oriented perpendicularly to a virtual imaginary edge. 4. For the case of the oblique incidence on a 2D edge, we find a relationship that couples the vector diffraction coefficients of the rigorous and physical optics solutions. 5. In the physical optics solution, we replace the physical optics vector diffraction contribution by the vector diffraction contribution of the rigorous solution.

4.2 Solution in physical optics approximation We calculate the scattered field using the well-known formulas which can be found readily in textbooks on electromagnetic theory (see, e. g., [32]). According to the Huygens electrodynamic principle, the field at the observation point located outside surface S enclosing sources can be expressed as the superposition of the waves of secondary sources located on surface S. Inside S, the field is determined by primary sources and the superposition of the secondary sources is zero. This condition is sometimes referred to as the absence of a backward wave [1, 45, 46]. Consider a plane electromagnetic wave incident on a perfectly conducting scatterer having the shape of a plane polygon. Vectors and of the scattered filed are expressed by equation (3.1) [1, 32]. The electric and magnetic potentials A⃗ e and A⃗ m (see (3.2)) involved in equa⃗ (see (3.3)), k = 2π/λ is the wave tion (3.1) depend on the surface currents j e⃗ and j m number; i is the imaginary unit; r is the distance from a point on a scatterer to the observation point; R is the distance from the origin to the observation point; R n⃗ 󸀠󸀠 and ρ ⃗ are the position vectors of the observation point and of a point lying on the scatterer, 󵄨 󵄨 respectively; r = 󵄨󵄨󵄨R n⃗ 󸀠󸀠 − ρ ⃗ 󵄨󵄨󵄨. Formulas (3.1) and (3.2) are not the only possible representation of the solution for the scattered field. We apply this form because it has been verified repeatedly after publication in books [1, 32] (originally published in Russian). In other books [3, 30] (published abroad, but then translated into Russian), the solution is represented in another form, which directly follows from Green’s formula. In book [45] (published in English and, later, in Russian), which is a modern version of [1], instead of the old representation of the solution, an alternative form (similar to that from [3, 30]) is used, which, however, does not affect the final results. When the primary sources are located outside surface S, and a scattering body rather than the sources is located inside S, the scattering problem can be reduced to that involving the appearance of additional sources on surface S. A solution to the diffraction problem can be obtained by adding the field of additional sources to the primary field. In particular, for the case of diffraction by a perfectly conducting plate (i. e., a plate with the boundary conditions [n⃗ × E]⃗ = 0 on its surface), we can contract

88 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

the surface S to the plate surface and obtain in the physical optics approximation [32] i e⃗ =

c [n⃗ × H⃗ inc ] , 2π

(4.2)

where the normal n⃗ to the surface is oriented from the shaded side of the plate toward its illuminated side. When the observation point is located far from the scatterer of finite dimensions, the far-zone condition (3.14) is satisfied. When the incident wave is not plane, this condition also depends on the source location. The far-zone condition can be explained with the use of the sight angle at which the scatterer is seen from the observation point. When the distance between the observation point and a real finite-dimension scatterer is increased (i. e., when kr → ∞), the sight angle decreases to zero. However, when the distance from a model scatterer (half-plane, wedge, quarter-plane) is increased, this angle never decreases to zero. Solutions to model problems also differ from solutions to real problems. In particular, the signal scattered by a model structure becomes constant on the shadow boundary irrespective of the factor (kr)−1 whereas for a real structure the scattered signal field decreases proportionally to (kr)−1 over the entire observation region, including the neighborhood of the shadow. When a solution is sought in the form of a contour integral, then, in the direction towards the shadow boundary (i. e., forward direction) and in the mirror reflection direction, the contribution of each edge element is singular, while the complete integral over the closed contour yields the correct solution, proportional to the area of the scatterer [36]. In the far zone, equations (3.1) and (3.2) are simplified substantially. Under conditions (3.14), potentials A⃗ depend only on R as exp(ikR)/R, while expressions for fields (3.1) are reduced to (3.4) [32]. Taking into account relations (3.14) and (3.35), it is possible to write equations (3.2) in the form analogous to (3.37). For a perfectly conducting plate, the magnetic vector m ⃗e potential is zero, A⃗ m = 0, A m φ = 0, A ϑ = 0, while the electric potential A is 1 exp (ikR) A⃗ e = [n⃗ × H⃗ 0 ] I, 2π R ⃗ ds. I = ∬ exp {ik (Δ,⃗ ρ)}

(4.3)

S

Here, the currents are taken from (4.2), the normal n⃗ is directed from shadow to illuminated side of the scattering surface. Auxiliary vector Δ⃗ is defined in the same way as in (3.37). An important feature of relations (4.3) is that the integral I over the scatterer surface enters the expression for potential in the form of a separate multiplier. This feature is characteristic for physical optics solutions which mainly depend on the scatterer surface, and to a lesser extent on its shape. It will be shown later that GTD and MEW solutions depend on T-polarization of the vector incident on the edge (i. e., on the relative orientation of the incident wave polarization vector and tangent vector

4.2 Solution in physical optics approximation

|

89

to the scatterer). The physical optics solutions depend on the relative orientation of the incident wave polarization vector and normal vector to the scatterer. In the case of normal incidence on the edge, the tangent vector to the edge and the normal vector to the scatterer surface are perpendicular, therefore, one can obtain the relationship between polarizations of rigorous and physical optics solutions. For oblique incidence, separation of polarizations in rigorous and physical optics solutions occurs differently. This fact must be taken into account when constructing heuristic solutions.

Diffraction by polygonal plate in the physical optics approximation When assessing the suitability of an electrodynamic model for engineering analysis of a certain object, it is important to have a simple method for constructing a solution and simple final formulas. Despite the fact that the diffraction theory has been substantially advanced during the last hundred years, the physical optics approximation remains a very popular approach to the solution of various problems. In the diffraction theory, the physical optics approximation is constructed on the basis of geometrical optics surface currents. First, the part shaded by a scatterer is cut from an unperturbed geometrical optics wave and turned toward the reflected wave. Sometimes, the absence of the cut part in the primary wave is called a shadow column [1, 45, 46]. After the geometrical optics structure of the solution is constructed, the physical optics approximation is applied.

4.2.1 Contour integral with enforced far zone condition In the physical optics approximation, a solution to the diffraction problem can be represented as an integral of the fields or functions of the fields specified on the surface of a scatterer. When the integral over surface S is calculated, fields E⃗ 0 and H⃗ 0 on the scatterer surface are chosen in formulas (3.3) to be the same as the geometrical optics incident fields: E⃗ 0 = E⃗ inc and H⃗ 0 = H⃗ inc . If necessary, the reflected field is also taken into account. Since an unperturbed wave cannot be reflected by free space, a physical optics solution should not contain a backward wave either. The check of physical optics integrals over the scatterer surface for the absence of backward waves is a good correctness test for the solution. The far zone condition is not satisfied for a plane angular sector, however, we “enforce” this condition (see Appendix B, text between formulas (B.13) and (B.14)). When the far zone condition (3.13) and equality (3.14) are satisfied, a solution to the scattering problem is found from formulas (3.4) and (4.3). Integral I in these expressions is of the main interest. The character of the scattered signal is determined by the spatial dependence of the amplitude and phase of this integral on the scatterer geometry.

90 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

According to the Stokes theorem, the integral over surface S can be reduced to the integral along the contour C bounding this area (3.38). Let the scatterer be a polygon. Then, for the j-th side of length a j bounded by the j − 1-th and j-th vertices we obtain equation (3.40). On the diffraction cone (Δ,⃗ ρ ⃗ 󸀠 ) = 0, and we obtain (see also (3.42)) Ij =

a j exp {iΦ j } . ik cos β1 (cos φ + cos φ0 )

(4.4)

4.2.2 Inputs of edges and vertices In formulas (3.40), I j is the input of j-th edge. This expression contains two phases at the vertices bounding the edge. The sum in (3.40) can be transformed in the following way (see also Appendix A): N

N

I = ∑ Ij = ∑ I j , j=1

j=1

where I j =

j j+1 exp {iΦ j } (Δ,⃗ n⃗ ) (Δ,⃗ n⃗ ) ]. [ − 󸀠 󸀠 󵄨 󵄨2 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 [ (Δ,⃗ ρ ⃗ j ) (Δ,⃗ ρ ⃗ j+1 ) ]

(4.5)

The terms I j are sometimes referred to as “vertex waves” because each term corresponds to the phase of one of the vertices. However, in reality the input of vertices is not taken into account at all in the physical optics integral, since the contour integral I describes the input of edges only. The input of edge element does not depend on the distance at which this element is located with respect to the vertex. It is deemed more accurate to reserve the term “vertex wave” for the solution component which does not depend on the input of the edge. The influence of vertex is then described by additional terms or factors appearing in the solution.

Factorization of diffraction coefficients from physical optics solution Let us show that the obtained 3D solution for the far field scattered by a side of a polygon contains the same diffraction coefficients as in the classical 2D solution. Consider the field (3.4), substituting into it the potential (4.3) on the side (4.4), corresponding to the diffraction cone condition in the case where the incidence and observation plane is located normally to the edge (i. e., for β = 0): ik exp (ikR) I j [n⃗ × H⃗ 0 ]φ 2π R ϑ ϑ [n⃗ × H⃗ 0 ]φ 1 a j exp (ikR) 1 a j exp (ikR) f 0 ϑ = = ( 0) . 2π R cos φ + cos φ0 2π R g

E φ = ikA eφ = ϑ

(4.6)

This expression corresponds to three-dimensional diffraction by the finite-length edge a j .

4.2 Solution in physical optics approximation

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91

Equation (4.6) admits quite simple physical interpretation. The field scattered by j-th side is equal to the product of the factor describing spherical divergence of the field and diffraction coefficient f 0 or g0 in the physical optics approximation. To consider a two-dimensional case, one has to perform integration over the whole edge between −∞ and ∞. Taking into account expressions for the Hankel function and its asymptotic form, we obtain: ∞ exp {ik √ r 2 + z 2 } 1 a j exp (ikR) dz ⇒ ∫ R √ r21 + z2 −∞ (1)

= iπH0 (kr1 ) ≅ √2πi

(4.7)

exp (ikr1 ) . √ kr1

Here, r1 is the distance from the edge to observation point in the plane normal to the edge. The variable r1 is similar to r from two-dimensional space considered in [1]. In the coordinates used in [1] (see comments to equation (4.4)) the vector products for TH and TE-polarizations (when H or E-vector is normal to the edge) are equal sin φ0 and sin φ, respectively. As a result, we obtain [n⃗ × H⃗ 0 ]φ √2πi exp (ikr1 ) i exp (ikr1 ) f 0 Eφ ϑ =√ ( )= ( 0) . 2π 2π √ kr1 Eϑ g √ kr1 cos φ + cos φ0

(4.8)

Here, f 0 and g 0 are diffraction coefficients in the physical optics approximation introduced in [1]. The just obtained expressions (4.8) coincide exactly with the result from [1]. This means that the physical optics diffraction coefficients from the 2D problem remain the same in the 3D case. Below, we apply this result to construct a heuristic solution to the 3D problem. Note that the result (4.8) is obtained under the assumption that, relative to the observation point, the incident field vector is oriented identically for all the points of the scatterer plane. This assumption is valid only for the 2D problem. In the far zone approximation, it is valid for the 3D problem as well. Compare the physical optics (f 0 , g0 ) and rigorous (f, g) diffraction coefficients from [1]. φ φ φ φ −2 cos 2 cos 20 2 sin 2 sin 20 , g= , f = cos φ + cos φ0 cos φ + cos φ0 (4.9) sin φ0 − sin φ 0 0 , g = . f = cos φ + cos φ0 cos φ + cos φ0

92 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

Note that the following statements are valid: f (φ, φ0 ) =

1 1 1 − ( φ+φ0 ) , 0 2 − cos φ−φ − cos 2 2

g (φ, φ0 ) =

1 1 1 + ( ), φ−φ 0 0 2 − cos 2 − cos φ+φ 2

0 0 sin φ+φ sin φ−φ 1 2 2 − f (φ, φ0 ) = ( ), φ−φ φ+φ 2 − cos 2 0 − cos 2 0

(4.10)

0

g0 (φ, φ0 ) =

φ−φ0 2 φ−φ0 2

sin 1 ( 2 − cos

+

φ+φ0 2 φ+φ − cos 2 0

sin

).

Formulas (4.10) agree with [3] where it was shown that the physical optics solution differs from the rigorous one only by a factor equal to the cosine of half the difference of the direction angles of the incident and scattered waves. Taking into account the way in which the angle φ0 of the incident wave is reckoned in (4.10) (coinciding with that in [1]), these factors are sin[(φ − φ0 )/2] and sin[(φ + φ0 )/2]. These factors, amongst other things, ensure the absence of so-called “backward waves”, i. e., of waves traveling “straight backwards” in the physical optics approximation. The importance of checking this property of physical optics solutions was discussed earlier. Comparing expressions (4.4) for the scalar integral I j over the j-th side on the diffraction cone with expressions for the physical optics diffraction coefficients f 0 and g0 (4.9) in the two-dimensional case (i. e., for β = 0) we see that they have identical denominators. The numerators sin φ0 and − sin φ of the physical optics diffraction coefficients follow from the vector products entering expression (4.3) for A⃗ e . Only these numerators characterize the polarization of the incident and reflected waves. Therefore, we can conclude that the denominator of the diffraction coefficients is not related to polarization. In the expression for scalar integral I j , the denominators are determined only by the position of the scattering edge relative to the direction vectors n⃗ 󸀠 and n⃗ 󸀠󸀠 of the incident and scattered fields rather than by polarization. We assume that the denominator of diffraction coefficients f and g of the rigorous solution do not depend on the polarization, although their polarization is determined in a different way than via the vector products from (4.3), i. e., not as in the expressions for diffraction coefficients f0 and g0 . The above discussion allows us to separate the vector and scalar components in expressions for the diffraction coefficients.

Elementary integration strips In the physical optics approximation, the integral (3.38) over a polygonal plate can be taken exactly, becoming an analytical expression. Using transformations (4.6) in (4.8), we have shown that the physical optics diffraction coefficients appearing as a result

4.2 Solution in physical optics approximation

|

93

of integration along the edge coincide exactly with those obtained in the model 2D problem [1] (see (4.9) and (4.10)) or [3], where the physical optics formulas are obtained with the use of integration over a semi-infinite plane plate. In (3.38), integration is performed in two directions: along the edge and in the transverse direction. The integral along the infinite edge yields the Hankel function. Integration along the transverse direction from the edge to infinity over an infinitely thin strip (elementary strip of integration) is a nontrivial mathematical problem considered in many studies (e. g., see [1, 44–49]). Here, we deal with the integration of the rigorous solution for a wedge and more complex scatterers over an elementary strip rather than with the physical optics approximation. This integral is important, because heuristic solutions to diffraction problems are constructed on its basis. In the 2D problem, the integral over an elementary strip of integration is associated with an edge element and interpreted as the pattern of the “incremental length diffraction coefficient” (ILDC), an “elementary edge wave”, or an “elementary edge current”. Having performed integration (usually, numerically) over the edge of a 3D scatterer, we can find a heuristic solution to the 3D problem. The orientation of integration strips relative to the edge may change depending on the chosen approach. It is sometimes assumed that the elementary integration strip is oriented perpendicularly to the edge (in the ILDC method [47–49]) or along the diffraction cone of the incident wave (in the author’s version of the edge wave method [44, 45]. The physical optics solution differs from the rigorous one by the factors [3] (see comments after formulas (4.9) and (4.10)). Therefore, for the model problem of diffraction by a half-plane, the pattern of an elementary edge wave is expressed through the same Fresnel integral for both the rigorous and physical optics solutions. The Fresnel integral takes the value of 1/2 on the boundary of the shadow. In contrast to this situation, the pattern of an elementary edge wave is singular on the boundary of the shadow in the far zone approximation. The rigorous solution for a wedge is integrated separately for each side in [47], where, as a result, a singularity in the direction toward the shadow is also obtained. For a 2D problem, this singularity sometimes seems to be a drawback, and the author of [47] tries to overcome it, replacing the solution by the Fresnel integral [48]. However, in a 3D problem, singularities should not be eliminated [36, 37]. With these methods of choosing the integration strip, its position is independent of the position of the observation point. It is assumed that the incident and scattering fields are primary and secondary, respectively. By virtue of the reciprocity principle, the source and observation points have equivalent positions, and should enter the analytical formulas symmetrically. This circumstance is, apropos, an additional method for checking a solution. In this Section, an alternative way of using 2D solutions in 3D problems is proposed. Instead of considering the incident wave as a primary one and, then, integrate it at the observation point, we will follow the mathematical method applied to the

94 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

analysis of the scattered signal in the physical optics approximation [58]. In that work, a solution for the scattered field is constructed depending on the behavior of phase function (3.39) on the surface of a scatterer. The phase function entering the integrand of integral I (3.38) depends simultaneously on the incident and scattered wave directions. Note that in [58] the authors consider rather large scatterers, such that the far zone condition (3.14) is not satisfied. When this condition is satisfied, it is recommended in [58] to apply methods of numerical integration. However, it turns out to be quite efficient to use strips coinciding with the vector of the gradient of phase function (3.39) (vector Δ⃗ in our case) as an elementary strip of integration. With this approach, the known pattern of an elementary edge wave from a 2D problem can be readily used instead of integrating the surface currents over a scatterer.

Features of rigorous and physical optics 2D solutions When a plane electromagnetic wave is obliquely incident on a cylindrical edge with an infinitely long (e. g., along the z-coordinate) generatrix, the field (both physical optics and rigorous) is scattered in a single way: along the diffraction cones. This property is determined by the Maxwell equations and does not depend on the shape of the cylinder directrix. There are no other solutions for this structure. At the same time, the field scattered by a fragment (having a finite length along the z-axis) of such a structure propagates in all directions. Nevertheless, the known pattern of an elementary edge wave of 2D solution can be used only in the direction of the diffraction cone according to the technique similar to the one we applied to transform 3D solution (4.6) into 2D solution (4.8). In this situation, we have to perform the same operations in the reverse order. This procedure yields an incorrect result for other directions. Apparently, it was decided to apply the edge current method [1] to the analysis of stealth objects [46], just because 2D solutions cannot be used directly according to the method recommended in the GTD. In order to obtain the pattern of the scattered signal in a direction beyond the diffraction cone, integration can be performed over an elementary edge wave, but, as has been mentioned above, this operation encounters mathematical difficulties. It is necessary to know currents over the entire integration strip, including the field near the edge. Therefore, it does not suffice to know just the pattern of an elementary edge wave. A method involving the application of 2D solutions in 3D problems with the use of complex angles is considered in Section 3.1, but this method can be employed in the case when an analytical 2D solution has already been obtained. In this section, it is proposed to apply a 2D problem solution without integration over an elementary edge wave in the case when one knows only the pattern of the 2D solution, rather than the currents over the entire surface of the scatterer.

4.2 Solution in physical optics approximation

|

95

“Imaginary edge” Consider diffraction by an edge in the case where the direction vectors n⃗ 󸀠 and n⃗ 󸀠󸀠 of the incident and scattered waves do not lie on the diffraction cone. Then, vector Δ⃗ is not oriented at the right angle to the edge. Introduce the concept of an “imaginary edge”. This is the spatial direction oriented perpendicularly to the vector such that vectors n⃗ 󸀠 and n⃗ 󸀠󸀠 are observed as if, relative to this direction, they were lying on the diffraction cone. With this choice of the direction of the integration strip, it is possible to use the known solution to a 2D problem of scattering by the imaginary edge and, thus, avoid direct integration over the elementary strip. The solution for the imaginary edge is considered in detail in Appendix C. In Figure 4.3, the top view of a plane angular sector is presented, with the real edges (shown by thick lines) are oriented at angles β/2 with respect to direction (−z). By the dashed lines, the imaginary edge oriented at an angle 𝛾/2 with respect to direction (−z) and the elementary integration strip are shown. By the vectors, the surface currents jφ⃗ and jϑ⃗ are shown for different incident field polarizations, as well as de⃗ and perpendicular compositions of these currents into the components parallel (cp) ⃗ to the imaginary edge. The problem geometry and directions of E-vectors of the (cr) incident field correspond to Fig. 4.1. The surface currents are excited by the magnetic field vectors directed along the unit vectors with corresponding subscripts. −z cr (a) jφ cp y

cr

(b) jϑ cp

γ 2 β/2

Fig. 4.3: Real and imaginary edges.

If the real and imaginary edges coincided, the observation point would lie on the diffraction cone. This would mean that a solution on an edge fragment could be found with the use of diffraction coefficients from the corresponding 2D problem for the imaginary edge. Since the real and imaginary edges generally do not coincide, the use of diffraction coefficients pertaining to the imaginary edge will yield a wrong result. However, if the diffraction coefficients are decomposed into the numerator and denominator, we can construct a new diffraction coefficient that has the numerator of the imaginary edge and the denominator of the real edge. For physical optics, this procedure is a fortiori correct, because the result does not depend on how the incident field vector is divided into components.

96 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

The following step is to apply the same approach to the rigorous solution. We use the 2D solution on the imaginary edge in same way as that employed for the case of physical optics. Thus, we avoid integration over elementary strips, because we have in hand the known diffraction coefficients. Thus, we have formulated a heuristic approach to the analysis of diffraction by a plane polygonal perfectly conducting plate. In the case of a plane scatterer, this approach raises no questions. However, difficulties can be encountered in the cases where scatterers have more complex shapes and where more complicated boundary conditions are imposed. Evidently, the shape of a virtual 2D scatterer associated with an imaginary edge differs from that of a real scatterer. We can assume that the shape changes depending on the position of the section of a real scatterer by the plane perpendicular to the imaginary edge. The change of boundary conditions should be analyzed more comprehensively. What remains now is to implement the proposed algorithm.

4.3 Solution in EECM approximation 4.3.1 Rigorous solution for oblique incident wave Heuristic solution for polygonal plate Let us construct a heuristic solution for diffraction by a plane angular sector. To this end, we substitute the vector component of the rigorous solution for the vector component of the physical optics solution. In order to obtain the vector component of the rigorous solution, we consider 2D diffraction of an electromagnetic wave obliquely incident on an infinite edge (see [30], Section 11.6). The problem geometry from [30] is presented in Fig. 4.4. We substitute the parameters indicated in the figure into the diffraction formulas for an infinite edge, with the far zone conditions imposed. Then, we factor out of these formulas the polarization component which is further substituted into the physical optics formulas for a plane angular sector. As a result, the physical optics formulas are transformed into EECM approximation. The expressions for vector components from [30] are presented in Appendix B, see equation (B.8). Considering the fields under far zone condition (3.14), which is satisfied at large distances from the edge (i. e., when kr → ∞), we obtain expressions which are much simpler than (B.8). Let the wave U1 = exp (−ikS1 ) = exp [−ik (x cos α1 cos β1 + y sin α1 cos β1 + z sin β1 )] be incident on a half-plane having its edge along the z-axis.

(4.11)

4.3 Solution in EECM approximation

|

97

y

β α x

Fig. 4.4: Problem geometry from [30].

z

Let us single out the T-polarization vectors: vectors t ⃗ perpendicular to the edge and direction vectors n⃗ 󸀠 and n⃗ 󸀠󸀠 of the incident and scattered fields, as well as vectors p⃗ perpendicular both to vectors t ⃗ and direction vector n⃗ 󸀠 or n⃗ 󸀠󸀠 . It is convenient to introduce new notation t i⃗ , p⃗ i , t s⃗ , and p⃗ s for the above vectors in the cases of incident (subscript i) and scattered (subscript s) fields. In the coordinates from [30], these vectors are t i⃗ = (sin α1 , − cos α1 , 0) , t s⃗ = (− sin ϑ1 , cos ϑ1 , 0) ,

p⃗ i = (− cos α1 sin β1 , − sin α1 sin β1 , cos β1 ) ; p⃗ s = (sin β1 cos ϑ1 , sin β1 sin ϑ1 , cos β1 ) .

(4.12)

The angles α1 , β1 , and ϑ1 are now related not the the global, but to local edge coordinates. Thus, in the case of two polarizations, we accordingly obtain two plane waves E⃗ = p⃗ i cos β1 U1 } } (TH-polarization), H⃗ = −t i⃗ cos β1 U1 } (4.13) E⃗ = t i⃗ cos β1 U1 } } (TE-polarization). H⃗ = p⃗ i cos β1 U1 } Solution of the diffraction problem is described by the formulas from [30] (see equation (B.8) from Appendix B. For oblique incident wave we obtain (simplifying the solution (B.8)) in the far zone (see Appendix B, equations (B.9)–(B.22)): E⃗ H⃗ E⃗ H⃗

= p⃗ s cos β1 U THβ } } = −t s⃗ cos β1 U THβ } ⃗ = t s cos β1 U TEβ } } = p⃗ s cos β1 U TEβ }

(TH-polarization), (4.14) (TE-polarization).

Rigorous, f and g, and physical optics, f 0 and g 0 , 2D vector coefficients for a halfplane are determined from expressions 4.9 and 4.10 [1] (the correspondence of the angles from [1] and [30] is as follows: φ0 ↔ α1 , φ ↔ ϑ1 ).

98 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

4.3.2 Substitution of polarization components of diffraction coefficients Comparing formulas (4.13) and (4.14), we note that they are symmetrical and describe the diffraction phenomenon in quite a simple way. In the case of oblique incidence on the edge, all components (4.14) of the scattered field can be obtained by replacing in the incident field expressions (4.13) the incidence angle α1 by the scattering angle ϑ1 , and function U1 by function U THβ or U TEβ , depending on which component of the incident field is T-polarized. It follows from formulas (4.12) and (4.14) that in the rigorous 2D solution Tpolarization of incident wave (4.13) is retained for all oblique incidence angles. A different situation is observed for the physical optics solution: polarizations are getting “mixed” . This circumstance determines the fundamental difference between the rigorous and physical optics solutions. Rigorous expressions for the 3D case differ from the physical optics ones only in vector components, whose values on the imaginary edge are determined from (4.14). We substitute these components for the vector component of the physical optics solution. The position angle 𝛾/2 of the imaginary edge (Fig. 4.3) enters all the formulas for the vector component of the solution. The field components are determined similarly to formula (4.6), and the edge contribution I j is replaced by vertex contribution from (4.6): E φ = ikA eφ = ϑ

ϑ

ik exp (ikR) j I [n⃗ × H⃗ 0 ]φ 2π R ϑ

[n⃗ × H⃗ 0 ]φ 1 a j exp (ikR) ϑ = . 2π R cos φ + cos φ0

(4.15)

The vector component of physical optics diffraction coefficient [n⃗ × H⃗ 0 ]φ on the imagϑ inary edge is replaced by the corresponding contribution from the rigorous solution. For the case of the oblique incidence, the polarizations of the rigorous and physical optics solutions are formed in different ways. For all angles of the oblique incidence and observation points on the diffraction cone, the T-polarization of the rigorous solution is retained. At the same time, the T-polarization of the physical optics solution is admixed to another polarization. Therefore, in order to obtain a heuristic solution on the integration strip, we replace both polarizations simultaneously rather than separately (i. e., the complete vector component of the solution that is characterized by the diffraction coefficients is replaced). Thus, instead of the vector component of the physical optics solution on the imaginary edge we use the numerators of diffraction coefficients (4.9) of the 2D solution 󸀠󸀠 󸀠󸀠 U THβ and U TEβ with allowance for the coefficients (depending on the edge orientation) of the decomposition of the TE and TH-polarizations in the polarizations of the incident and scattered fields. Consider the coordinates φ𝛾 and φ𝛾0 (similar to φ and φ0 from [1]) that are the angles between the projections of direction vectors −n⃗ 󸀠 and n⃗ 󸀠󸀠 on

4.3 Solution in EECM approximation

|

99

the plane perpendicular to the imaginary edge and the inward normal to the imaginary edge. In these coordinates, the numerators of the diffraction coefficients take the form φ𝛾0 φ𝛾 sin , 2 2 φ𝛾0 φ𝛾 = g (φ𝛾, φ𝛾0 ) (cos φ𝛾 + cos φ𝛾0 ) = −2 cos cos . 2 2

󸀠󸀠 U THβ = f (φ𝛾, φ𝛾0 ) (cos φ𝛾 + cos φ𝛾0 ) = 2 sin 󸀠󸀠 U TEβ

(4.16)

Denoting the coordinate vectors φ and ϑ of the incident (inc) and scattered (s) inc s s fields from [16] by e ⃗ inc φ , e ⃗ ϑ , e ⃗ φ , and e ⃗ ϑ , we obtain heuristic expressions for coefficients D: D ϑϑ =

H φ∞ E∞ kR kR ϑ = exp (ikR) E inc exp (ikR) H φinc ϑ

D φφ

ik2 j s ⃗ inc ⃗ 󸀠󸀠 󸀠󸀠 = I [(e ⃗ sϑ , p⃗ s ) (e ⃗ inc ϑ , p⃗ i ) U THβ + ( e ⃗ ϑ , t s ) ( e ⃗ ϑ , t i ) U TEβ ] , 2π E∞ −H ϑ∞ kR kR φ = = exp (ikR) E inc exp (ikR) H inc φ ϑ

D ϑφ

ik2 j s inc ⃗ 󸀠󸀠 󸀠󸀠 ⃗ = I [(e ⃗ sφ , p⃗ s ) (e ⃗ inc φ , p⃗ i ) U THβ + ( e ⃗ φ , t s ) ( e ⃗ φ , t i ) U TEβ ] , 2π H φ∞ E∞ kR kR ϑ = = exp (ikR) E inc exp (ikR) H inc φ

(4.17)

ϑ

D φϑ

ik2 j s ⃗ inc ⃗ 󸀠󸀠 󸀠󸀠 = I [(e ⃗ sϑ , p⃗ s ) (e ⃗ inc φ , p⃗ i ) U THβ + ( e ⃗ ϑ , t s ) ( e ⃗ φ , t i ) U TEβ ] , 2π E∞ −H ϑ∞ kR kR φ = = exp (ikR) E inc exp (ikR) H φinc ϑ

ik2 j s inc ⃗ 󸀠󸀠 󸀠󸀠 ⃗ = I [(e ⃗ sφ , p⃗ s ) (e ⃗ inc ϑ , p⃗ i ) U THβ + ( e ⃗ φ , t s ) ( e ⃗ ϑ , t i ) U TEβ ] . 2π These formulas yield a solution in the equivalent edge current (EEC) approximation, and coefficients D are coupled with the fields in the same way as in formulas (4.1). More details on derivation of equations (4.17) can be found in Appendix D. In expressions (4.17), the scalar integral I j from (4.5) is determined in the coordinates associated with the real edges forming a vertex, wheras the diffraction 󸀠󸀠 󸀠󸀠 coefficients U THβ and U TEβ are determined from formulas (4.17) in terms of the angles φ𝛾 and φ𝛾0 associated with the real edge, taking into account the correspondence φ𝛾0 ↔ α1 , φ𝛾 ↔ ϑ1 . Polarization vectors of the incident and scattered waves t i⃗ , p⃗ i , t s⃗ , and p⃗ s are also determined in the local coordinates associated with the imaginary inc edge (Fig. 4.3). The unit direction vectors of incident and scattered waves e ⃗ inc φ , e⃗ ϑ , e ⃗ sφ , and e ⃗ sϑ are defined in the coordinates from [16] (Fig. 4.1). It is assumed in expressions (4.17) that the ray aligned with vector Δ⃗ begins on one of the edges forming the angular sector and goes to infinity along this sector. We may observe another situation, when the ray begins on one edge and ends on another one. Then, formulas (4.17) become a bit more complicated. On different sides

100 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

of the angular sector, the diffraction coefficients are different, because the vectors of the inward normals to real edges being located on the imaginary edge are oriented in different directions from it. The end face edge position is determined by the direction vectors n⃗ 󸀠 and n⃗ 󸀠󸀠 of the incident and scattered waves and does not depend on the positions of real edges. Figure 4.5 shows the results of calculation according to modified formulas (4.17) compared to the data from [16]. In Fig. 4.2 from [16], functions D from (4.1) are plotted by the solid lines, whereas the points and dashed lines represent the functions plotted according to formulas (4.17). 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

2 1.5 1 0.5 0 0

45

(a) 2

90 φ→

135

180

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

1.5 1 0.5 0 0 (c)

45

90 φ→

0

45

90 φ→

135

180

0

45

90 φ→

135

180

(b)

135

180 (d)

Fig. 4.5: Comparison of coefficients D calculated by heuristic formulas with results from [16] (see Figs. 4.1 and 4.2). Errors of physical optics method (a), (b) is larger because cross-polarizations D φϑ and D ϑφ vanish in the coordinate system of Fig. 4.1. Curves corresponding to equivalent edge currents (c) and (d) agree better with results of rigorous calculation.

As we see from Fig. 4.5, EECM approximation (Fig. 4.5 (c), (d)) adequately describes the the solution. EECM performs substantially better than the physical optics approximation (Fig. 4.5 (a), (b)). In the physical optics approximation, two of four coefficients D (namely, cross-polarization coefficients D φϑ and D ϑφ ) are equal to zero over the entire range of angles φ.

4.4 Modified EECM |

101

4.4 Modified EECM Solution correction at the edge endpoint The solution obtained in EECM approximation is quite close to that from [16]. It can well be applied to many diffraction problems where high accuracy is not required. However, the EECM solution can be substantially improved if the influence of the finite length of the edge is accounted for. The finiteness of the edge is not considered in EECM where it is assumed that the equivalent current is constant over the whole span of the edge. This influence can, however, be taken into account by considering the edge as an elementary strip in the physical optics approximation. It can be evaluated quantitatively by comparing the approximate, physical optics, and rigorous coefficients on an elementary strip. As was mentioned above, the numerators of physical optics coefficients f 0 and g0 characterize the vector potential A⃗ e for different polarizations. Introduce the correcting coefficients as the ratio of the rigorous and physical optics coefficients. Multiplying the correcting and physical optics coefficients, we can obtain a rigorous analytical solution to the problem of diffraction by a polygonal plate in the EECM approximation in the case of normal incidence. Comparing expressions (4.9) for the diffraction coefficients, we obtain the correcting coefficients cp𝛾 and cr𝛾 for TH and TE polarizations in the case of normal incidence on the edge: φ𝛾

sin 2 f (φ𝛾, φ𝛾0 ) → cp𝛾 = φ𝛾 0 f (φ𝛾, φ𝛾0 ) sin 2 0

φ𝛾

and

sin 2 0 g (φ𝛾, φ𝛾0 ) 𝛾 = → cr φ𝛾 . g0 (φ𝛾, φ𝛾0 ) sin 2

(4.18)

These coefficients characterize the difference between the rigorous integral over an elementary integration strip and the physical optics integral. However, when talking about the replacement of the physical optics diffraction coefficient by the rigorous one, we do not mean that the correcting coefficient (4.18) is to be applied. This operation can only be performed in the case of normal incidence, when there is only one polarization in the solution. It is sometimes believed that the larger the scatterer size, the better agreement between the physical optics and rigorous solutions. However, formulas (4.18) proves this statement wrong. Differences between the physical optics and rigorous solutions depend not on the scatterer size, but on how close the observation point is to the shadow boundary. In the forward and mirror directions, the physical optics solution coincides with the rigorous one. This fact is well-known; it is used, for example, in the mirror antenna theory for evaluation of the gain factor for the main beam and side lobes. As the distance from shadow boundary increases, the differences between physical optics and rigorous solutions is growing for any scatterer size. The correcting coefficients (4.18) describe field variation near the edge of halfplane on the elemental integration strip when passing from physical optics to rigorous

102 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

solution. It follows from formulas (4.6)–(4.8) that relationship between the diffraction coefficients remains intact when we pass from 3D to 2D diffraction. Therefore, we can assume that the same coefficients are also applicable in 1D case. The words “1D case” mean here that in EECM the amplitude of equivalent contour current is assumed to be constant over the whole length of straight edge. It is clear from physical considerations that the amplitude must vary near the edge end, and we can assume that this variation will be similar to that exhibited by the field amplitude on the elementary strip in the case of rigorous solution, as compared with the constant amplitude of physical optics field. Therefore, the correcting coefficients (4.18) can also be interpreted as a measure of amplitude variation of equivalent contour current at the edge end due to finite length of the edge. On the other hand, it is clear that this variation of amplitude must depend upon the plane sector apex angle. Also, it can be dependent on the sector orientation with respect to the incident and scattered waves. In [16], results are presented only for quarter-plane sector and one incidence angle on the bisecting plane. These data are insufficient for comprehensive study of the solution. However, we can try to find out a correcting coefficient in a particular case considered in [16]. Let us normalize function cp𝛾 so that it is equal to unity on the shadow boundary (thus obtaining the function cp𝛾n) and let us adjust the power to which the function cp𝛾n is to be raised. The results are shown in Fig. 4.6. 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

2 1.5 1 0.5 0 0 (a)

45

90 φ→

135

180 (b)

0

45

90 φ→

135

180

Fig. 4.6: Solution in EECM approximation corrected by the coefficients D pre-normalized by (cp𝛾n)±0.7 ; agreement of results which those from [16] is improved significantly.

Clearly, multiplication of coefficients D by the same function (cp𝛾n)±0.7 with the positive (left graph) or negative (right graph) power exponents improves the agreement with the EECM results (Fig. 4.5) significantly for all the four plots. We can assume that this function depends in a certain way on the apex angle β of the angular sector because the amplitude correction should vanish as this angle grows to π. For example, it is quite reasonable to check the dependence of each coefficient D as a whole, or

4.4 Modified EECM |

103

individual contributions of the sides, on the function (cp𝛾n)cos(β/2) . On the other hand, the correction coefficient can also depend on the orientation of the angular sector with respect to the direction vectors of the incident and scattered fields. However, in order to reveal these dependencies, a reliable verification method (i. e., based on a reliable analytical or numerical solution) must be applied, with further analysis required. Nevertheless, the fact that the same function improves substantially the agreement between all four curves indicates indirectly the correctness of the chosen solution correction technique and confirms the validity of the physical assumptions on which this technique was based.

Conclusions It has been shown that a correcting coefficient can be applied for refining the EECM solution in the case of diffraction by a plane angular sector. The EECM solution has been obtained without direct integration over an elementary strip. The correcting coefficient has been obtained by comparing the patterns of an elementary strip for the rigorous and physical optics solutions.

Commentaries on the imaginary edge concept Prior to obtaining the solution on “imaginary edge”, one has to analyze the physical optics field behavior on a plate. Since the physical optics field is independent of the contour shape, solution can be written for the imaginary edge and factored out. One factor is related to the edge shape and location, the other one involves the vector component of the solution. Then, in the rigorous 2D solution for the imaginary edge we factor out the input of the edge, and substitute it by the edge input from 2D physical optics solution. At first glance, it may seem that the imaginary edge concept lacks sufficient mathematical rigor. The cylindrical edge cross-section shape varies with the angle. In the given cross-section, the wave number can also be variable. We, however, do not claim that integration over the elementary strip is performed observing all mathematical formalities. On the contrary, in this section a method is proposed which allows one to obtain a solution skipping mathematical complexities. In doing so, we rely on the fact that the scatterer considered has a simple cross-section. For diffraction by a wedge or a scatterer with a size parameter, it is necessary to analyze the imaginary edge solution, taking into account the results obtained in work [47] and in Section 3.1. However, we consider diffraction by half-plane, and results of such analysis will not affect the solution. Since the above-mentioned study would be of purely academic interest in this case, we left it aside.

104 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

Heuristic solutions for diffraction by a plane angular sector are derived sometimes by considering multiple reflection of rays off the angular sector sides. This approach does not guarantee high accuracy as well, but it adds complexity. Introduction of a heuristic plate end face correction factor allows one to simplify the solution. The simple form of resulting formulas enables us to increase the range of problems to which the proposed approach can be applied. As a validation criterion, we propose to consider agreement of the final result with a test solution. Moreover, resulting from such a check, additional refinement of the solution can be achieved heuristically, i. e., again without observing mathematical formalities. By comparing the EECM-IE and rigorous solutions, having analyzed the solution and introducing correction factors, we obtain a modified solution EECM-IE (MEECM). This solution describes the vertex wave in the best possible way.

Commentaries on the plate end face correction A signal scattered by a polyhedron consists primarily of vertex waves. Neither physical optics, nor EECM approximations describe these waves correctly. In formulas (4.18) we resorted again to heuristic approach by introducing the plate end face correction. A more rigorous solution would be to integrate the surface currents with amplitude depending on the distance to edge over the quarter-plane. However, in 2D problems such an integration does not give correct result. The corresponding physical optics integral differs from the rigorous solution due to the presence of correcting coefficients (4.18). Therefore, it was decided to avoid direct integration (which would unlikely give a correct result) and, instead, simply introduce a plate end face correction factor. An ultimate criterion for the validity of the heuristic solution can only be based on its comparison with more rigorous result, and the solution obtained satisfies indeed such a criterion. The main difference between the rigorous and physical optics solutions is that the former one takes into account T-component of the field (normal to the edge). Physical optics solution does not possess this property and can give false zeroes in some directions, resulting in substantial loss of accuracy. The physical optics approximation is constructed by “assigning” currents on the scatterer surface according to geometrical optics of the incident wave. Directions of diffraction cones in EECM and physical optics solutions coincide. In the case of normal incidence on the edge, polarizations are separated. For each of the two polarizations, coefficients cr or cp relating the physical optics and EECM solutions can be introduced. In the case of oblique incidence, polarizations of EECM and physical optics solution components are separated differently. In the EECM solution, T-polarizations are preserved. In the rigorous solution, if the E or H vector is T-polarized, it will also be T-polarized in the scattered wave. In the physical optics solution, T-polarizations

4.4 Modified EECM |

105

“mix” with each other. If E or H-vector of physical optics solution is T-polarized, the scattered wave vector will be T-polarized only in the case of normal incidence. For oblique incidence, T-polarization is not preserved. On the other hand, physical optics can give give zeroes for cross-polarized waves, regardless of the scattering wave position. When the incident wave currents lie in the plane of observation angles or are normal to it, the input to the cross-polarized wave vanishes. On the diffraction cone, EECM solution is accurate enough, while physical optics solution can suffer from significant drawbacks (for example, zeroing of crosspolarizations). For normal incidence on a vertex, the EECM differs from the solution from [16] by a coefficient which depends on polarization of the wave incident in the bisecting plane and which affects equally both vector components. In the physical sense, this coefficient takes into account that the edge has finite size and, therefore, the edge current amplitude varies near the edge end. What happens if the wave is incident not in the bisecting plane is not yet known, this question needs further study. It may well be that polarization dependence will become more complex. In the same way, upon oblique wave incidence on a two-dimensional edge, dependence of the solution upon wave polarization will be much more complex than in the case of normal incidence where the amplitude coefficients act independently on each of the two (E and H) Tcomponents of the solution. Possibly, an important point is that for coordinates [16], projections of all basis vectors of the incident and scattered fields on the angular sector plane possess geometrical symmetry with respect to this sector. On the other hand, symmetry with respect to the imaginary edge is yet different. Each method for obtaining heuristic solutions has each drawbacks. Sometimes, quite complicated integration over an elementary strip is required. In other cases, questions arise on the shape of virtual scatterer and boundary conditions for it. Nevertheless, acceptability of a heuristic solution is established by its comparison with more accurate results. Since mathematical inaccuracies are inherent to heuristic solutions, it makes sense to evaluate the accuracy of the solution as a whole, rather than of its separate components. Having applied multiplication by a normalized correcting coefficient raised to some power, we offered the simplest way of correcting a heuristic solution on the bases of physical assumptions on its behavior near the edge end. By resorting to a more sophisticated correction method, one could have achieved even better agreement of the heuristic solution with that from [16]. However, at the current stage further improvement of solution correction does not seem reasonable. The data we have in hand allow us to validate the heuristic solution for a single apex angle of the plane angular sector, single incidence direction and one plane of scattering angles. In order to study solution behavior when these parameters are varied, more extensive data obtained by reliable calculation methods are necessary.

106 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

4.5 Applicability limits of heuristic approaches in problems of electromagnetic wave diffraction by polygons and other scatterers In this section, we analyze suitability of different approaches for derivation of heuristic analytical solutions to diffraction problems. For the sample problem of electromagnetic wave scattering by a plane angular sector, various heuristic approaches are analyzed (method of physical optics, equivalent edge current method and modified equivalent edge current method). By comparing the results obtained with the rigorous solution [16], we show that in the diffraction problems for polyhedral scatterers the physical optics method can lead to significant errors, the equivalent edge current method provides qualitatively correct solution, while the modified equivalent edge current method gives the solution closest to the rigorous one.

4.5.1 Solution algorithm Method of physical optics is applied widely to approximate solution of diffraction problems. Some authors claim that physical optics approximation provides the higher solution accuracy, the larger is the scatterer relative to the wave length. As a matter of fact, this statement is wrong because accuracy of physical optics approximation is the higher, the closer is the observation point to the direction of geometrical optics light–shadow boundary. There are just two such directions in the whole space, one is forward and the other one is in the reflected wave direction. The physical optics approximation works pretty well in the angular neighborhoods of these directions, corresponding to the main beam and one or two side lopes. The larger the body sizes in comparison with the wave length, the smaller is the angular size of this neighborhood. Outside these neighborhoods, solution will deviate from that given by physical optics. In particular, physical optics method gives fairly good results for design of angular reflectors (which always reflects the geometrical optics signal in the backward direction), as well as for the determination of aperture antenna gain coefficient (as well as field pattern near the main beam). Also, physical optics method can give acceptable accuracy for signals scattered by convex objects with many reflecting points. In this respect, one may be tempted to apply this method to solution of different problems, for example, to diffraction by polyhedrons. However, in this case the method can give significantly erroneous results because solution accuracy decreases away from the shadow boundary. The irradiation maximums given by the physical optics solution are also valid for the rigorous solution, though the amplitude and polarization agree with those of the rigorous solution only near the light-shadow boundary. Recall now the steps towards the derivation of a heuristic solution for scattering by a plane angular sector. We start with the physical optics heuristic solution; this

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107

solution is obtained rigorously from the mathematical point of view, but the problem formulation on which it is based (see equations (3.1)–(3.4) from Section 3.1.1) corresponds to the physical optics approximation, rather than to the rigorous statement of the problem. Then, we obtain from formulas (3.2) the surface integral I (see equation (A.2) in Appendix A) which is further reduced to the contour integral (A.16). After this, inputs of individual edges (4.6) are factored into two multipliers: the diffraction coefficient (which, it turn, involves the nominator describing the input of polarization, and denominator related to edge orientation) and the factor depending on the distance to the edge. The physical optics solution is then refined in two steps. At the first step, the polarization input given by physical optics is substituted by that available from the rigorous solution; as a result, we obtain a solution by the equivalent edge current method (ECCM), see (4.17). In doing so, we use the solution for imaginary edge, see Appendix C, which is specific to EECM-IE solution. This approach allows us to avoid integration over elementary strips and use the readily available diffraction coefficient for the imaginary edge. The polarization input of rigorous solution is taken from book [30]. Formulas are transformed according to (4.14), assuming that the far zone condition is satisfied. Comparison of the EECM-IE solution with the rigorous one shows that they differ only by the multiplier functions which can be expressed in terms of 2D physical optics and rigorous solutions on an infinite edge. On the second step, applying these modifying multiplier functions to EECM-IE solution, we obtain the MEECM solution. Comparing the MEECM solution with the rigorous solution from [16], we conclude that MEECM solution is superior to other heuristic solutions, PO and EECM (EECM-IE), in terms of its accuracy. In the 2D case, application of multiplier functions leads to refinement of physical optics solution to the rigorous one in the case of infinite edge. In 3D case, application of multiplier functions (which are similar to 2D ones, except for some differences) refines the EECM solution to MEECM one. Without the multiplier functions, we obtain a solution for less accurate heuristic approximation. The multiplier functions are equal to unity in those directions where the stationary phase condition is satisfied. Therefore, we conclude that less accurate heuristic approximations can give exact solutions in selected directions (namely, in those where the multiplier functions are equal to unity). Thus, we can see quite a consistent behavior picture of diffraction solutions which became revealed, though, only now that the rigorous solution [16] allowing comparison with various 3D heuristic solutions became available. Thus, heuristic MEECM approach to solution of diffraction problems gives a new possibility to obtain relatively simple analytical formulas for diffraction by polygons. As any heuristic method, this approach requires availability of a rigorous solution for verification and calibration. First, a solution is obtained in the approximation of equivalent edge current method. Then, on the basis of its comparison with the rigorous solution, correction factors are found. This is the main difference of the proposed

108 | 4 Diffraction by a Plane Perfectly Conducting Angular Sector (Heuristic Approach)

method from the approach [86] which is similar the the method of edge waves: in the latter approach, additive correction to approximate physical optics solution is introduced, rather than correcting factors. The additive correction is as complex as the rigorous solution, whereas the correcting factors render the final formulas only slightly more complicated.

4.5.2 Applicability limits of heuristic solutions As was discussed in Section 4.5.1, in some cases heuristic solutions give results of very high accuracy sufficient for solving scientific and engineering problems. Nevertheless, this does not mean that the same applied to diffraction problem in its general formulation. When applying heuristic methods, one must have clear understanding of their applicability limits. Here we present few examples. 1. It is known that geometrical optics allows one to find the fields in the near zone and two selected directions only. However, by geometrical optics one can i) find out the signal reflected by mirror points of a scatterer having complex shape; ii) calculate the field in the aperture of mirror antenna by solving a complex problem of field interaction with curvilinear surface. 2. It is known that physical optics allows one to find the solution in the whole space for any scatterers and boundary conditions. However, this solution is inaccurate because the polarization component is calculated incorrectly. 3. It is known that geometrical theory of diffraction allows one to find the exact field only on the rays satisfying Fermat’s principle (i. e., on the diffraction cones). However, with GTD the total scattered signal can be obtained as a reflection from curvilinear edges, if they are available. This is similar to the method of mirror points; the difference being that there are many more stationary phase edges in space. A mirror point on the scatterer surface (point of steady phase) corresponds to a single direction in space (i. e., to a single point on the surface of an infinitely distant sphere). At the same time, a stationary phase edge (or a point on curvilinear edge) corresponds not to a single direction, but to a diffraction cone (i. e., not to a single point on an infinitely distant sphere, but to a whole line). 4. It is known that physical theory of diffraction is suitable for arbitrary threedimensional scatterers having a two-dimensional solution on the edge, i. e., by PTD solutions can be found in the whole space. However, accurate enough solution can be obtained only along the diffraction cones, while in other directions (where only vertex waves exist) the solution is inaccurate. However, vertex waves can be corrected by amplitude factors, thus making the accuracy of heuristic solution to that of rigorous reference solution.

5 Propagation of Radio Waves in Urban Environment (Deterministic Approach) 5.1 Relevance of the problem Preliminary analysis of signal level aimed at achieving required performance (e. g., establishment of communication, object detection etc.) is a problem which emerged at the very beginning of electromagnetic wave usage. However, solution of such a problem often encounters significant difficulties. In particular, this is true for the analysis of signals scattered by objects of large size and complex shape, typical for urban built environment. Development of mathematical models for the analysis of paths and strengths of VHF waves at the receiving points in urban environment is difficult due to a number of reasons specific to radio wave propagation in cities, where the conditions differ significantly from those of near-ground open pathways. Urban environment is responsible for complexity of wave propagation mechanisms and their integrated effect on the analysis results. There exist various approximate methods for the analysis of VHF signal propagation in urban environment. Empirical models are based on large number of experiments in different urban area types. Statistical models are derived from approximate concepts of signal propagation in urban environment. Partially deterministic approach allows the analysis of signal blockage near large-scale objects (city blocks). There exist models combining the empirical, statistical and partially deterministic approaches. The above models are suitable for approximate field evaluation, but none of them allows one to obtain the arrival angles for the given positions of the receiver and transmitter, which is very important in direction finding problems. Therefore, one has to resort to purely deterministic methods (numerical or analytical). For the reasons discussed above, even these, the most accurate, methods cannot provide high accuracy of signal power prediction, though they predict sufficiently accurately the wave arrival angles important for the direction finding task. Hybrid numerical methods allow one to obtain accurate enough solutions for large objects. In contrast to rigorous numerical methods, hybrid methods provide an optimal balance between the accuracy and rigor of the solutions. However, for the reasons to be discussed below, application of numerical methods to simulation of radio wave propagation in the urban environment conditions is not reasonable. It follows that the most suitable for deterministic analysis of VHF signal level in urban environment are approximate analytical methods, namely, the method of physical optics. According to this method, to find out the field scattered by an object of complex shape, perturbation introduced by this object to the incident field is first calculated, after which this perturbed field is integrated over the object surface. The

110 | 5 Propagation of Radio Waves in Urban Environment (Deterministic Approach)

essence of method of physical optics is that the field scattered by a finite-dimension object is taken on the object surface (including near its edges) to be such as it would be in the absence of edges, i. e., for an infinite-size object. In this way, we neglect the incident field variation caused by edge effects, while taking into account finite geometrical sizes of the scatterers, their shape and material properties. Physical optics approximation does not give accurate quantitative values, but it allows the positions of points of stationary phase and maximum signal directions to be determined correctly. These parameters of scattered signal coincide for rigorous and approximate calculations. Among the advantages of physical optics approximation is that there exist formulas reducing the surface integral of a plane wave to an integral over the contour bounding this surface. For plane polygons (of which most of urban area objects consist), these formulas take an especially simple form. Also, even if more accurate analytical or numerical approaches are applied, the physical optics solution will be used as the initial approximation. It can be refined by adding the edge waves (if known, of course). To be fair, we should say that the question of how reasonable it is to refine the physical optics solution needs further study because the refined solution may well deviate from the true one more significantly that the approximate one due to the difference of the real edge shapes from those employed in the model. It is known that a VHF signal propagating in urban environment becomes distorted in comparison with a signal propagating in open space due to the following phenomena: 1. Scattering by building walls and roofs, including scattering by windows, cornices, architectural elements, rain-water pipes, roof guard rails, stacks, etc. 2. Scattering by other static elements of urban environment, including road paving, lamp posts, wires, etc. 3. Scattering by moving objects. 4. Scattering and attenuation upon propagation through canopy, smokes, etc. In the presence of vast shadow zones, the multiply reflected and scattered waves form multi-beam fields with complex interference patterns, also, the signal levels are characterized by rapid and deep spatial and temporal variation. Due to these features, models for VHF wave propagation in urban environment were mostly developed on the basis of statistical approach [67–69]. Empirical models for field strength [88, 89] were based on quite extensive experiment data obtained in urban areas of various types. In these models, the principal attention was paid to statistics of spatial and temporal variation of field strength. Statistical models (for example, [90]) are based on statistical description of urban environment, with approximate calculation of field strength by the Kirchhoff formula and its subsequent averaging. Since the statistical methods do not provide sufficient predictive accuracy necessary for practical applications, deterministic methods for calculation of VHF field in

5.2 Specifics of radio wave propagation in urban environment | 111

urban environment were developed [91–94]. The highest prediction accuracy could have been achieved by numerical a hybrid methods. However, rigorous statement of the problem for numerical simulation for urban area conditions is practically impossible due to the complexity, large sizes and multitude of scattering urban objects, as well as absence of exact data on their geometrical parameters and electrodynamic properties of their construction materials. Therefore, simulations on the basis of numerical methods are not viable because they require sophisticated software and powerful computational resources. A partially deterministic method was developed in [91] in application to shadow zone calculation near large-scale objects (city blocks). In [92], a model combining the empirical, statistical, and partially deterministic approaches was proposed. The models described above are suitable for approximate calculation of field level, but neither of them allows the arrival direction angles corresponding to the given positions of transmitter and receiver to be calculated. Therefore, it is more preferable to apply purely deterministic methods which allow the above field characteristics to be obtained with high accuracy. In [93, 94], a deterministic three-dimensional computational method was developed for the determination of paths, arrival angles and field strength for millimeter wave length range. Topographic city map is used to describe the shape and sizes of buildings. Reflection from building surfaces with periodic inhomogeneous structure is described by the effective reflection coefficient determined from method of physical optics, whereas scattering by rough surfaces is described by the Rayleigh scattering theory. In this section, we present a unified deterministic method which allows one, by the method of physical optics, to take into account such propagation mechanisms as reflection, scattering, and diffraction by buildings and other urban environment elements, as well as to formulate technical requirements to calculation of paths and arrival angles. By deterministic, we understand such an analysis method in which only deterministic wave propagation mechanisms (scattering by particular objects and attenuation by various media) are taken into account. No a priori assumptions are made upon statistical averaging of signal structure on its way from the transmitting to receiving antennas.

5.2 Specifics of radio wave propagation in urban environment Urban environment has a number of specific features which have to be taken into account when developing predictive tools for radio wave propagation. It is not reasonable to apply rigorous calculation methods because of large object sizes, uncertainties in the input parameters related to geometry and material properties. Therefore, it is clear beforehand that accurate calculation of the signal is not feasible (by the

112 | 5 Propagation of Radio Waves in Urban Environment (Deterministic Approach)

accuracy, we understand the degree of agreement between the calculated data and experimental measurements). On the other hand, the advanced analytical and numerical predictive methods of the diffraction theory developed recently allow one to obtain more accurate results than those given by statistical formulas or purely geometrical optics. In the urban environment, there exist numerous shadow zones which are of interest from the radio wave propagation point of view, including interference and multi-beam propagation. Among other urban area features, we point out large number of polyhedrons and linear edges, which makes the results obtained in this book very relevant. Upon multi-beam signal propagation in urban environment, several groups of signals of different powers can be distinguished: 1. Line-of-sight signal. If such a signal exists (only one such signal is possible), it will be the strongest one. 2. Signals reflected specularly off walls and road cover. The smaller the number of sequential reflections, the stronger the signal. These signals are of interest in the absence of the signal described in item 1. Many currently available programs simulating radio wave propagation in urban environment are limited to signals described in items 1 and 2, i. e., diffraction is not taken into account. Therefore, zones of geometrical shadow are left out of consideration. 3. Signals scattered by the edges, i. e., by building walls and roofs. In the absence of signals described in items 1 and 2, it is these type of signals is transmitted to the receiver antenna. In this case, the accessible urban area becomes much larger. One can take into account signal behavior in the shadow zone, including interference and multiple beams. It is this kind of calculations that can benefit from the results presented in this book. 4. Signals scattered by vertices. Objects of urban environment contain many vertices which provide a limited input into the scattered signal. On the first stage, we leave vertices out of consideration because in urban area conditions there available many signals from plane surfaces and edges (items 1–3), while the vertices give an input of minimum amplitude. Taking into account the input of vertices can improve the accuracy of calculations in the parts of geometrical shadow inaccessible neither by line-of-vision signals, nor signals from emitting surfaces and edges. Nevertheless, if refined calculations taking into account the inputs of vertices are required, results of this book will also be handy.

5.3 Design formulas 5.3.1 Zone significant for radio wave propagation Since the sizes of urban area objects are very large compared to the wave length, solution of the electrodynamic boundary-value problem for such objects cannot be found

5.3 Design formulas

| 113

not only rigorously, but even in the physical optics approximation, because numerical integration of rapidly oscillating signals over large surfaces is very computationally expensive. We make use of the known fact that the scattered signal is mostly determined by the areas and edges of stationary phase. Therefore, we can consider only the first Fresnel zone around the geometrical optics ray, leaving aside all other spatial zones. It is the first Fresnel zone that we consider the zone significant for wave propagation (ZSWP). Everything that does not fall within the first Fresnel zone is dropped from the analysis. Ray tracing is performed according to geometrical optics. We assume that the ray can propagate from the source to the receiver only by one of the three paths: (1) Along the line of sight, (2) By specular reflection off surfaces (for example, tar, walls, windows, etc.), and (3) Along the diffraction cones determined by relative location of the source and receiver points, as well as spatial position of an edge (for example, roof edge, or building corner). Such an approach allows us to account comprehensively for multi-beam propagation of radio signals in urban environment conditions. Though the mathematical problem of ray tracing is the key element of our approach, we consider it beyond the scope of this book. Namely, we assume that ray tracing (according to items (1), (2), and (3) in the previous paragraph) has already been performed, and we focus our efforts on the calculation of the diffraction coefficients for reflection off surfaces and diffraction by edges (Fig. 5.1). This problem is by far nontrivial because the effect of zone significant for wave propagation (ZSWP) has to be taken into account. As we stated above, limiting the urban area objects to those falling within this zone allows the computational complexity of the problem to be decreased dramatically.

(a)

(b)

Fig. 5.1: Geometry of three rays in urban environment. Around the ray reflected specularly by the windowed wall, ZSWP is shown.

114 | 5 Propagation of Radio Waves in Urban Environment (Deterministic Approach)

In Fig. 5.1 (a), a fragment of urban area is presented; the source (upper point) and receiver (lower point) are located away from the line-of-sight zone, the signal paths corresponding to the stationary phase condition (see Section 5.3.6) are shown by the dashed straight lines. Three path types are shown, corresponding to i) reflection off a plane windowed wall, ii) scattering by a roof edge, and iii) scattering by a building corner. The ray paths are traced according to the Fermat principle (as in GTD method). Therefore, with respect to the source point, the receiver point is located on the diffraction cone. To keep the drawing clear, the zone significant for wave propagation is shown for one ray only (the one reflected off the wall). According to the calculation method, we take into account only those parts of urban environment which fall within ZSWP, while influence of all other areas is neglected. This is illustrated in Fig. 5.1 (b), where intersection of ZSWP with the wall is shown by the thick oval, and the piece of window falling within ZSWP is drawn by thick lines. Formulas for the sizes of ZSWP will be given later. ZSWP is taken into account in the following way. First, we find out diffraction objects falling within ZSWP (surface and edge fragments). Then we discard everything beyond ZSWP, and consider the remaining part in exactly the same manner as we treat a scatterer within ZSWP on which a stationary phase area (surface or edge) exists. This latter object was studied in detail in the previous chapters. All heuristic formulas necessary to calculate power attenuation of reflected or scattered beam on its path from the transmitter to the receiver are available in this book.

5.3.2 Reference solutions As we said before, in order to derive the scattered field solutions we rely on the formulas for diffraction by plane scatterers, including polygons. This solution is expressed in terms of a surface integral which can be reduced to a contour integral. In turn, in the “straight forward” and “mirror reflection” directions (where inputs of all edges have singularities), we can obtain easily the power level at the observation point using directly the physical optics surface integral. In the absence of obstructions (i. e., in the line-of-sight zone), the signal at the observation point depends on the distance between the transmitter and receiver. On the other hand, this signal can be obtained by multiplying the power flux density transmitted through the first Fresnel zone by the area of this zone. If some part of the Fresnel zone is obstructed by an obstacle diminishing the area through which power is transmitted, the signal at the observation point will be diminished proportionally to the area reduction. In this case, the scattered signal power is proportional to the area of that part of he scatterer with falls within the Fresnel zone. Also, power of the signal scattered by an individual edge in the forward and mirror directions is proportional to the ratio of the length of this edge and perimeter of the scatterer. In other directions

5.3 Design formulas | 115

lying on the diffraction cone, signal power decays with angular distance to the forward and mirror directions. Although we do possess two-dimensional solutions for diffraction by an infinite edge, think that it is more correct to use the three-dimensional solution for diffraction by a plane polygon bounded on one side by the scatterer contour, and on the other side by the first Fresnel zone boundary. The Fresnel zone is elliptic in cross-section, but if we approximate it by a polygon, the analysis can be performed completely by the formulas available in this book. The reference solution which can be used in order to study radio wave propagation in urban environment are those pertaining to surfaces, edges, and vertices. For simplicity, we do not make use of the vertex solutions; rather, we take into account only reference solutions for surfaces and edges. Because our approach relies on the concept of ZSWP, we never use solutions for semi-infinite scatterers (half-plane and angular sector). Semi-infinite scatterers with the far zone condition imposed give infinite signal in the “straight forward” and “mirror” directions. This circumstance prevents power normalization in these directions. Instead of solutions for semi-infinite scatterers, we apply the solution describing diffraction by a three-dimensional polygonal plate of finite size. Solutions presented in this book imply that the far zone conditions are satisfied. If this condition is not satisfied for a large plate, it can be subdivided into smaller pieces, so that for each of them the far zone condition holds true. These considerations, however, are relevant to an abstract scatterer of large size, rather than to the chosen calculation method for the analysis of radio wave propagation in urban area conditions because we can assume that for the first Fresnel zone the far zone condition is always satisfied. In the heuristic approach to the analysis of radio wave propagation in urban environment, we rely on the physical optics approximation. If necessary, it can be refined by the methods presented in this book. Such a refinement can be applied to solutions for diffraction by an edges (including the one located on a scatterer with non-ideal boundary conditions), as well as to vertex waves.

5.3.3 Mutual coupling between two antennas In this section, a formula for mutual coupling between two antennas is presented, its applicability condition is discussed as well as the reflection and transmission coefficients on a boundary between two media; also, properties of some materials relevant to urban area constructions are presented. Upon electromagnetic wave reflection off a plane homogeneous surface (for example, a wall, or road covering), the reflected wave power is determined by the formula for mutual coupling between two antennas (see [95], page 26): 󵄨󵄨 ⃗ 󵄨󵄨 󵄨󵄨E R 󵄨󵄨 λ2 G1 G2 |V0 |2 󵄨 󵄨, P1 = P0 A , = (5.1) |V | 12 0 󵄨󵄨 ⃗ 󵄨󵄨 󵄨󵄨E0 󵄨󵄨 (4π)2 (R1 + R2 )2 󵄨 󵄨

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where P0 and P1 are signal powers in the transmitting and receiving antennas, λ is the wave length, G1 and G2 are transmitting and receiving antenna directivity indices, R1 and R2 are the distances between the mirror point and the transmitting and receiving antennas, A12 is the attenuation coefficient on the path with reflection (in the current approximation, A12 = 1), V0 is the Fresnel reflection coefficient of the wall, E⃗ 0 and E⃗ R , H⃗ 0 and H⃗ R are electric and magnetic fields (subscript 0 is for incident fields, R for reflected, T for transmitted) determined by formulas (see page 28 in [96]): E⃗ R = V3 E⃗ 0 −

⃗ (E⃗ 0 , n)

H⃗ R = V2 H⃗ 0 − V3 =

sin2 ϑ ⃗ (H⃗ 0 , n)

[(V2 cos 2ϑ + V3 ) n⃗ + cos ϑ (V2 + V3 ) n⃗ 󸀠 ] ,

(5.2)

[(V3 cos 2ϑ + V2 ) n⃗ + cos ϑ (V2 + V3 ) n⃗ 󸀠 ] ,

(5.3)

sin2 ϑ

cos ϑ − √ ε − sin2 ϑ , cos ϑ + √ ε − sin2 ϑ

V2 =

ε cos ϑ − √ ε − sin2 ϑ , ε cos ϑ + √ ε − sin2 ϑ

(5.4)

where ϑ is the angle between the surface normal n⃗ and direction to the receiver, n⃗ 󸀠 is a unity vector in the incident wave propagation direction, ε = ε󸀠 + iε󸀠󸀠 is the complex dielectric permeability, V2 and V3 are the reflection coefficients for vertically and horizontally polarized waves. Under a vertically polarized wave, we understand the wave for which vector E⃗ 0 has only components lying in the incidence plane XOZ, while for horizontally polarized wave the vector E⃗ 0 is normal to the incidence plane. For the refracted (i. e., passing from a medium with dielectric permeability ε1 to that with permeability ε2 ) waves E⃗ T and H⃗ T , we have: E⃗ T = W3 E⃗ 0 −

⃗ (E⃗ 0 , n)

{[nW3 − W2 (n sin2 ϑ2 + cos ϑ2 cos ϑ)] n⃗ n sin2 ϑ + (nW3 cos ϑ − W2 cos ϑ2 ) n⃗ 󸀠 } ,

H⃗ R = W2 H⃗ 0 −

⃗ (H⃗ 0 , n)

{[W2 − nW3 (n sin2 ϑ2 + cos ϑ2 cos ϑ)] n⃗ sin2 ϑ + (W2 cos ϑ − nW3 cos ϑ2 ) n⃗ 󸀠 } ,

W3 =

(5.5)

2 cos ϑ , cos ϑ + √ ε − sin2 ϑ

W2 =

(5.6)

2 cos ϑ , ε cos ϑ + √ ε − sin2 ϑ

(5.7) ε2 , sin ϑ = n sin ϑ2 , ε1 where W2 and W3 are transmission coefficients for vertically and horizontally polarized waves, ε1 = ε, while n and ϑ2 are determined by equation (5.7). The applicability condition for formula (5.1) is that the ellipse enveloping the first Fresnel zone, having semi-axes a x and a y with center located at the mirror point, must all be contained by the reflecting surface. The sizes of ellipse are determined by n=√

2a x =

λ R1 R2 2 √ , cos ϑ 2 R1 + R2

2a y = 2√

λ R1 R2 , 2 R1 + R2

(5.8)

5.3 Design formulas | 117

where the distances R1 and R2 must satisfy the condition R1 , R2 ≫ √ λ ⋅ min(R1 , R2 ),

(5.9)

i. e., not to be too close to the reflecting surface. If the Fresnel zone (5.8), (5.9) falls completely within the scatterer surface, the reflected signal power corresponds to (5.1). If the condition (5.9) is not satisfied, i. e., some part of the ellipse goes beyond the reflecting surface, the reflected signal power P1 determined from (5.1) is decreased in the same proportion as the remaining “illuminated” surface of the scatterer is decreased with respect to the area of the original ellipse. The Fresnel reflection coefficient V0 in equation (5.1) depends on the complex dielectric permeability if the reflecting surface material. For a perfectly conducting surface, V2 = 1, V3 = −1, and W3 = W2 = 0, while properties of some other materials employed in urban construction are given in Table 5.1. Table 5.1: Dielectric permeability of some materials used in urban constructions Material

Wave length, mm

ε󸀠

ε󸀠󸀠

Concrete Concrete Concrete Cement plaster Red brick Red brick Silica brick Silica brick Glass Chalk plaster Pine woodboard

8 6 2 6 6 2 6 2 2 6 2

5.5 5.7 5.55 5.7 4.2 3.2 3.4 3.3 3.8 2.9 2.0

0.5 0.3 0.36 0.14 0.03 0.11 0.09 0.14 0.19 0.02 0.08

To derive heuristic formulas, we describe diffraction by semi-transparent scatterers by the method offered in [37, 60, 62]. Formulas for diffraction by semi-transparent plates in the physical optics approximation are presented in Appendices A, G, and H.

5.3.4 Energy relationships When applying two-dimensional solutions to three-dimensional problems, one has to ensure that energy relationships are observed. Edge diffraction solutions are expressed in terms of two-dimensional formulas which possess different divergence properties than three-dimensional ones. Also, two-dimensional formulas are derived

118 | 5 Propagation of Radio Waves in Urban Environment (Deterministic Approach)

assuming excitation by a plane wave, corresponding to an infinitely distant source of infinite power, i. e., a mathematical abstraction. In these circumstances, energy relationships have to be chosen quite carefully in order to obtain correct amplitude of scattered signal in the heuristic formulas. Since we make use of the reference solutions for semi-infinite scatterers and, also, assume the far zone condition to be satisfied, the scattered signal tends to infinity in the “straight forward” and “mirror reflection” directions. Therefore, special care must be taken with respect to energy normalization. This normalization is based on the solution to three-dimensional problem in the “straight forward” and “mirror reflection” directions, while upon deviation from these directions the scattered signal decays according to the diffraction solution for a plane polygonal plate. Far away from the scatterer, the power flux density is calculated at the point where the second antenna is located by the formula 󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨E ϑ (ϑ, φ)󵄨󵄨󵄨 + 󵄨󵄨󵄨E φ (ϑ, φ)󵄨󵄨󵄨 S2 (ϑ, φ) = 󵄨 , 2

(5.10)

where E ϑ (ϑ, φ) and E φ (ϑ, φ) are calculated by formulas (3.4), (G.20), and (G.19), (R, ϑ, φ) are spherical coordinates related to the stationary phase point. In (G.19), we have R = R 2 , while for E0x and H0x we substitute the fields propagating from the first antenna satisfying the condition G1 |E0x |2 + |H0x |2 , = S1 (ϑ, φ) = p1Σ 2 4πR21

(5.11)

where p1Σ is the total output power of the first antenna. Fields E0x and H0x correspond to two polarization components for a scattering screen in plane XOY. To calculate signal scattering by a polygonal screen, one has to apply formulas for electromagnetic potentials of the form (3.37), substituting into them integral I evaluated by (3.40). The latter formula is valid for all observation points, except the “straight forward” and “mirror reflection” directions. In these directions, terms (3.42) from equation (3.40) become singular. Therefore, in these directions one has to evaluate the integral I by formula (3.43). For scattering in the “mirror” direction, it is useful to compare the result obtained by formula (3.40) into which I from (3.43) has been substituted, with the result obtained by formula (5.1). They must coincide provided that in (3.43) SF λR1 R2 πλ R1 R2 I= = , where S F = . (5.12) 2 R1 + R2 (π/2) R1 + R2 Here, S F is the area of the first Fresnel zone. More details on this substitution will be given below (see equation (5.15)). If we approximate the shape of oval Fresnel zone by a polygon, we will obtain the better agreement, the larger number of sides has the polygon filling ZSWP. Formula (3.40) is applied in the case where the polygonal screen completely falls within the Fresnel zone determined by equation (5.8). In this case, by R1 and R2 the

5.3 Design formulas | 119

distances from the edges to the source and observation points are implied. If the screen sizes exceed those of the Fresnel zone, the edges must be analyzed individually, making use of expressions involved in formula (3.40) with a j taken to be the linear size of the corresponding Fresnel zone, rather that the total side length. Input of the contour bounding the Fresnel zone is also added to the solution. Thus, the reflecting surface size does not affect the reflection coefficients (in the sense of equation (5.4)), but it can be shown that the relationship between the sizes of reflecting surface and of corresponding Fresnel zone affects signal divergence. If the Fresnel zone fits inside the scattering screen, the divergence will be of the order of (R1 + R2 )−2 , as in formula (5.1); in the opposite case, where the whole scattering polygon its inside the Fresnel zone, the divergence will be of the order of (R1 ⋅ R2 )−2 , as follows from equations (G.19), (3.4), and (3.40).

5.3.5 Fresnel zone In this section, we consider the concept of using the Fresnel zone in calculation of diffraction by fragments of urban environment. Also, applicability limits of heuristic formulas are obtained, depending on whether the far zone condition holds true or not. Consider two aperture-type antennas with aperture areas S1 and S2 , separated by a distance R (see Fig. 5.2). As is known from the aperture antenna theory, the field patterns in the main beam direction are G1 = 4πS1 /λ2 and G2 = 4πS2 /λ2 , where λ is the wave length. On the other hand, the total power in the aperture of the second antenna p2Σ is equal to the product of power density at the distance R from the first antenna (i. e., power density in the second antenna aperture, p2S ) and the second antenna surface area: p1Σ G1 λ2 S = p G 2 1Σ 2 4πR2 4πR2 4π λ2 G1 G2 S1 S2 = p1Σ , = p1Σ 2 2 (4π) R (λR)2

p2Σ = p2S S2 = G1

(5.13)

which agrees with formula (5.1) if we substitute into it R = R1 + R2 , where R1 and R2 are the distances between the antennas and the signal-scatterer interaction point. In the radio wave propagation theory, the zone significant for wave propagation is considered to be the first Fresnel zone. It bounds the spatial domain in which the phase differs from the phase on the central line not more than by π/2. If we assume

Fig. 5.2: ZSWP in free space between two aperture antennas

120 | 5 Propagation of Radio Waves in Urban Environment (Deterministic Approach)

that a scatterer is represented by the Fresnel zone of area S F (5.12) located at distances R1 from the first, and R2 from the second antenna, then, according to the concept being discussed, the power arriving to the Fresnel zone from the first antenna is p FΣ , while power p2Σ arrives to the second antenna not from the first antenna, but from the Fresnel zone: S1 S F , (λR1 )2 S F S2 S1 S F S F S2 S1 S2 = p FΣ = p1Σ = p1Σ S2 , 2 F (λR2 )2 (λR1 )2 (λR2 )2 (λ2 R1 R2 )

(5.14)

π 2 (λR1 R2 )2 π 2 S1 S2 = p1Σ ( ) ( ) . 2 2 [λ (R1 + R2 )] 2 (λ2 R1 R2 ) 2 (R1 + R2 )

(5.15)

p FΣ = p FS S F = p1Σ p2Σ therefore p2Σ = p1Σ

S1 S2

2

Comparing the latter formula with expression (5.13) we see that, up to a constant coefficient (π/2)2 , the power transmitted through the first Fresnel zone is equal to the power transmitted from the first to the second antenna. Diminishing the linear size of the Fresnel zone by a factor of (π/2)1/2 ≈ 1.253 , we obtain the zone which transmits the power exactly equal to that transmitted from the first antenna to the second one (see also the reduction in lineal size in formula (5.12)). When analyzing radio wave propagation in urban environment one can, as the first approximation, consider only this zone, assuming that at all its points the wave amplitude and phase are constant, whereas beyond this zone the amplitude vanishes. In higher-order approximations, subsequent zones can be included; this would improve accuracy but, at the same time, increase the amount of calculations necessary. We refer to the zone described above as the zone of stationary phase (ZSP). ZSP is located near the scatterer, it depends on the distances to the transmitting and receiving antennas. If the scatterer obstructs ZSP completely or partially, one has to apply the formula for reflected signal, correcting the amplitude accordingly. If the point of stationary phase belongs to the straight line connecting the transmitting and receiving antennas, the power of signal reaching the receiving antenna is equal exactly to the corresponding fraction of ZSP power, namely, corresponding to the surface part of ZSP in this particular reflection or diffraction act. This is the essence of the amplitude correction. For all other parts of ZSP (located beyond the scatterer and, therefore, containing its edges), one has to apply the formula for diffracted signal. Before, we defined the zone significant for radio wave propagation (ZSWP) around the geometrical optics path, and the reason was rather technical (to limit the amount of computations). We can say now that this zone coincides with the zone of stationary phase (ZSP) which was introduced from physical considerations. If a scatterer falls within ZSP, or, equivalently, within ZSWP, one has to apply the formula for polygon. If the scatterer obstructs ZSP (ZSWP) partially, one has to consider, as the scattering surface, only that part of the scatterer surface which falls

5.3 Design formulas

| 121

within ZSWP; in this case, the effective scatterer boundary consists of two parts: the part of actual scatterer boundary falling within ZSWP, and the part of ZSWP boundary lying on the actual scatterer surface.

5.3.6 Derivation of heuristic formulas We follow the plan discussed above. First, we obtain the power of waves reflected from an infinite plane surface. This power is expressed in terms of reflection, refraction,and transmission coefficients. Then, we bound the surface fragment and construct, in the physical optics approximation, a solution for a plane polygonal plate with non-ideal boundary conditions. Thus, calculation of scattering by an edge in the physical optics approximation is performed in the following steps: 1. Find out the points of stationary phase on the scatterers. If such a point corre󵄨 󵄨 sponds to mirror reflection, scattering occurs on a plane fragment and 󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨 = 0. Two cases are possible: the scatterer is small in size (obstructs ZSWP only partially), or large in size (obstructs ZSWP completely). In diffraction calculations we consider, as the diffracting object, the intersection of the spatial domains occupied by the scatterer and ZSWP. If the observation point is not on line of sight from the source, but is located on the diffraction cone, the scatterer is equivalent to a “separate” edge. Therefore, condition (Δ,⃗ ρ)⃗ = 0 holds true. The object boundaries are determined in the same way as in the case of scattering by a plane fragment. 2. Determine the geometrical and physical parameters involved in the formula. 3. Substitute the parameters into the formula and obtain the diffraction coefficient. We first place an auxiliary observation point at the same distance from the scatterer in the line-of-sight area, determine the reference signal level according to the energy relationships (see Sections 5.3.4 and 5.3.5), and then determine its attenuation due to angular deviation from the line of sight according to the formulas for diffraction by a plane polygon.

5.3.7 Solution algorithm In this section, a solution algorithm for radio wave propagation in urban environment is presented. 1. Ray tracing between the transmitting and receiving antennas is performed. Ray paths are polygonal chain lines in space, starting at the transmission and ending at the receiving antennas, with the nodes corresponding to the points of stationary phase. The line segments are directed, i. e., they are vectors. The points of stationary phase (PSP) are located on the plane, or on the ends of scatterers. If a PSP

122 | 5 Propagation of Radio Waves in Urban Environment (Deterministic Approach)

2.

belongs to the scatterer plane, the line segments joining at this point are oriented according to specular reflection law. If a PSP is located on the screen edge, the line segments joining at this point are located on the diffraction cone. In order to limit the number of objects taken into account in the scattering analysis, introduce the zone significant for radio wave propagation (ZSWP). This approach is justified from the physical point of view because electromagnetic wave energy traveling from the transmitting to receiving antenna is mainly contained in this zone. In order to simplify the formulas, assume that the cross-section of ZSWP is not a circle, but a regular 24-gon inscribed in the circle bounding ZSPW, so that the vertices of the polygon lie on the circle at equal angular intervals of 15 degrees. The radius of the circle is proportional to that of the first Fresnel zone (5.8): R ZSWP = a y √

2 λ R1 R2 , =√ π π R1 + R2

then π (R ZSWP )2 =

λR1 R2 R1 + R2

(5.16)

(see also (5.12)). We consider scattering by only those elements which cross ZSWP. Consider the urban environment elements as an ensemble of polyhedrons of various shapes. When signal scattering along the ray path is calculated in physical optics approximation, we can substitute the polyhedrons by plane semi-transparent polygonal screens. The “illuminated” screens (i. e., those which cross ZSWP, at least partially) are assigned the properties of corresponding urban construction material, whereas the “shadow” parts of screens falling within ZSWP are assumed, in the first approximation, to be perfectly absorbing, so that the physical optics solution for a plane plate was applicable in the shadow zone. In this way, it is possible to calculate approximately diffraction by a building of finite thickness (Fig. 5.3).

Fig. 5.3: Ray geometry for diffraction by a building of finite thickness. The shadow side of the building falling within ZSWP is assumed to be perfectly absorbing, so that calculations can be performed on the basis of physical optics approximation for plane screens.

Note on the heuristic approach. In the previous paragraph, the word assumed implies that we introduce some systematic error into the method in order to be able to use the available solution. When simplifying the calculations, one has to be fully aware of the problem features that are left aside.

5.3 Design formulas | 123

Intersections of semi-transparent polygonal screens ans polygonal ZSWP form polygons located on the ray path, i. e., in the neighborhood of the points of stationary phase. Application of formulas for scattering by polygonal screens allows us to calculate beam attenuation along the path. In the case of mirror reflection, the “mirror” formula is applied, while scattering by edges is evaluated by the “diffraction” formula. The same formula can be used whenever it is necessary to analyze signal propagation in directions deviating from ZSWP.

6 Analytical Heuristic Solution for Wave Diffraction by a Plane Polygonal Scatterer 6.1 Introduction Development of heuristic solutions for diffraction of generic wave fields, including elastic waves, is important for a number of reasons. Firstly, this phenomenon is being studied quite actively in view of various engineering problems relevant to remote sensing of seismic objects, including solution of inverse problems. Secondly, these solutions are important because of high complexity of two-dimensional and, in particular, three-dimensional elastodynamic problems. Thirdly, availability of numerical methods allows one to obtain verification solutions. From the physical point of view, diffraction of elastic wave is significantly different from the electrodynamic case. The differences lie not only in physical nature of the field, but also in polarizations, field reflection and propagation with different polarizations etc. An elastic wave can propagate at a given frequency with different polarizations, with particles of the medium moving in the wave propagation direction (longitudinal polarization) or normal to it (transverse polarization). Longitudinal and transverse waves propagate at different speeds. Upon the incidence of a longitudinal wave on an interface between two media, its reflection and transmission occurs at various polarizations and in various directions (including those different from the mirror and forward directions). This situation reminds electromagnetic wave refraction on the interface between two media. In Fig. 6.1, elastic wave interaction with a plane boundary between two different media is sketched. Upon incidence of elastic wave (1) on the boundary, there appear two reflected waves (2) and (4) with longitudinal and transverse polarizations, as well as two transmitted waves (3) and (5) with longitudinal and transverse polarizations, respectively. Accordingly, the directions in which the particles of elastic medium are moving in the cases of longitudinal and transverse polarizations are shown by arrows (6) and (7). Despite the above-mentioned differences, wave fields of different physical nature possess common features, which allows one to use solutions for a field of some physical nature as a basis for solving problems for fields of another nature. Therefore, results presented in this book can be applied to a very wide spectrum of problems. Rigorous solution of a boundary-value problem describing diffraction of elastic waves is quite complicated. In order to obtain the diffraction coefficients of approximate solution, we choose another approach in this section. Taking the solution of electromagnetic field diffraction by a perfectly conducting half-plane as the basis, we obtain the diffraction coefficients from heuristic reasoning, and then correct them by comparison with numerical solutions.

126 | 6 Analytical Heuristic Solution for Wave Diffraction by a Plane Polygonal Scatterer

1

2

4 7

6

6

6 7 5

3

Fig. 6.1: Elastic wave incidence on a plane interface between two media: 1. incident longitudinal wave; 2. reflected longitudinal wave; 3. transmitted longitudinal wave; 4. reflected transverse wave; 5. transmitted transverse wave; 6. particle motion direction in longitudinal wave; 7. particle motion direction in transverse wave.

6.2 Problem formulation for elastic wave diffraction Consider diffraction of an elastic wave by a perfectly conducting plate. The problem geometry is shown in Fig. 6.2. Let the elastic wave propagate in a cubic domain with the edge length of 1000 m. The scatterer is located at the coordinate origin, the source is located at the point (−500, 0, −500). We restrict ourselves by the case where the source and observation points belong to the plane y = 0; to this end, we place the observation points evenly on the interval [(−500, 0, −500), (−500, 0, 500)] with the spacing of 100 m [61]. x(m) −500 −500

0

500

z(m) 0

500 −500 y(m)

0 500

Fig. 6.2: Problem geometry for elastic wave diffraction.

We choose the wave length such that the far zone condition be applicable in this geometry. This condition is satisfied when the distance between the scatterer and source or observation point exceeds 2D2 /λ, where D is the maximum size of the scatterer, λ is the wave length. In this case, the incident wave, which is actually spherical, can be treated as a plane one. This allows all formulas to be simplified significantly without affecting the solution accuracy.

6.2 Problem formulation for elastic wave diffraction

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127

As the source of longitudinal elastic wave, we take perturbation of elastic body particles velocity v⃗ L0 . Temporal dependence of the longitudinal wave amplitude corresponds to the Ricker wavelet [98] with dominant frequency f i = 40 Hz. Upon incidence on a surface with “linear slip” boundary condition [102], the longitudinal wave excites two reflected waves: a longitudinal one, and a transverse one. Propagation speeds of longitudinal and transverse waves in the elastic medium are c L = 4000 m/s and c T = 2500 m/s, respectively; the wave lengths are λ L = c L /f i and λ T = c T /f i , while the wave numbers are k L = 2π/λ L and k T = 2π/λ T . The density of elastic medium is ρ = 2 g/cm3 [103]. It follows from the problem geometry that the source and receiver are positioned so that the far field condition is satisfied. This means that both the source and receiver are located at distances not less than 2D20 /λ i from the source, where D0 is the maximum size of the scatterer in the direction transverse to the direction to the source or observation point, λ i = min(c L , c T )/f i . Under this assumption, a spherical wave can be considered as a plane one, and all formulas are simplified significantly. Construct a heuristic analytical solution for the total velocity vector of elastic body particles v,⃗ equal to the sum of incident, v⃗ L0 , and scattered, v⃗ s , fields. v⃗ = v⃗ L0 + v⃗ s .

(6.1)

In turn, the scattered field vector v⃗ s is equal to the sum of velocity components of elastic fields having longitudinal and transverse polarizations v⃗ s = v⃗ L + v⃗ T .

(6.2)

Components v x , v y , and v z of the total velocity vector v⃗ are related to the components of longitudinal v⃗ L and transverse v⃗ T polarizations of the scattered field, as well as components of the initial excitation v⃗ L0 by relations: v x = v L0x + v Lx + v Tx ,

v y = v L0y + v Ly + v Ty ,

v z = v L0z + v Lz + v Tz .

(6.3)

In turn, vectors v⃗ L0 , v⃗ L , and v⃗ T are expressed by the formulas −1 r0z F R (t − , fj ) , cL √4πk L r0z 1 1 v⃗ L = n⃗ s (D L1 AI L1 + D L3 AI L3 ) , √4πk L r0 √4πk L r z 1 1 v⃗ T = v⃗ Ts (D T1 AI T1 + D T3 AI T3 ) . √4πk T r0 √4πk T r z

v⃗ L0 = n⃗ 0z

(6.4)

Here, n⃗ 0z is the unit polarization vector of the incident longitudinal wave, F R (t, f j ) is the Ricker wavelet (describing the longitudinal elastic wave amplitude), t is the current wave at the observation point, r0z is the distance between the source and observation point.

128 | 6 Analytical Heuristic Solution for Wave Diffraction by a Plane Polygonal Scatterer The expressions for velocities v⃗ L and v⃗ T are valid for the given problem geometry and have clear physical interpretation. Unit vectors n⃗ s and v⃗ Ts determine the polarization of medium particle velocity vectors v⃗ L and v⃗ T , respectively. Vector n⃗ s is parallel to the direction in which the longitudinal scattered wave propagates, vector v⃗ Ts is normal to the propagation direction of the transverse scattered wave. All other multipliers of vectors n⃗ s or v⃗ Ts are scalar functions describing the scattered wave amplitudes. The first two multipliers describe damping of signals propagating from the source to the scatterer, and from the scatterer to the observation point. Distances from the scatterer 󵄨 󵄨 to the source and to the observation points are r0 and r z , respectively, where r0 = 󵄨󵄨󵄨N⃗ 0 󵄨󵄨󵄨, 󵄨󵄨 󵄨󵄨 ⃗ 󵄨󵄨 󵄨󵄨 ⃗ ⃗ ⃗ ⃗ r z = 󵄨󵄨N z 󵄨󵄨, and r0z = 󵄨󵄨N z − N0 󵄨󵄨. Here, N0 and N z are position vectors of the source and observation point In Fig. 6.3, elastic wave incidence on a plane polygonal scatterer located in XOYplane is sketched. The angle φ0 corresponds to the position vector N⃗ 0 (the direction vector of the incident wave is −N⃗ 0 ), the angle φ zs corresponds to the position vector of the observation point N⃗ z , the upper and lower edges of the scatterer are denoted by numbers 1 and 3.

N⃗z

z 3

φzs 1 –N⃗0

φ0

Fig. 6.3: Incidence of elastic wave on a plane polygonal scatterer.

The expression in parentheses in (6.4) is the sum of inputs from two edges (first and third). For the given geometry, inputs of two other edges are insignificant because the highest input is provided by the edges lying on the axis of the diffraction cone. In this particular case, the diffraction cone surface degenerates into the plane determined by the incident wave direction and direction to the observation point, while the cone axis is given by the normal to this plane. For more complex geometries, solution will also be more complex, however, the approach remains intact.

6.2 Problem formulation for elastic wave diffraction

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129

Functions AI describe the influence of geometrical parameters of the edge on the scattered signal: r0 + r z ), cL rz r0 = AI T1s (t − − ), cL cT

r0 + r z ), cL rz r0 = AI T3s (t − − ). cL cT

AI L1 = AI L1s (t −

AI L3 = AI L3s (t −

AI T1

AI T3

(6.5)

Functions D L and D T describe the influence of surface properties (boundary conditions) on the scattered signal. Heuristic expressions for functions AI and D will be derived later on the basis of similar functions appearing in the solution to the scattering problem for an electromagnetic wave. In order to obtain the diffraction coefficients of heuristic solution for elastic wave diffraction, we make use of the electromagnetic wave diffraction coefficients (Appendix H). To this end, we need to obtain the transmission T and reflection R coefficients for the interaction of elastic wave with infinite plane non-uniformity in the wave propagation medium. The formulas obtained in the previous chapters allow us to construct heuristic solutions to quite complicated diffraction problems, including diffraction of elastic waves. This requires the following steps: i) to construct heuristic solution formulas in the frequency domain for elastic wave diffraction by a plane rectangular nonuniformity; ii) to construct heuristic formulas for the diffraction coefficients of the elastic wave; iii) to convert the solution from frequency to time domain by the Fourier transformation. Diffraction on plane polygons was considered in Chapter 4. Diffraction by semitransparent plates in PO approximation was described in Chapter 5 and Appendices G and H. On the basis of these results, we will obtain in this chapter the generalized diffraction coefficients for the elastic wave, as well as apply the Fourier transformation in order to convert the solution from frequency to time domain. The solution is determined by a number of factors: a) Problem geometry: size and shape of the scatterer, position of the source and receiver. b) Type of exciting signal: its shape and spectrum. c) Boundary conditions on the scatterer surface: specific physical features of the propagating wave, polarization of the incident and scattered signals.

Excitation by the Ricker wavelet Excitation of seismic waves is often described by the Ricker wavelet. It is very convenient for calculations because it belongs to the class of functions which are almost finite in time and possess almost finite spectrum.

130 | 6 Analytical Heuristic Solution for Wave Diffraction by a Plane Polygonal Scatterer

The Ricker wavelet time dependence is determined by [98] F R (t, f j ) = (1 − 2π2 f j2 t2 ) exp (−π2 t2 f j2 ) ,

(6.6)

while its spectral density is 2 f2 f2 exp (− 2 ) . F̄ R (f, f j ) = 3 √π f j fj

(6.7)

Here, f j is the Ricker wavelet dominant frequency.

6.3 Approach to derivation of formulas In the previous chapters, electromagnetic wave diffraction by three-dimensional objects (including semi-transparent polygonal plates) was considered in the physical optics approximation, as well as in the framework of the method of equivalent edge currents. These solutions are also of fundamental value for elastic wave diffraction. The difference between elastic and electromagnetic waves lies in the direction of polarization vectors, in the directions of “light–shadow” boundaries for various wave types, as well as in the boundary conditions on the surface. In the heuristic solutions offered here for the electromagnetic wave, influence of all these components is localized in the form of constant factors, of functional factors. Substituting in the heuristic formulas for an electromagnetic wave the factors characterizing the electromagnetic wave by those relevant to an elastic wave, one can obtain a heuristic solution for the elastic wave. Another important point is that out heuristic formulas for the electromagnetic wave were obtained in the “frequency domain”, i. e., they are frequency-dependent, while heuristic formulas for the elastic wave must be obtained in the “time domain”, i. e., time-dependent solutions are required. This problem is tackled by the Fourier transformation. The elastic wave diffraction coefficients are obtained by solving the problem of electromagnetic field diffraction by half-plane, taking into account that, upon incidence of a longitudinally polarized elastic wave on a plane boundary of the medium, the reflected signal splits into two components. Each of these components propagates in its own direction and possesses its own polarization, speed and reflection coefficient. To describe the surface properties, the generalized diffraction coefficient is used. Its derivation in the case of electromagnetic wave is given in Appendix H (equations (E.1)–(G.12)).

6.3 Approach to derivation of formulas |

131

Fourier transformation Simplicity of the form of diffraction coefficients allows one to apply the Fourier transformation and convert the solution for diffraction coefficients from frequency to time domain. In particular, this simplicity is due to the fact that the distance R from the coordinate origin to observation point can be factored out of the general integral. The pair of Fourier transformations is defined by f ̂(ω) =

∞

1 ∫ f (x) exp (−ixω) dx, √2π −∞

∞

f(x) =

1 ∫ f ̂ (ω) exp (ixω) dω. √2π

(6.8)

−∞

Here, f ̂(ω) = w R (f, f j ), f(x) = w R (t, f j ); the change of variables in formulas is performed according to the following scheme: x ↔ t, f ↔ ω, f j ↔ f02 Hz. At high frequencies, the far zone conditions for the considered scatterer can cease to be valid because the wave length is decreased. Applying the Fourier transformation, one has to keep this circumstance in mind because it can affect the solution quality. Time dependence of the Ricker wavelet F R (t, f j ) ↔ f(x) and its spectral density F̄ R (f, f j ) ↔ f ̂(ω) are determined by equations (6.6) and (6.7). In these transformation, we take time t for the variable x. As a result, we obtain a pair of “time-frequency” and “frequency–time” variable transformations. Input of an individual edge upon diffraction of electromagnetic wave by a polygonal plate is described by equation (H.13) (Appendix H): E=

1 ka j exp (ikR0 ) exp {iΦ j } fg (φ, φ0 , R, T) . 2π kR0

Apply the same equation in order to obtain an expression for one of the components of an elastic wave field v L or v T (namely, velocity of elastic body particles) for a given polarization (

1 k (f ) a j exp (ikR0 ) vL DL exp {iΦ j } ( ) . )= 2π kR0 vT DT

(6.9)

This equation describes the frequency dependence of elastic wave. Of interest to us, however, is the time dependence. These two dependencies can be matched by the Fourier transformation. Diffraction coefficients D L and D T introduced above in equation (6.4) are determined in the same way as the generalized diffraction coefficients (GDC), see Appendix H: D L = fg (φ L , φ0 , R L , T L ) , (6.10) D T = fg (φ T , φ0 , R T , T T ) . When deriving the diffraction coefficients, we took into account the fact that the reflection angle φ, reflection coefficient R, and transmission coefficient T depend on elastic wave polarization (longitudinal, L, or transverse, T).

132 | 6 Analytical Heuristic Solution for Wave Diffraction by a Plane Polygonal Scatterer

A question may arise: on what grounds did we write down formulas (6.9) and (6.10)? They can be justified by the following reasoning. Both electromagnetic and elastic waves represent solutions to the Helmholtz equation for homogeneous isotropic medium. Wave polarization is determined by its physical nature, it enters the heuristic solution as a separate factor. The surface properties are defined individually for each wave type and polarization in the form of reflection and transmission coefficients R and T at an infinite plane surface. It is natural to assume that the scattered wave amplitudes will be determined by the same scalar functions. For example, this assumption is entirely valid for acoustic waves [45]. Of course, our solutions for both electromagnetic and elastic waves are heuristic, they possess some inherent inaccuracies. Presumably, for some particular problem formulations the boundary conditions will be satisfied with higher accuracy on the illuminate than on the shadow side. For example, if the coefficients R and T depend on the incidence angle (which is true for electromagnetic wave in the case of impedance boundary conditions [32], as well as for elastic wave in the case of “linear slip” conditions [102]), then this angle will be varying weaker on the illuminated side than on the shadow one. However, effect of this inexactness in the boundary conditions on the final result may be quite limited. One has to keep in mind that our elastic wave solution is heuristic. It can be constructed in very complex cases, its advantageous feature is its simplicity, but, in any case, it requires verification independently of how rigorously it is substantiated. Verification is the very process which highlights adequacy of our concept of the process being studied, as well as shows how far the heuristic solution deviates from the rigorous one. We obtained the diffraction coefficients in the frequency domain, whereas the excitation function (The Ricker wavelet F R (t, f j )) is defined in the time domain. To perform transformation from the frequency to time domain, we take the frequencydependent heuristic solution for finite-size edge (6.9) and integrate with spectral density F̄ R (f, f j ) the frequency-dependent component of diffraction solution obtained in the frequency domain and described by the formula for elastic wave component v L or v T (6.9): i (6.11) k (f ) exp { [Φ L1 (f, φ0 , φ zs ) + Φ L4 (f, φ0 , φ zs )]} . 2 Frequency-independent components are factored out of the Fourier integral. As a result, we obtain functions AI (6.5) in the frequency domain:

6.3 Approach to derivation of formulas |

133

∞

a AI L1s (t) = ∫ F̄ R (f 󸀠 , f j )ik L (f 󸀠 ) √2π −∞

i × exp { [Φ L1 (f 󸀠 ) + Φ L4 (f 󸀠 )]} exp (−i2πf 󸀠 t) df 󸀠 , 2 ∞

AI L3s (t) =

a ∫ F̄ R (f 󸀠 , f j )ik L (f 󸀠 ) √2π −∞

i × exp { [Φ L3 (f 󸀠 ) + Φ L2 (f 󸀠 )]} exp (−i2πf 󸀠 t) df 󸀠 , 2 ∞

(6.12)

a AI T1s (t) = ∫ F̄ R (f 󸀠 , f j )ik T (f 󸀠 ) √2π −∞

i × exp { [Φ T1 (f 󸀠 ) + Φ T4 (f 󸀠 )]} exp (−i2πf 󸀠 t) df 󸀠 , 2 ∞

AI T3s (t) =

a ∫ F̄ R (f 󸀠 , f j )ik T (f 󸀠 ) √2π −∞

i × exp { [Φ T3 (f 󸀠 ) + Φ T2 (f 󸀠 )]} exp (−i2πf 󸀠 t) df 󸀠 . 2 Here, f 󸀠 is the integration variable having the dimension of frequency. The phases of edge ends (vertices) are denoted by Φ. The arguments of exponential functions entering the integrands (half-sums of vertex phases) are equal to the phases at the edge mid-points. Subscripts L and T denote the inputs of longitudinal and transverse polarizations, respectively. Subscripts 1 and 3 in the functions AI denote the scatterer edges. To perform numerical integration, one can use an approximate formula for an integral between finite limits, i. e., bound the integration limits. This is possible for excitation by the Ricker wavelet because the function F R (t, f j ) is almost-finite in time, whereas the function F̄ R (f, f j ) has an almost-finite spectrum. 4f02Hz

a AIL1s(t) = √2π

∫

F̄ R (f 󸀠 , f j )ikL (f 󸀠 )

−4f02Hz

i × exp { [Φ L1 (f1 , φ0 , φ zs ) + Φ L4 (f1 , φ0 , φ zs )]} 2

(6.13)

× exp (−i2πf 󸀠 t) df 󸀠 . Substitute into these relations the signal phase at the j-th vertex, Φ j = k (Δ,⃗ ρ⃗ j ), position vector of the j-th vertex, ρ⃗ j , and edge length, a j . In our case, the two sides contributing to the scattered signal are of the same length (a1 = a3 = a), and this parameter can be factored out of the integrals as a constant. Additional symbols L and T in the subscripts of Φ j and corresponding subscripts of wave numbers k denote

134 | 6 Analytical Heuristic Solution for Wave Diffraction by a Plane Polygonal Scatterer the longitudinal and transverse polarizations. Auxiliary vector Δ⃗ (A.3) is the projection on the scatterer surface of the difference between the incident field vector n⃗ 󸀠 and direction to the observation point n⃗ 󸀠󸀠 .

6.4 General form of the solution Heuristic solution for diffraction of an elastic wave by a plane polygonal object is obtained by the following algorithm. 1. Derive the rigorous solution in the physical optics approximation for electromagnetic wave diffraction by a perfectly conducting polygonal plate under the far zone conditions (Appendices A and D). 2. Obtain solution to the electromagnetic wave diffraction problem by the equivalent edge current method. EECM solution can be derived from the physical optics solution for normal or oblique wave incidence on the edge by formulas from Appendices B and D. Correcting coefficients can be introduced in the physical optics field pattern in order to take into account the variation of pattern due to perturbation of the currents by the edge. As a result, we obtain an EECM solution with uniform edge currents along the edge. Variation of the edge currents due to finite length of the edge (“vertex waves”) are not yet taken into account. In our geometry, influence of the vertex waves is not too noticeable, but in the future their effects must be taken into account. 3. In the expression for diffraction coefficients, substitute the electromagnetic wave coefficients R and T by respective elastic wave coefficients (6.10). 4. Perform transformation from frequency to time domain (6.12), (6.13). 5. Carry out test calculation by heuristic formulas (6.4) and tune the solution against the verification calculation data. As any heuristic formula, elastic wave diffraction solution requires verification. Approximate formulas for elastic wave diffraction by a plane polygonal plate were tested against the data obtained on the cluster of Massachusetts Institute of Technology. Comparison of heuristic results with rigorous ones demonstrated that the heuristic approach [61] is very promising. However, definite conclusions on adequacy of heuristic formulas can only be drawn after the work on their construction has been completed, and their applicability limits have been established. Such research has yet to be carried out, therefore we do not present verification data for elastic waves in this book.

7 Conclusion In this book, a number of approaches is proposed enabling one to obtain new heuristic solutions in the diffraction theory of electromagnetic waves and waves of other types. The book is based on the physical (in contrast to rigorous mathematical) approach to diffraction theory. Heuristic solutions derived by physical approach are less accurate, but are simpler and more understandable. Heuristic solutions need to be verified, i. e., checked against more rigorous solutions. Compared to engineering formulas which provide just a simpler form of rigorous solutions and applicable only to a single verification solution, heuristic solutions can be derived even for those problems for which verification solutions are not yet available. Of course, one has to be fully aware that such an approach bears some risk because accuracy of a heuristic solution can only be guaranteed by its verification. The method of generalized eikonal (Chapter 1) allows one to obtain new twodimensional solutions on the edges, including the edges possessing a size parameter. MGE is based on the integral representation of the solution, generalizing the Sommerfeld integral representation of the solution for diffraction by a wedge. On the basis of MGE, integral representations are obtained for diffraction by half-plate and truncated wedge (Chapter 2). From these solutions one can factor out functions that allow known 2D diffraction solutions for scatterers of other shapes (for example, those consisting of several half-planes) to be refined, provided that veritices are located at the same points. The functions in question are pN functions (2.45) for a wedge, and similar functions for other scatterer shapes. A number of new approaches to application of 2D diffraction solution to 3D problems were developed (Chapter 4). A method is proposed by which the solutions available for the line segments of two-dimensional scatterer boundary, valid on the diffraction cones, can be applied (in a rather non-trivial way) to find out the solutions in the directions different from those of the diffraction cones. This solutions can be applied directly to three-dimensional problems provided that the angular variables are substituted by analytical formulas obtained from geometrical considerations; these angular variables can take not only real, but also complex values. Properties of the rigorous solution obtained in the physical optics approximation for a plane polygonal scatterer are studied (see Section 3.2.1 and Appendix A); this solution can also be applied to three-dimensional diffraction problems. A heuristic solution is obtained (Chapter 4) for diffraction by a perfectly conducting angular sector. As its basis, a PO solution is taken; it is then refined to a solution in EECM approximation. At the final stage, the EECM solution is refined to a modified EECM (MEECM) one. MEECM solution is as simple as PO one, however its accuracy is close to that of a rigorous solution that accuracy of all known heuristic methods, including GTD and PTD (MEW). When deriving the EECM solution, the imaginary edge

136 | 7 Conclusion

solution (EECM-IE) was used (Section 4.3.2, Appendix C) which allowed us to avoid integration of equivalent edge currents over the edge. The diffraction theory methods developed in Chapters 1–4 can be applied to problems of practical importance. In Chapter 5, radio wave propagation in urban environment is considered. Approaches to derivation of deterministic heuristic formulas for scatterers of very large sizes are proposed. Energy relationships are obtained by which the scattered signal amplitude can be calculated correctly. The approach developed allows one to analyze radio signal interference and multi-beam propagation in the urban environment conditions. In Chapter 6, diffraction of a wave field of arbitrary nature by a plane polygonal scatterer is considered. In particular, we analyze elastic wave diffraction by an inhomogeneity having the shape of a plate polygonal scatterer located in the medium where the elastic wave is propagating. The surface properties are taken into account via ODC (Appendix H). Solution is converted from the frequency to time domain by Fourier transformation. Due to their simplicity, heuristic formulas can be integrated analytically, if necessary. Some approaches presented in the book are mathematically rigorous. This applies to the integral representation in MGE (Section 1.1.1) and to the way in which two-dimensional solutions are used in three-dimensional problems where complexvalued angles are introduced (Section 3.1). Other (heuristic) approaches developed in this book, in contrast to known heuristic methods (GTD and PTD) do not rely on the availability of rigorous two-dimensional solutions, but predict behavior of rigorous solutions. This applies to functions pN in MGE (Section 2.3.2), to MEECM (Section 4.4), as well as to method of ODC (Appendix H). Summarize now the general concepts fundamental for the approaches developed in this book. When deriving new heuristic approaches, we are not searching for new mathematically rigorous solutions, but we find out the most efficient ways in which existing solutions of lower dimensionality (simpler ones) can be applied to research and engineering problems of higher dimensionality (see also discussion between formulas (H.11) and (H.12) in Appendix H). Influence of various factors (edge shape, vertices, boundary conditions) is represented in the form of simple factors. It is assumed that their addition to the physical optics solution allows the solution accuracy to be increased significantly. Validity of this assumption is confirmed by verification performed for several 2D and 3D problems. These include: diffraction by a 2D edge in the form of half-plate or a truncated wedge; diffraction by a plane angular sector; diffraction by an impedance wedge; elastic wave diffraction by a plane polygonal scatterer. Derivation of news solution is really important by itself, however, of equal importance is to be able to apply these solution to scientific and engineering problems. One can say that we are improving the physical optics approximation which is very popular due to the problems inherent in rigorous analytical or numerical solutions:

7 Conclusion

|

137

i) these can be absent for different reasons (no analytical formulas found, “curse of dimensionality”, etc.); ii) they can be very complex; iii) specialized software and highskill researchers capable of handling this software may be required; iv) many other reasons, including necessary computing power and so on. In other cases, we derive heuristic formulas for problems which have not be solved so far. For example, attempts to supplement the analytical solution for a plane angular sector by a “tiny thickness” or non-ideal boundary conditions encounter significant difficulties. In heuristic solutions, similar problems are resolved relatively easily. Our approach allows one to construct the solution as a product of several factors, each having its clear physical meaning, i. e., separation of variables is enabled in the formulas. This speaks in favor of clear physical sense of our solutions. A cursory reader of this book might get an impression that the author advocates for dropping rigorous solutions in favor of heuristic ones. This is by no means right. Rigorous analytical solutions must be derived, analyzed and applied wherever possible. However, if some arising difficulties could be overcome by using heuristic solutions, this research method should be taken into account.

A Application of Stokes Theorem to Diffraction Problems Consider the formulation of diffraction problem corresponding to equations (3.1)– (3.4). The formulas for potentials involve an integral over the scatterer surface (3.2). In this appendix, we evaluate this integral and analyze its properties.

A.1 Stokes theorem. Relationship between the surface and contour integrals Consider a region S bounded by a contour C, both lying in the plane XOY. Let the normal n⃗ to S be directed along the z-axis, and the contour C is oriented counterclockwise with respect to region S. Thus, the normal n⃗ matches the orientation of contour C by the rule of a screw. Denote the unit vector tangent to the contour by ρ ⃗ 󸀠 . Introduce a unit inner normal n⃗ j to contour C in such a way that vectors (n,⃗ ρ ⃗ 󸀠 , n⃗ j ) form the right triad. e⃗ y 0 ρ󸀠y

e⃗ x [n⃗ × ρ ⃗ 󸀠 ] = ( 0 ρ󸀠x

e⃗ z 1 ) = −e⃗ x ρ󸀠y + e⃗ y ρ󸀠x = n⃗ j . 0

(A.1)

Evaluate the integral ⃗ ds. I = ∬ exp {ik (Δ,⃗ ρ)}

(A.2)

S

Here, the auxiliary vector Δ⃗ Δ⃗ = (n⃗ 󸀠 − n⃗ 󸀠󸀠 ) − n⃗ [(n⃗ 󸀠 − n⃗ 󸀠󸀠 ) , n]⃗

(A.3)

is the projection on the scatterer surface (in our case, on the plane XOY) of the difference (n⃗ 󸀠 − n⃗ 󸀠󸀠 ) between the incident wave and observation point direction vectors. To this end, apply the Stokes theorem [104] ⃗ = ∮ (d r,⃗ F⃗ (r)) ⃗ . ∬ (d S,⃗ rot F⃗ (r)) S

(A.4)

C

Here, d S⃗ is the surface element vector. Its normal is aligned with the orientation of contour C according to the right-hand screw rule. Let d S⃗ = e⃗ z ds. A curve vector element d r ⃗ (differential of the position vector) is directed along the contour C at each regular point, ρ ⃗ 󸀠 is the unit tangent vector to the contour: ρ ⃗ 󸀠 = e⃗ x ρ ⃗ 󸀠x + e⃗ y ρ ⃗ 󸀠y . Introduce the coordinate t along the contour and write in terms of t the position vector differential: d r ⃗ = e⃗ x dx + e⃗ y dy = ρ ⃗ 󸀠 dt = (e⃗ x ρ󸀠x + e⃗ y ρ󸀠y ) dt.

(A.5)

140 | A Application of Stokes Theorem to Diffraction Problems

Introduce an auxiliary function ⃗ (Δ y e⃗ x − Δ x e⃗ y ) = F x e⃗ x + F y e⃗ y F⃗ = exp {ik (Δ,⃗ ρ)}

(A.6)

and apply the Stokes theorem (A.4). The curl components are e⃗ x rot F⃗ = (∂/∂x Fx

e⃗ y ∂/∂y Fy

e⃗ z ∂/∂z) Fz

(A.7)

∂F y ∂F x ∂F z ∂F y ∂F z ∂F x − − − = e⃗ x ( ) − e⃗ y ( ) + e⃗ z ( ). ∂y ∂z ∂x ∂z ∂x ∂y The quantities denoted by r ⃗ in [104] and ρ⃗ in (A.6) are equivalent and denote the position vector of a point in space. Note that function F⃗ (A.6) is independent of coordinate z. This means that F z = 0, ∂/∂z = 0. Then, instead of (A.7), we obtain: rot F⃗ = e⃗ z (

∂F y ∂F x − ). ∂x ∂y

(A.8)

Write down the integrand from the left-hand side of the Stokes theorem (A.4) for function F⃗ (A.6). ⃗ = ( ∂F y − ∂F x ) ds. (A.9) (d S,⃗ rot F) ∂x ∂y ⃗ = Δ x x + Δ y y, write down the derivatives of funcTaking into account (Δ,⃗ ρ) tion (A.6) involved in (A.9): ∂F y ⃗ , = −Δ x ikΔ x exp {ik (Δ,⃗ ρ)} ∂x ∂F x ⃗ . = Δ y ikΔ y exp {ik (Δ,⃗ ρ)} ∂y

(A.10)

With reference to (A.8) and (A.10), we obtain: ∂F y ∂F x 󵄨 󵄨2 ⃗ , − = −ik 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 exp {ik (Δ,⃗ ρ)} ∂x ∂y

(A.11)

while taking into account (A.9), we have: ⃗ = −ik 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨2 exp {ik (Δ,⃗ ρ)} ⃗ ds. (d S,⃗ rot F) 󵄨 󵄨

(A.12)

On the other hand, with reference to (A.5) and (A.6): ⃗ = exp {ik (Δ,⃗ ρ)} ⃗ (Δ y dx − Δ x dy) (d r,⃗ F⃗ (r)) ⃗ (Δ y ρ󸀠x − Δ x ρ󸀠y ) dt. = exp {ik (Δ,⃗ ρ)}

(A.13)

However, it follows from (A.1) that (Δ y ρ󸀠x − Δ x ρ󸀠y ) = (Δ,⃗ n⃗ j ). Therefore, ⃗ = exp {ik (Δ,⃗ ρ)} ⃗ (Δ,⃗ n⃗ j ) dt. (d r,⃗ F⃗ (r))

(A.14)

A.2 Integral over the surface of a finite-size polygon

|

141

As a result, with reference to (A.4), (A.12), and (A.14) we obtain: 󵄨 󵄨2 ⃗ ds = ∮ exp {ik (Δ,⃗ ρ)} ⃗ (Δ,⃗ n⃗ j ) dt. − ik 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 ∬ exp {ik (Δ,⃗ ρ)} S

(A.15)

C

Finally, we obtain ⃗ ds = I = ∬ exp {ik (Δ,⃗ ρ)} S

i ⃗ ⃗j ⃗ ⃗ 󵄨󵄨 ⃗ 󵄨󵄨2 ∮ (Δ, n ) exp {ik (Δ, ρ)} dt. k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 C

(A.16)

where n⃗ j = [n⃗ × ρ ⃗ 󸀠 ]/|ρ ⃗ 󸀠 | is the unit normal vector to the contour C enveloping the scat⃗ d ρ/dt is the unit tangent vector to the contour, t is the coordinate measured terer, ρ ⃗ 󸀠 = |d ρ/dt| ⃗ along the contour. The integral (A.16) represents an alternative form of the relationships obtained in [36] independently of an earlier work on the same subject [35]. The key role in the integral appearing in (A.16) is played by the phase of the integration point in the exponent: ⃗ . Φ = k (Δ,⃗ ρ) (A.17) Equation (A.16), which reduces the area integral to contour integral in the case of linear phase, is quite simple and well-known in mathematics. However, it is very important for diffraction theory, which is why we paid so much attention to it.

A.2 Integral over the surface of a finite-size polygon Consider a finite-size polygon on which a plane electromagnetic wave is incident (Fig. A.1). Use the following notations: n⃗ 󸀠 is the direction vector of the incident wave, n⃗ 󸀠󸀠 is the direction vector of the observation point, n⃗ is the normal to the scatterer, n⃗ j is the unit inner normal vector to the contour C encircling the scatterer, ρ ⃗ 󸀠 is the unit tangent vector to the contour, N is the number of sides and vertices of the polygon. On the j-th straight side connecting j − 1-th and j-th vertices, the vector n⃗ j remains constant, while the current position vector ρ⃗ of the contour point is related to the position vector of the initial vertex ρ⃗ j−1 , coordinate along the contour t and constant n⃗ʺ

n⃗ʹ n⃗ ⃗j n

j–1 ρ⃗ʹj

aj + 1

aj j

j+1 ρ⃗ʹj+1

Fig. A.1: Incidence of electromagnetic wave on a plane polygon.

142 | A Application of Stokes Theorem to Diffraction Problems tangent vector ρ ⃗ 󸀠j by ρ⃗ = ρ⃗ j−1 + t ρ ⃗ 󸀠j . Then, for the j-th side of length a j , we obtain: aj

Ij =

i

⃗ ⃗j ⃗ ⃗ 󵄨 󵄨2 ∫ (Δ, n ) exp {ik (Δ, ρ)} dt k 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 0 aj

i (Δ,⃗ n⃗ j ) ⃗ ⃗ ⃗󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 ∫ exp {ik (Δ, ρ j−1 + t ρ j )} dt k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 0 aj i (Δ,⃗ n⃗ j ) ⃗ ⃗ ⃗ ⃗󸀠 = 󵄨󵄨 ⃗ 󵄨󵄨2 exp {ik (Δ, ρ j−1 )} ∫ exp {ik (Δ, t ρ j )} dt 󵄨 󵄨 k 󵄨󵄨Δ󵄨󵄨 0 j ⃗ ⃗ i (Δ, n⃗ ) exp {ik (Δ, ρ⃗ j−1 )} 󵄨t=a j exp {ik (Δ,⃗ t ρ ⃗ 󸀠j )}󵄨󵄨󵄨󵄨t=0 . = 󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 ik (Δ,⃗ ρ ⃗ j ) k 󵄨󵄨󵄨Δ󵄨󵄨󵄨

=

(A.18)

Therefore, taking into account that ρ⃗ j = ρ⃗ j−1 + a j ρ ⃗ 󸀠j , we obtain: i (Δ,⃗ n⃗ j ) exp {ik (Δ,⃗ ρ⃗ j−1 )} [exp {ik (Δ,⃗ a j ρ ⃗ 󸀠j )} − 1] 󸀠 󵄨 ⃗ 󵄨󵄨2 ⃗ 󵄨 ⃗ ik (Δ, ρ j ) k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 ia j (Δ,⃗ n⃗ j ) exp {ik (Δ,⃗ ρ⃗ j )} − exp {ik (Δ,⃗ ρ⃗ j−1 )} . = 󵄨 󵄨2 ik (Δ,⃗ ρ⃗ j − ρ⃗ j−1 ) k 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

Ij =

(A.19)

Denote the phase difference at the ends of the interval by Φ j − Φ j−1 = k (Δ,⃗ ρ⃗ j − ρ⃗ j−1 ). Then, factoring the mean phase (the phase at the midpoint of the interval) in the parentheses as well as

Φ j +Φ j−1 2

exp (iΦ j ) − exp (iΦ j−1 ) = i (Φ j − Φ j−1 )

× exp (i

=

exp (i

k(Δ,⃗ ρ⃗ j +ρ⃗ j−1 ) 2

and applying formulas sin α =

e iα −e−iα , 2i

Φ j − Φ j−1 Φ j − Φ j−1 ) − exp (−i ) 2 2

Φ j + Φ j−1 )= 2

2i

Φ j −Φ j−1 2

Φ j − Φ j−1 ) Φ j + Φ j−1 2 exp (i ), Φ j − Φ j−1 2 2

sin (

(A.20)

we obtain by summation over all sides of the polygon: N

I = ∑ Ij , j=1

Ij =

ia j (Δ,⃗ n⃗ j ) sin [(Φ j − Φ j−1 ) /2] exp {i (Φ j + Φ j−1 ) /2} , 󵄨 󵄨2 (Φ j − Φ j−1 ) /2 k 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

(A.21)

where Φ j = k (Δ,⃗ ρ⃗ j ) is the phase of the signal from j-th vertex with the direction vector ρ⃗ j , while a j is the length of j-th side of the polygon (connecting j − 1-th and j-th vertices).

A.3 Integral over the surface of a plane angular sector |

143

Equation (A.21) represents the scattered field as the sum of inputs, each of which is related to a particular side of the polygon and has the phase of the mid-point of this side. If the diffraction cone condition (Δ,⃗ ρ ⃗ 󸀠 ) = 0 is satisfied, it is possible to derive from (A.21) the relationship given by (3.42), from which, in turn, it is possible to derive the 3D diffraction coefficient in the physical optics approximation, exactly equal to the classical expression for 2D physical optics diffraction coefficient (4.9) from [62]. Consider now the vertex wave in the physical optics approximation.

A.3 Integral over the surface of a plane angular sector Consider a plane angular sector on which a plane electromagnetic wave is incident (Fig. 4.1). To obtain the scattered field, apply equation (A.18) for the contour integral over a polygon (Fig. A.1) to a finite-length side: aj

i

⃗ ⃗j ⃗ ⃗ 󵄨󵄨 ⃗ 󵄨󵄨2 ∫ (Δ, n ) exp {ik (Δ, ρ)} dt k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 0 aj i (Δ,⃗ n⃗ j ) ⃗ ⃗ ⃗ ⃗󸀠 = 󵄨󵄨 ⃗ 󵄨󵄨2 exp {ik (Δ, ρ j−1 )} ∫ exp {ik (Δ, t ρ j )} dt. 󵄨 󵄨 k 󵄨󵄨Δ󵄨󵄨 0

Ij =

(A.22)

Now tend the side length to infinity a j → ∞ and apply the formula ∞

∫ exp (−px) dx =

1 , p

[Re(p) > 0]

0

from [105]. As a result, we obtain from (A.22): aj i (Δ,⃗ n⃗ j ) ⃗ ⃗ ⃗ ⃗󸀠 Ij = 󵄨󵄨 ⃗ 󵄨󵄨2 exp {ik (Δ, ρ j )} ∫ exp {ik (Δ, −t ρ j )} dt 󵄨 󵄨 k 󵄨󵄨Δ󵄨󵄨 0 j ⃗ i (Δ, n⃗ ) 1 ⃗ ⃗ 󳨀󳨀󳨀󳨀󳨀→ 󵄨󵄨 ⃗ 󵄨󵄨2 exp {ik (Δ, ρ j )} ik (Δ,⃗ ρ ⃗ 󸀠 ) a j →∞ k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 j j ⃗ ⃗ exp {ik (Δ, ρ⃗ j )} (Δ, n⃗ ) = I j∞ . = 󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 ⃗ 2 ⃗ Δ, ρ ( ) k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 j

(A.23)

Here, integration along the j-th side is performed from vertex j to infinity (i. e., along the vector −ρ ⃗ 󸀠j ).

144 | A Application of Stokes Theorem to Diffraction Problems Integrate now along another half-line (i. e., from vertex j along the vector ρ ⃗ 󸀠j+1 ). aj

i (Δ,⃗ n⃗ j+1 ) ⃗ ⃗ ⃗ ⃗󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 exp {ik (Δ, ρ j )} ∫ exp {ik (Δ, t ρ j+1 )} dt k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 0 i (Δ,⃗ n⃗ j+1 ) −1 ⃗ ⃗ 󳨀󳨀󳨀󳨀󳨀→ 󵄨 ⃗ 󵄨󵄨2 exp {ik (Δ, ρ j )} ik (Δ,⃗ ρ ⃗ 󸀠 ) a j →∞ 󵄨 k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 j+1 exp {ik (Δ,⃗ ρ⃗ j )} (Δ,⃗ n⃗ j+1 ) ∞ . = I j+1 = 󸀠 󵄨 ⃗ 󵄨󵄨2 ⃗ 󵄨 2 ⃗ (Δ, ρ j+1 ) −k 󵄨󵄨󵄨Δ󵄨󵄨󵄨

I j+1 =

(A.24)

j

As a result, we obtain for the vertex wave I∞ a sum of inputs from two semi-infinite sides of a plane angular sector: ∞ = I∞ = I j∞ + I j+1 j

exp {ik (Δ,⃗ ρ⃗ j )} (Δ,⃗ n⃗ j ) (Δ,⃗ n⃗ j+1 ) ]. [ − 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j ) (Δ,⃗ ρ ⃗ 󸀠j+1 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 ] [

(A.25)

A.4 Vertex waves for a finite-size polygon Equation (A.25) describes the scattered field for a plane angular sector. This expression consists of a product of the vertex phase and a scalar function depending on problem geometry (i. e., on the plane angular sector shape, as well as the source and receiver positions. The contour integral over a plane polygon can be written in terms of the sum of inputs having the phases of mid-points of polygon sides (A.21), but, alternatively, it cam be written in terms of inputs of vertices. To this end, the summation procedure for inputs to the scattered field has to be changed. As the starting point, equation (A.19) can be taken. Note that the following relations are valid: ρ⃗ j = ρ⃗ j−1 + a j ρ ⃗ 󸀠j , ρ⃗ j−1 = ρ⃗ j − a j ρ ⃗ 󸀠j , and ρ⃗ j+1 = ρ⃗ j + a j+1 ρ ⃗ 󸀠j+1 . For j + 1-th side, we have (see (A.19)): exp {ik (Δ,⃗ ρ⃗ j−1 )} (Δ,⃗ n⃗ j ) [exp {ik (Δ,⃗ a j ρ ⃗ 󸀠j )} − 1] 󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 ⃗ 2 ⃗ Δ, ρ ( ) 󵄨 󵄨 k 󵄨󵄨Δ󵄨󵄨 j j ⃗ 1 (Δ, n⃗ ) ⃗ ⃗ ⃗ ⃗ = 󵄨󵄨 ⃗ 󵄨󵄨2 (Δ,⃗ ρ ⃗ 󸀠 ) [exp {ik (Δ, ρ j )} − exp {ik (Δ, ρ j−1 )}] . 2 󵄨 󵄨 k 󵄨󵄨Δ󵄨󵄨 j

Ij =

(A.26)

A.4 Vertex waves for a finite-size polygon

| 145

For j + 1-th side we have (see (A.19)): exp {ik (Δ,⃗ ρ⃗ j )} (Δ,⃗ n⃗ j+1 ) [exp {ik (Δ,⃗ a j+1 ρ ⃗ 󸀠j+1 )} − 1] 󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 ⃗ 2 ⃗ Δ, ρ ( ) k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 j+1 j+1 1 (Δ,⃗ n⃗ ) = [exp {ik (Δ,⃗ ρ⃗ j+1 )} − exp {ik (Δ,⃗ ρ⃗ j )}] 󸀠 󵄨 󵄨2 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 (Δ,⃗ ρ ⃗ j+1 )

I j+1 =

(A.27)

Obtain now for a finite-size polygon expressions for the vertex waves similar to formulas (A.19). To this end, change the order in which integrals over the intervals of contour C are summed up. Instead of summing the inputs of total-length intervals (A.26) and (A.27), we divide each interval into two halves, and sum up the inputs of the halves connecting at each vertex. Denote by ρ⃗ j− 1 the position vector of mid-point of j-th side, by ρ⃗ j+ 1 the position 2 2 vector of mid-point of j + 1-th side, by I j− 1 and I j+ 1 the integrals over the half-intervals 2 2 connecting at the j-th vertex. Then we can write: exp {ik (Δ,⃗ ρ⃗ j−1/2 )} (Δ,⃗ n⃗ j ) aj 󸀠 ρ ⃗ )} − 1] [exp {ik (Δ,⃗ 󸀠 2 󵄨 󵄨 ⃗ 2 j 󵄨 󵄨 (Δ, ρ ⃗ j ) k2 󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨 j 1 (Δ,⃗ n⃗ ) = [exp {ik (Δ,⃗ ρ⃗ j )} − exp {ik (Δ,⃗ ρ⃗ j−1/2 )}] , 󸀠 󵄨 󵄨2 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 (Δ,⃗ ρ ⃗ j )

I j− 12 =

(A.28)

exp {ik (Δ,⃗ ρ⃗ j )} (Δ,⃗ n⃗ j+1 ) a j+1 󸀠 ρ ⃗ )} − 1] [exp {ik (Δ,⃗ 󸀠 󵄨󵄨 ⃗ 󵄨󵄨2 ⃗ 2 j+1 2 ⃗ Δ, ρ ( ) k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 j+1 (A.29) j+1 1 (Δ,⃗ n⃗ ) = [exp {ik (Δ,⃗ ρ⃗ j+ 12 )} − exp {ik (Δ,⃗ ρ⃗ j )}] . 󸀠 󵄨 󵄨2 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 (Δ,⃗ ρ ⃗ j+1 ) Summing up the integrals over halves of the sides, we obtain the vertex wave: I j+ 12 =

I j− 12 + I j+ 12 j 1 (Δ,⃗ n⃗ ) [exp {ik (Δ,⃗ ρ⃗ j )} − exp {ik (Δ,⃗ ρ⃗ j− 12 )}] 󸀠 󵄨 󵄨2 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 (Δ,⃗ ρ ⃗ j ) j+1 1 (Δ,⃗ n⃗ ) + [exp {ik (Δ,⃗ ρ⃗ j+ 12 )} − exp {ik (Δ,⃗ ρ⃗ j )}] 󸀠 2 󵄨 󵄨 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 (Δ,⃗ ρ ⃗ j+1 ) exp {ik (Δ,⃗ ρ⃗ j )} (Δ,⃗ n⃗ j ) (Δ,⃗ n⃗ j+1 ) ] [ = − 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j ) (Δ,⃗ ρ ⃗ 󸀠j+1 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 ] [ j ⃗ ⃗ exp {ik (Δ, ρ⃗ j− 12 )} (Δ, n⃗ ) exp {ik (Δ,⃗ ρ⃗ j+ 12 )} (Δ,⃗ n⃗ j+1 ) + . − 󵄨 󵄨2 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j ) (Δ,⃗ ρ ⃗ 󸀠j+1 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

=

(A.30)

146 | A Application of Stokes Theorem to Diffraction Problems j

One can see that equation (A.30) contains the vertex wave I∞ (see (A.25)) evaluated over semi-infinite sides of angular sector. Thus, inputs of individual sides are expressed in terms of “one-half” phases. Each “one-half” phase enters two integrals. j We single out now the vertex wave I∞ (A.25) from the sum I j− 1 + I j+ 1 and compare 2 2 the result with that obtained if the vertex wave is singled out from the sum I j−1 + I j+1 . If the sides are not divided into two halves and the intervals connecting at the j-th vertex are taken at full length, the residual will be exp {ik (Δ,⃗ ρ⃗ j−1 )} (Δ,⃗ n⃗ j ) 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 exp {ik (Δ,⃗ ρ⃗ j+1 )} (Δ,⃗ n⃗ j+1 ) . + 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j+1 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 If, however, we divide the intervals, the residual will be j

I j−1 + I j+1 − I∞ = −

exp {ik (Δ,⃗ ρ⃗ j− 1 )} (Δ,⃗ n⃗ j ) 2 󵄨󵄨 ⃗ 󵄨󵄨2 2 (Δ,⃗ ρ ⃗ 󸀠j ) k 󵄨󵄨󵄨Δ󵄨󵄨󵄨 exp {ik (Δ,⃗ ρ⃗ j+1/2 )} (Δ,⃗ n⃗ j+1 ) . + 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j+1 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

j

(A.31)

I j− 1 + I j+ 1 − I∞ = − 2 2

(A.32)

Compare now equation (A.32) with the vertex waves for j−1-th and j+1-th vertices. The residual for the wave from j − 1-th vertex I j− 3 + I j− 1 will take the form: 2

2

exp {ik (Δ,⃗ ρ⃗ j− 3 )} (Δ,⃗ n⃗ j−1 ) 2 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j−1 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 exp {ik (Δ,⃗ ρ⃗ j− 12 )} (Δ,⃗ n⃗ j ) . + 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

j

I j− 32 + I j− 12 − I∞ = −

(A.33)

The second term in (A.33) is equal to the first term in (A.32) taken with opposite sign. The residual for vertex wave I j+ 1 + I j+ 3 will take the form: 2 2 j

exp {ik (Δ,⃗ ρ⃗ j+1/2 )} (Δ,⃗ n⃗ j+1 ) 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j+1 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 exp {ik (Δ,⃗ ρ⃗ j+3/2 )} (Δ,⃗ n⃗ j+2 ) . + 󵄨 󵄨2 (Δ,⃗ ρ ⃗ 󸀠j+2 ) k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨

I j+ 12 + I j+ 32 − I∞ = −

(A.34)

The first term in (A.34) is equal to the second term in (A.32) taken with an opposite sign. Thus, we proved that integration of the residuals (A.32) over the closed contour C gives zero. Therefore, equation (A.25) can be considered a vertex wave for both

A.5 Phase function and far zone condition

|

147

semi-infinite and finite-size structures; it can be used in heuristic formulas. However, rigorous expression for the vertex wave will be different from the one we used above, it will be described by equation (A.25), rather than (A.30).

A.5 Phase function and far zone condition It was shown in Chapter 3 (see formula (3.15) and relevant text) that the integral (A.2) can be extracted from expressions (3.2), provided that the far zone conditions are satisfied: r ≥ 2D2 /λ, (A.35) where D is the characteristic size of the object (for example, its maximum transverse size). Under this conditions, many expressions are simplified. For example, all distances except those related to the position of a point on the scatterer surface are factored out of the integral. In the general case, expressions take much more cumbersome form. The far zone condition leads to relation (3.14), from which it follows that, as the observation point moves away from an object, the angular sizes of this object tend to zero. The phase function (A.17) can be written as ⃗ = k (Δ x x + Δ y y) . Φ = k (Δ,⃗ ρ)

(A.36)

This functions depends on the positions of the source and observation point. These parameters enter the phase function symmetrically, therefore, the reciprocity principle holds true. Find out the gradient of phase function. grad Φ = e⃗ x

dΦ dΦ + e⃗ y = k (e⃗ x Δ x + e⃗ y Δ y ) = k Δ.⃗ dx dy

(A.37)

Thus, vector Δ⃗ has the physical meaning of phase gradient. If the far zone condition (A.35) holds true, the phase is linear and, therefore, the gradient is a constant function. This property is a consequence of the far zone condition. If the far zone condition is not satisfied, the phase function is more complicated. In the paper by mathematicians Karatygin and Rozov [58], the Kirchhoff integral for a plane angular sector is evaluated by the two-dimensional stationary phase method. In [58], a function φ is introduced related to our phase Φ (see (A.36)) by Φ = kφ, therefore Δ⃗ = grad φ. It is interesting to note that in paper [58] (Section 2) it is stated that if the phase on the integration contour is constant, the method of stationary phase becomes inapplicable and numerical integration must be performed. In our approach, on the contrary, this case results in the simplest expressions corresponding to “forward” and “mirror” directions. Instead of numerical integration, we apply the Stokes theorem (A.16), (3.43).

B Rigorous Two-dimensional Solution for Diffraction by Half-plane In order to obtain a solution for oblique incidence on a perfectly conducting halfplane, we make use of the formulas from [30] (see Section 11.6 therein). Substitute the angles α and β from [30] (Fig. 4.4) by the angles α1 and β1 , related to local, rather than global, edge coordinates. Other variables are also denoted by the same subscript. Let a plane wave with the phase factor U1 = exp (−ikS1 ) = exp [−ik (x cos α1 cos β1 + y sin α1 cos β1 + z sin β1 )] ,

(B.1)

be incident on the scatterer, with a perfectly conducting screen occupying the halfplane y = 0, x > 0. This phase factor is obtained from the two-dimensional form corresponding to β1 = 0 by substituting k cos β1 for k and multiplication by exp(−ikz sin β1 ). This method, applicable to any two-dimensional solution of wave equation, allows one to obtain a solution to three-dimensional wave equation in which z enters only via the factor exp(−ikz sin β1 ). Also, if U1 is such a solution to three-dimensional wave equation, it can be easily shown that two electromagnetic fields satisfying the Maxwell equations are written as E⃗

= (− i sink β1

H⃗

1 = (− ki ∂U ∂y ,

E⃗

i ∂U1 1 = ( ki ∂U ∂y , − k ∂x , 0)

H⃗

=

i sin β1 ∂U1 ∂U1 2 ∂x , − k ∂y , cos i ∂U1 k ∂x , 0)

i sin β i sin β1 ∂U1 2 1 (− k 1 ∂U ∂x , − k ∂y , cos

β 1 U 1 )} }, }

(B.2)

} }. β1 U1 ) }

(B.3)

For β1 = 0, equation (B.2) corresponds to two-dimensional field with TH-polarization, while (B.3) corresponds to TE-polarized field. If U1 has the form (B.1), then equations (B.2) and (B.3) give, accordingly, two plane waves E⃗ H⃗ E⃗

= (− cos α1 sin β1 , − sin α1 sin β1 , cos β1 ) exp(−ikS1 )} }, = (− sin α1 , cos α1 , 0) exp(−ikS1 ) }

(B.4)

= (sin α1 , − cos α1 , 0) exp(−ikS1 )

} (B.5) }. = (− cos α1 sin β1 , − sin α1 sin β1 , cos β1 ) exp(−ikS1 ) } Here, the factor cos β1 was omitted through. Solution to diffraction problem is given by equation (B.2) in the case where the incident wave is described by (B.4), and by equation (B.3) for the incident wave described by (B.5); the function U1 is obtained from E z sec β1 in the two-dimensional case by substituting k cos β1 for k and multiplying by exp(−ikz sin β1 ). The explicit form of the scattered field occurring upon incidence of wave (B.2) is H⃗

U1 =

sec β1 exp [ik (r cos β1 − z sin β1 )] {G (p1 ) − G (q1 )} , √πi

(B.6)

150 | B Rigorous Two-dimensional Solution for Diffraction by Half-plane

where p1 = −√2kr cos β1 cos

θ1 − α1 , 2

q1 = −√2kr cos β1 cos

θ1 + α1 . 2

(B.7)

Components of the scattered field take the following form in the Cartesian coordinates: exp [ik (r cos β1 − z sin β1 )] } cos β1 {G (p1 ) − G (q1 )} } } } √πi } } } } exp [ik (r cos β1 − z sin β1 )] } } } Hx = − } } √πi } } } } α1 θ1 } } } cos } } × {sin α1 [G (p1 ) + G (q1 )] + i√2/ (kr cos β1 ) sin 2 2 }. } } } exp [ik (r cos β1 − z sin β1 )] } } Hy = } } } √πi } } } } α1 θ1 } } cos }} × {cos α1 [G (p1 ) − G (q1 )] − i√2/ (kr cos β1 ) sin } 2 2 } } } } E x = −H y sin β1 , E y = H x sin β1 , H z = 0 } Ez =

(B.8)

A similar result can also be obtained in the case where the incident wave is determined by (B.5). Function G(a) is defined by ∞

G (a) = exp (−ia2 ) F (a) ,

F (a) = ∫ exp (iμ2 ) dμ.

(B.9)

a

The expression for scattered field can be simplified significantly if the Fresnel integral is substituted by its asymptotic form −x

∫ e

∞ iq2

2

2

dq = ∫ e iq dq ≈ −

−∞

x

e ix for x ≫ 1, 2ix

(B.10)

or by asymptotic expansion known from handbooks e ix e ix ∞ i n (2n)! + ∑ 2ix 2ix n=1 (2x)2n (n)! 2

F (x) = −

2

2 e ix e ix ∞ i n (2n)! e ix 2i 1 −1 =− + + + ≅ e ix ( ∑ ). 2ix 2ix (2x)2 2ix n=2 (2x)2n (n)! 2ix 4x3 2

2

2

Therefore,

−1

G (a) ≅ − (2ia)−1 + (4a3 )

.

(B.11)

(B.12)

Also, the following is true: 1 −1 1 dG (a) −i = . = −1 − 2iaG (a) ≅ −1 − 2ia ( + )= da 2ia 4a3 2a2 2ia2

(B.13)

B Rigorous Two-dimensional Solution for Diffraction by Half-plane | 151

Note that the derivative is smaller by absolute value than G (a) ≅ − (2ia)−1 for large values of argument a. From the physical point of view, substitution of asymptotic formulas (B.12) and (B.13) for the rigorous ones (B.9) in equation (B.8) corresponds to imposing the far zone conditions (3.13) in the problem of diffraction by a plane angular sector. In the case of finite-dimension scatterers, the far zone condition is satisfied when, as the distance from the observation point to the scatterer is increasing, its angular sizes are diminishing and shrink to a point. On the contrary, angular sizes of any semi-infinite scatterer never shrink to a point, whatever the distance to the observation point. Therefore, e. g., solution to the diffraction problem for a three-dimensional polygon of finite dimension is described in the far zone by the sum of inputs of each its edge, with all these inputs being singular, see equations (A.16), (A.21), (3.42), and (4.4) [36]. At the same time, solution to diffraction problem for a half-plane is written in terms of the Fresnel integral (see, e. g., (1.57), (2.40), (B.8), and (H.1) [1]). On the “light–shadow” boundary, the diffraction coefficients are singular, but the Fresnel integral offsets this singularity, and, as a result, the solution on the boundary is equal to 12 , see (E.5). After we impose the far zone condition in the diffraction problem for a plane angular sector, we obtain, instead of a special function similar to the Fresnel integral, singular inputs of edges in the direction of “light–shadow” boundaries. It seems that a similar technique was applied in [16], because the reference solution given therein also has singularity in the direction of “light–shadow” boundary, and coefficients D involve fields taken “at infinity”. In the rigorous solution of diffraction problem for a plane angular sector, the geometrical optics field value on the “light–shadow” boundary must be equal to A(1 − S⊥ ), it is bounded between A/2 and A. Here, A is the geometrical optics field value in the absence of angular sector, S⊥ is the area of a sector cut from a disk of unity area. The opening angle of the cut sector S⊥ is equal to the projection of the opening angle of the plane sector on the plane normal to the incident wave propagation direction. Imposing the far zone condition, we obtain two advantageous features of the reference solution. Firstly, the reference solution is simplified. Secondly, no further adaptation of this solution is needed in application to three-dimensional problems in which the far zone condition is satisfied. Return now to the solution for oblique incidence on the edge of a half-plane. Introduce the vectors of T-polarizations, i. e., vectors t ⃗ orthogonal to the edge and to the direction vectors of the incident and scattered fields n⃗ 󸀠 or n⃗ 󸀠󸀠 . Introduce vectors p⃗ orthogonal both to vectors t ⃗ and to the direction vectors n⃗ 󸀠 or n⃗ 󸀠󸀠 . For these vectors, in the cases of incident (subscript i) and scattered (subscript s) fields, it is convenient to introduce new notation: t i⃗ , p⃗ i , t s⃗ , and p⃗ s . In the angular coordinates [30], these vectors are t i⃗ = (sin α1 , − cos α1 , 0) , t s⃗ = (− sin θ1 , cos θ1 , 0) ,

p⃗ i = (− cos α1 sin β1 , − sin α1 sin β1 , cos β1 ) p⃗ s = (sin β1 cos θ1 , sin β1 sin θ1 , cos β1 ) .

(B.14)

152 | B Rigorous Two-dimensional Solution for Diffraction by Half-plane

The angles θ1 and α1 are the same as in [30], albeit they are considered in the global, rather than local coordinates related to the edge. The angle θ1 is reckoned in the same direction as α1 . In the case of “forward” scattering, the directions to the source and observation point are opposite (in contrast to the direction vectors n⃗ 󸀠 or n⃗ 󸀠󸀠 which coincide), while the respective angles differ by π. Therefore, the signs of trigonometric functions of θ1 and α1 in the expressions for vectors t i⃗ and t s⃗ , as well as p⃗ i and p⃗ s are opposite. Thus, for the two polarizations the obliquely incident plane waves can be expressed via vectors t i⃗ and p⃗ i : E⃗ = p⃗ i cos β1 U1 } } H⃗ = −t i⃗ cos β1 U1 } ⃗ ⃗ E = t i cos β1 U1 } } H⃗ = p⃗ i cos β1 U1 }

TH-polarization, (B.15) TE-polarization.

For oblique incidence, we obtain in the far zone (by simplifying the solution [30]): U THβ = (cos β1 )−1 U TH0 (k cos β1 ) exp (−ikz sin β1 ) , U TEβ = (cos β1 )−1 U TE0 (k cos β1 ) exp (−ikz sin β1 ) .

(B.16)

Here, U THβ and U TEβ is the solution for oblique incidence, U TH0 and U TE0 are solutions for normal incidence (with β1 = 0) [30]: U TH0 = E z = (πi)−1/2 exp (ikr) {G (u1 ) − G (v1 )} ,

(B.17)

U TE0 = H z = (πi)−1/2 exp (ikr) {G (u1 ) + G (v1 )} ,

θ1 +α1 1 √ where u1 = −√2kr cos θ1 −α 2 , v 1 = − 2kr cos 2 . Equations (B.14) and (B.15) describe the incident field in the form adopted in [30]. However, we can simplify substantially the expressions for the scattered field from [30] by taking advantage of the fact that we are interested in the far zone field only. At large distances from the edge, we have a → ∞, so that asymptotic formulas (B.12) and (B.13) can be used. Apply now equations (B.10) or (B.12) to find out the asymptotic formulas for the sums G(u1 ) ± G(v1 ).

G (u) − G (v) ≅ = =

−1 −1 i − = 2iu 2iv −2√2kr cos θ+α 2

−

i −2√2kr cos

− cos

θ−α 2 ) θ−α −2√2kr cos 2 cos θ+α 2 i (−2 sin 2θ sin 2α )

i (cos

θ−α 2

−√2kr (cos θ + cos α)

= i√

θ+α 2

(B.18) 2 sin 2θ sin 2α , kr cos θ + cos α

B Rigorous Two-dimensional Solution for Diffraction by Half-plane | 153

G (u) + G (v) ≅ =

−1 −1 i + = 2iu 2iv −2√2kr cos i (cos

θ+α 2

+ cos

−2√2kr cos

θ−α 2 ) θ−α θ+α 2 cos 2

θ−α 2

=

+

i √ −2 2kr cos

θ+α 2

i (2 cos 2θ cos 2α ) −√2kr (cos θ + cos α)

(B.19)

2 cos 2θ cos 2α . kr cos θ + cos α In the asymptotic formulas (B.19), the radial and angular variables (r and φ, respectively) are separated, therefore, each of two sums G(u1 ) ± G(v1 ) can be combined into a single expression. Thus, the sums G(u1 ) ± G(v1 ) are reduced to well-known expressions for the diffraction coefficients f and g [1]: = −i√

G (u1 ) ∓ G (v1 ) ≅

i f ( ). √2kr g

(B.20)

In the sums G(u1 ) ± G(v1 ), f and g are rigorous diffraction coefficients for halfplane, coinciding with known expressions from [1]: f =

2 sin φ2 sin φ20 , cos φ + cos φ0

g=

−2 cos φ2 cos φ20 . cos φ + cos φ0

(B.21)

Correspondence between the angles in the expressions from [1] and [30] is given by φ0 ↔ α1 and φ ↔ θ1 . One can show that it is only correct to apply functions G(u1 ) and G(v1 ) in the form (B.9) to semi-infinite scatterers, whereas for three-dimensional finite-dimension scatterers under the far zone conditions it is more correct to apply their asymptotic forms (B.20). Also, if the asymptotic formulas (B.10) or (B.12) are applied, relations (B.17) for U TH0 and U TE0 are simplified significantly because the number of terms is reduced, and the expressions no longer contain the Fresnel integrals and involve elementary functions only. For a half-plane, the simplified relations (B.17) for U TH0 and U TE0 are valid only far away from the “light–shadow” boundary, whereas for three-dimensional plane scatterers of finite size they are valid in the whole space. If the asymptotic formulas (B.12) and (B.13) are substituted into relations (B.8) from [30], the relations for the scattered field take a simple form, very similar to that for the incident field (B.15): E⃗ = p⃗ s cos β1 U THβ } } H⃗ = −t s⃗ cos β1 U THβ } ⃗ ⃗ E = t s cos β1 U TEβ } } H⃗ = p⃗ s cos β1 U TEβ }

TH-polarization, (B.22) TE-polarization.

In formulas (B.15) and (B.22), there appears a factor cos β1 which was omitted in relations (B.4) and (B.5).

154 | B Rigorous Two-dimensional Solution for Diffraction by Half-plane

It follows from formulas (B.14) and (B.15) that in the rigorous two-dimensional solution the T-polarizations of the incident wave (B.22) are conserved for all angles of oblique incidence. This is not true in the physical optics, where “mixing” of polarizations occurs. This is the principal difference between the rigorous and physical optics solutions. Consider now a number of useful formulas determining geometrical relationships between the vectors. Write down the T-vectors which are normal to the edge and described as vector products in the global coordinate system from [30]. Hereafter, vectors in the coordinates [30] are denoted by subscript BW. e⃗ x [n⃗ 󸀠BW × e⃗ z ] = (− cos β cos α 0

e⃗ y − cos β sin α 0

e⃗ z − sin β) 1

(B.23)

= cos β (− sin α, cos α, 0) , [n⃗ 󸀠󸀠 BW

e⃗ y cos β sin θ 0

e⃗ x × e⃗ z ] = (cos β cos θ 0

e⃗ z − sin β) 1

(B.24)

= cos β (sin θ, − cos θ, 0) . The scalar product of T-polarization vectors of incident and scattered waves is: 2 ⃗ ([n⃗ 󸀠BW × e⃗ z ] , [n⃗ 󸀠󸀠 BW × e z ]) = cos β (− sin α sin θ − cos α cos θ)

= − cos2 β cos (θ − α) .

(B.25)

In the local coordinate system, we deal with the imaginary edge. Find out the vectors which are normal both to the “imaginary edge” and to the direction vector of the incident wave in the coordinates [16]. For a plane angular sector, in the coordinates [16] we have n⃗ 󸀠 = (− sin ϑ0 , 0, − cos ϑ0 ) , ρ ⃗ 󸀠𝛾 = (0, − sin

n⃗ 󸀠󸀠 = (cos φ, sin φ, 0) ,

𝛾

𝛾

(B.26)

, cos ) . 2 2

T-polarization vector of the source wave is e⃗ x [n⃗ 󸀠 × ρ ⃗ 󸀠𝛾] = (− sin ϑ0 0

e⃗ y e⃗ z 0 − cos ϑ0 ) 𝛾 𝛾 − sin 2 cos 2

= (− cos ϑ0 sin

𝛾

𝛾

2

2

, sin ϑ0 cos

(B.27)

𝛾

, sin ϑ0 sin ) . 2

Change the sign of the vector tangent to the edge, so that the T-vector be pointing into the upper half-space x > 0. [n⃗ 󸀠 × −ρ ⃗ 󸀠𝛾] = (cos ϑ0 sin

𝛾

𝛾

2

2

, − sin ϑ0 cos

𝛾 , − sin ϑ0 sin ) = t i⃗ . 2

(B.28)

B Rigorous Two-dimensional Solution for Diffraction by Half-plane |

155

Polarization vector p⃗ i of the source wave, perpendicular to t i⃗ , is e⃗ y e⃗ z e⃗ x [n⃗ 󸀠 × t i⃗ ] = ( − sin ϑ0 0 − cos ϑ0 ) 𝛾 𝛾 𝛾 cos ϑ0 sin 2 − sin ϑ0 cos 2 − sin ϑ0 sin 2 = (− sin ϑ0 cos ϑ0 cos

𝛾

𝛾

2

2

, − sin

(B.29)

𝛾

, sin2 ϑ0 cos ) = p⃗ i . 2

T-polarization vector of the receiver wave is e⃗ x [n × −ρ ⃗ 𝛾] = (cos φ 0 ⃗ 󸀠󸀠

󸀠

e⃗ y e⃗ z sin φ 0 ) 𝛾 𝛾 sin 2 − cos 2

= (− sin φ cos

𝛾

𝛾

2

2

, cos φ cos

(B.30)

𝛾 , cos φ sin ) = t s⃗ . 2

Polarization vector p⃗ s of the receiver wave, perpendicular to t s⃗ , is e⃗ y e⃗ z e⃗ x ⃗ ) [n × t s ] = ( cos φ sin φ 0 𝛾 𝛾 𝛾 − sin φ cos 2 cos φ cos 2 cos φ sin 2 ⃗ 󸀠󸀠

= (sin φ cos φ sin

𝛾

𝛾

2

2

, − cos2 φ sin

𝛾

, cos ) = p⃗ s . 2

(B.31)

C Application of Imaginary Edge in Diffraction Problems In many heuristic method of diffraction theory (such as PO, MEW, ILDC, EECM) the scattered field is found as a linear integral over the edge. To evaluate this integral, the integrand related to the edge must first be found. This integrand may be known under different names in different methods: “integral over elementary strip” in PO or MEW, “incremental length diffraction coefficient” in ILDC, or “equivalent contour current” in EECM. In addition to the differences in terminology, there may be some differences in the implementation details, namely, in the length (finite or infinite) of the integration strip and its direction which can be either aligned with the normal, or be at an angle to it. Leaving aside the analysis of existing methods (which, however, can lead to the same results despite the differences in problem formulation), we offer here one more approach and explain the reasons for its choice. All the above-mentioned approaches are based on rigorous solutions (MEW, ILDC, EECM) and make use of available 2D solutions for diffraction by a semi-infinite structure with ideal or non-ideal boundary condition. Upon normal or oblique incidence on the edge, the elementary strip of integration is normal to the edge, while the observation point, whatever its spacial position, is always located on the diffraction cone. The latter fact is the property of two-dimensional solution and infinite-length edge. In 2D problem, for any relative positions of the source and observation point, there always exist a diffraction cone corresponding to these points, because on an infinite edge there always can be found a point of stationary phase located at the diffraction cone apex. If a stationary phase point is available on the edge, one can always make use of readily available 2D solutions. The scattering pattern of each element will then correspond to the pattern taken from the solution to 2D problem for the case of normal incidence on the edge. In the heuristic methods, two-dimensional solutions are applied to threedimensional problems; the edge of finite size is considered. If a stationary phase point is available on this edge, a heuristic solution to diffraction problem on the edge can be found performing integration along the edge. If the far zone condition is satisfied, the signal phase on the edge will be constant. However, there may be encountered a situation when the stationary phase point is absent on the edge. In this case, a heuristic solution can be constructed introducing an “imaginary edge” on the scatterer surface, i. e., a surface line which is the generatrix of the diffraction cone for the given source and observation point positions. On the imaginary edge, there appears a stationary phase point located at the diffraction cone apex. Under the far zone condition, the scattered signal phase on the “imaginary” edge will be constant. In the three-dimensional heuristic solution, one can use the known twodimensional solution on the diffraction cone of the imaginary edge, integrating this

158 | C Application of Imaginary Edge in Diffraction Problems

solution along the real edge with corresponding phase. On the real edge, the phase is no longer constant, it varies linearly along the edge. Consider now, in the physical optics approximation, a perfectly conducting plane angular sector on which a plane electromagnetic wave is incident. Without loss of generality of the approach, we use the coordinates introduced in [16]. Consider first the wave incident on a real edge oriented at an angle β/2 to z-axis. For the given direction to the observation point, we construct an imaginary edge at an angle 𝛾/2 to z-axis (Fig. 4.3). The real edge is shown by the solid line, the imaginary edge is by the dashed one. The imaginary edge is not needed to obtain the PO-solution, However, we will need it to obtain the formulas based on the solution [30] and subsequent comparison with the PO-solution. Therefore, some formulas below do not contain the angle 𝛾 which substitutes in all formulas the angle β, while other equations contain it. All discussions remain valid, whether the angle is present in the formulas or not. Consider a wave (a) or (b) incident on an edge of a plane angular sector, shown schematically in Fig. C.1 in the coordinates from [16] (see also Fig. 4.1). By the thick solid arrow, the incident magnetic field vector is drawn, the thick dashed arrow shows the surface current vector. First, we find out the surface current excited by this wave and decompose it into components parallel and perpendicular to the edge. Then we multiply these components by coefficients cp1 or cr1 taking into account the difference between the rigorous and physical optics solutions (for physical optics solution cp1 = cr1 = 1). After that we sum up the inputs of all components at the observation point. x

x φ

φ ⃗

H inc ⃗

⃗

j

H inc

⃗

j

E⃗ inc

⃗

E inc ϑ

ϑ

45˚

45˚ y

y

z

z (a)

(b)

Fig. C.1: Incidence of a wave on a plane angular sector: polarizations of the magnetic field vector of incident waves and induced surface currents.

C Application of Imaginary Edge in Diffraction Problems |

159

Consider excitation by wave (a), in which case the magnetic field has only y-component in Cartesian coordinates, or only φ-component in the angular coordinates: ⃗ = H0 (0, −1, 0) exp {ik (n⃗ 󸀠 , r)} ⃗ , H⃗ (i) = H⃗ 0 exp {ik (n⃗ 󸀠 , r)} ⃗ (i)

H

or

⃗ = −e⃗ φ H0φ exp {ik (n , r)} ⃗ . = H⃗ 0 exp {ik (n , r)} ⃗󸀠

⃗󸀠

(C.1)

Excitation by wave (a) results in the induction of current on the scatterer surface ⃗ = −e⃗ z H0φ . jφ1

(C.2)

Decomposition of the current induced by wave (a) into components parallel and perpendicular to the edge: β β β 󵄨 ⃗ 󵄨󵄨 󵄨󵄨 sin (0, − cos , − sin ) , jφ1r = 󵄨󵄨󵄨󵄨jφ1 󵄨 2 2 2

(C.3)

β β β 󵄨 ⃗ 󵄨󵄨 󵄨󵄨 cos (0, sin , − cos ) . jφ1p = 󵄨󵄨󵄨󵄨jφ1 󵄨 2 2 2 ⃗ at the observation point are Components of vector potential Aφ Aφ φ = jφ1 ⋅ cos φ β β β β sin cp1 − sin cos cr1) 2 2 2 2 β β Aφ ϑ = jφ1 ⋅ (cos2 cp1 + sin2 cr1) 2 2 × (cos

(projections on y-axis),

(C.4)

(projections on −z-axis).

Consider now excitation by wave (b): ⃗ , H⃗ (i) = H0 (cos ϑ, 0, − sin ϑ) exp {ik (n⃗ 󸀠 , r)}

or

⃗ . H⃗ (i) = e⃗ ϑ H0ϑ exp {ik (n⃗ 󸀠 , r)}

(C.5)

Excitation by wave (b) results in the induction of a current on the scatterer surface ⃗ = e⃗ y H0ϑ sin ϑ. jϑ1

(C.6)

Decomposition of the current induced by wave (b) into components parallel and perpendicular to the edge is β β β 󵄨 ⃗ 󵄨󵄨 󵄨󵄨 sin (0, − cos , − sin ) , jϑ1r = 󵄨󵄨󵄨󵄨jϑ1 󵄨 2 2 2 β β β 󵄨 ⃗ 󵄨󵄨 󵄨󵄨 cos (0, − sin , cos ) . jϑ1p = 󵄨󵄨󵄨󵄨jϑ1 󵄨 2 2 2

(C.7)

⃗ at the observation point are Components of vector potential Aφ β β cp1 + cos2 cr1) 2 2 β β β β Aϑ ϑ = jϑ1 ⋅ (sin cos cp1 − cos sin cr1) 2 2 2 2

Aϑ φ = jϑ1 ⋅ cos φ ⋅ (sin2

(projections on −y-axis), (C.8) (projections on z-axis).

160 | C Application of Imaginary Edge in Diffraction Problems

The vector potentials are decomposed as ⃗ = e⃗ φ Aϑ φ + e⃗ ϑ Aϑ ϑ . Aϑ

(C.9)

e⃗ ϑ = e⃗ x cos ϑ − e⃗ z sin ϑ,

(C.10)

⃗ = e⃗ φ Aφ φ + e⃗ ϑ Aφ ϑ , Aφ Taking into account relations e⃗ φ = e⃗ x sin φ + e⃗ y cos φ, we obtain

⃗ = (e⃗ x sin φ + e⃗ y cos φ) Aφ φ + (e⃗ x cos ϑ − e⃗ z sin ϑ) Aφ ϑ , Aφ ⃗ = (e⃗ x sin φ + e⃗ y cos φ) Aϑ φ + (e⃗ x cos ϑ − e⃗ z sin ϑ) Aϑ ϑ . Aϑ

(C.11)

Taking into account that x-components on the scatterer surface are zero, we obtain: ⃗ = e⃗ y cos φAφ φ − e⃗ z sin ϑAφ ϑ , Aφ

⃗ = e⃗ y cos φAϑ φ − e⃗ z sin ϑAϑ ϑ . Aϑ

(C.12)

For excitation types (a) or (b), the components parallel or perpendicular to the edge are: β β β 󳨀󳨀󳨀→ cap = cos (e⃗ y sin − e⃗ z cos ) , 2 2 2 β β β 󳨀󳨀→ car = sin (−e⃗ y cos − e⃗ z sin ) , 2 2 2 (C.13) 󳨀󳨀󳨀→ β β β ⃗ ⃗ cbp = sin (−e y sin + e z cos ) , 2 2 2 󳨀󳨀→ β β β cbr = cos (−e⃗ y cos − e⃗ z sin ) . 2 2 2 󳨀󳨀󳨀→ 󳨀󳨀󳨀→ For clarity, in Fig. 4.3, instead of vectors cap and cbp, the vectors cp⃗ and cr⃗ corresponding to excitation types (a) or (b) are shown. The angular arguments φ𝛾 and φ𝛾0 are determined (taking into account that the 𝛾 𝛾 vector of internal normal to imaginary edge is n⃗ 𝛾 = (0, − cos 2 , − sin 2 )) as follows: 𝛾 𝛾 n⃗ 󸀠󸀠 sin 𝛾 cos 2 + sin 2 (1 − cos 𝛾) 𝛾 φ𝛾0 = arccos ( 󵄨 󸀠 󵄨 , n⃗ 𝛾) = arccos ( ) 󵄨 󵄨 󵄨󵄨n⃗ 󵄨󵄨 2 󵄨󵄨󵄨󵄨n⃗ 󸀠𝛾󵄨󵄨󵄨󵄨 cos−1 ϑ0 󵄨󵄨 𝛾󵄨󵄨 𝛾

= arccos (

sin 2 cos ϑ0 √1 − cos2 2𝛾 cos2 ϑ0 n⃗ 󸀠󸀠 𝛾

),

φ𝛾 = arccos ( 󵄨 󸀠󸀠 󵄨 , n⃗ 𝛾) = arccos ( 󵄨󵄨n⃗ 󵄨󵄨 󵄨󵄨 𝛾 󵄨󵄨 𝛾

= arccos (

− cos 2 sin φ √1 − sin2 2𝛾 sin2 φ

).

𝛾

𝛾

𝛾

− 2 sin2 2 cos 2 ) 󵄨󵄨 󸀠󸀠 󵄨󵄨 −1 2 󵄨󵄨󵄨n⃗ 𝛾 󵄨󵄨󵄨 sin φ

−2 cos3

2

(C.14)

C Application of Imaginary Edge in Diffraction Problems |

161

Therefore, sin φ𝛾0 =

sin ϑ0 √1 − cos2 2𝛾 cos2 ϑ0

,

sin φ𝛾 =

cos φ √1 − sin2 2𝛾 sin2 φ

.

(C.15)

Coefficients cp𝛾1 or cr𝛾1 for y > 0 (i. e. on side (1) of the plane angular sector) are equal φ𝛾 sin 2 f (φ𝛾, φ𝛾0 ) 𝛾 1 = → cp φ𝛾 (TH) and f 0 (φ𝛾, φ𝛾0 ) sin 2 0 (C.16) φ𝛾 sin 2 0 g (φ𝛾, φ𝛾0 ) → cr𝛾1 = φ𝛾 (TE). g0 (φ𝛾, φ𝛾0 ) sin 2 Thus, coefficients D in the physical optics approximation (corrected by factors cp𝛾 and cr𝛾) are: D ϑϑ =

∞ kR E ϑ

e−ikR E inc ϑ

=

∞ kR H φ

e−ikR H φinc

=

e kR ikAφ ϑ

e−ikR H φinc

ik2 I β β [−H0φ (cos2 cp𝛾 + sin2 cr𝛾)] , 2π 2 2 ∞ e ∞ kR E φ kR −H ϑ kR ikAϑ φ = −ikR inc = −ikR inc = −ikR e e e Eφ Hϑ H ϑinc =

D φφ

ik2 I β β [H0ϑ sin ϑ0 cos φ (sin2 cp𝛾 + cos2 cr𝛾)] , 2π 2 2 ∞ e ∞ kR E ϑ kR H φ kR ikAϑ = −ikR inc = −ikR inc = −ikR incϑ e e e Eφ Hϑ Hϑ =

D ϑφ

(C.17)

ik2 I β β β β [H0ϑ sin ϑ0 (sin cos cp𝛾 − cos sin cr𝛾)] , 2π 2 2 2 2 ∞ e ∞ kR E φ kR −H ϑ kR ikAφ φ = −ikR inc = −ikR inc = −ikR e e e Eϑ Hφ H φinc =

D φϑ

=

ik2 I β β β β [−H0φ cos φ ⋅ (cos sin cp𝛾 − sin cos cr𝛾)] . 2π 2 2 2 2

By setting cp𝛾 = 1 and cr𝛾 = 1 in equation (C.17), one obtains the coefficients D in the physical optics approximation (uncorrected).

D Summary of Formulas for Diffraction by Plane Angular Sector In Appendix A, we considered the integral over a plane angular sector in the physical optics approximation. Then, in Appendix C, this integral was used to obtain the solution for plane electromagnetic wave diffraction by a perfectly conducting plane angular sector. This solution does not provide high accuracy, even with the corrections introduced. In this appendix, we consider the solution for a plane angular sector obtained by a different approach. The solution is constructed in several steps. 1. Integral over the plane angular sector is found in the physical optics approximation. 2. The PO integral is split into the edge inputs, with input of each individual edge additionally factored out into two multiplicands: an integral over the edge, and an integral over elementary integration strip (polarization input). 3. Polarization component is factored out of the rigorous two-dimensional solution [30] with enforced far zone condition; this solution was considered in Appendix B. 4. In each edge input factored out of the PO integral, the PO polarization component is substituted by the polarization component of rigorous solution. 5. It can be shown that the solution obtained corresponds to EECM approximation (or, equivalently, to MEW and ILDC approximations); its further refinement is achieved by application of the modified EECM (MEECM). The field scattered by an edge of length a j can be written in the physical optics approximation as (see Appendix A) ik exp (ikR) I j [n⃗ × H⃗ 0 ]φ 2π R ϑ ϑ 0 ⃗ ⃗ [ n × H ]φ 1 a j exp (ikR) ϑ = . 2π R cos φ + cos φ0

E φ = ikA eφ = ϑ

(D.1)

Here, we used the equation for I j for normal incidence on the edge Ij =

ia j exp {iΦ j } a j exp {iΦ j } . = j ⃗ ik φ + cos φ0 ) (cos ⃗ k (Δ, n )

(D.2)

The vertex wave (i. e., scattering by plane angular sector) in the physical optics approximation (Appendix A is obtained by changing the summation order for edge inputs N

N

I = ∑ Ij = ∑ I j , j=1

j=1

where I j =

j j+1 exp {iΦ j } (Δ,⃗ n⃗ ) (Δ,⃗ n⃗ ) ]. [ − 󸀠 󸀠 󵄨 󵄨2 k2 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 [ (Δ,⃗ ρ ⃗ j ) (Δ,⃗ ρ ⃗ j+1 ) ]

(D.3)

164 | D Summary of Formulas for Diffraction by Plane Angular Sector

Heuristic solution for plane electromagnetic wave diffraction based on the rigorous solution [30] (Appendix B) for a perfectly conducting edge is different from heuristic solution to the same problem based on the physical optics approximation (Appendix C), but its derivation follows the same concept, i. e., by substitution of polarization component of diffraction coefficient in the physical optics integral (Appendix A). ∞ ∞ kR E ϑ kR H φ D ϑϑ = ikR inc = ikR inc e Eϑ e Hφ ik2 j s ⃗ inc ⃗ 󸀠󸀠 󸀠󸀠 I [(e ⃗ sϑ , p⃗ s ) (e ⃗ inc ϑ , p⃗ i ) U THβ + ( e ⃗ ϑ , t s ) ( e ⃗ ϑ , t i ) U TEβ ] , 2π ∞ ∞ kR E φ kR −H ϑ = ikR inc = ikR inc e Eφ e Hϑ = D φφ

ik2 j s inc ⃗ 󸀠󸀠 󸀠󸀠 ⃗ I [(e ⃗ sφ , p⃗ s ) (e ⃗ inc φ , p⃗ i ) U THβ + ( e ⃗ φ , t s ) ( e ⃗ φ , t i ) U TEβ ] , 2π ∞ ∞ kR E ϑ kR H φ = ikR inc = ikR inc e Eφ e Hϑ =

D ϑφ

(D.4)

ik2 j s ⃗ inc ⃗ 󸀠󸀠 󸀠󸀠 I [(e ⃗ sϑ , p⃗ s ) (e ⃗ inc φ , p⃗ i ) U THβ + ( e ⃗ ϑ , t s ) ( e ⃗ φ , t i ) U TEβ ] , 2π ∞ ∞ kR E φ kR −H ϑ = ikR inc = ikR inc e Eϑ e Hφ =

D φϑ

ik2 j s inc ⃗ 󸀠󸀠 󸀠󸀠 ⃗ I [(e ⃗ sφ , p⃗ s ) (e ⃗ inc ϑ , p⃗ i ) U THβ + ( e ⃗ φ , t s ) ( e ⃗ ϑ , t i ) U TEβ ] . 2π Consider the component form of the vectors involved in formulas (D.4). These formulas contain unit direction vectors of the incident and scattered fields =

inc e ⃗ inc φ = (0, 1, 0) , e ⃗ ϑ = (cos ϑ 0 , 0, − sin ϑ 0 ) ,

e ⃗ sϑ = (0, 0, −1) , e ⃗ sφ = (− sin φ, cos φ, 0) , t i⃗ = (cos ϑ0 sin

𝛾

𝛾

2

2

t s⃗ = (− sin φ cos

, − sin ϑ0 cos

𝛾

, cos φ cos

𝛾

𝛾

, − sin ϑ0 sin ) , 2

𝛾

, cos φ sin ) , 2 2

2 𝛾 𝛾 𝛾 p⃗ i = (− sin ϑ0 cos ϑ0 cos , − sin , sin2 ϑ0 cos ) , 2 2 2 𝛾 𝛾 𝛾 p⃗ s = (sin φ cos φ sin , − cos2 φ sin , cos ) , 2 2 2 𝛾 𝛾 ⃗ (e ⃗ inc (e ⃗ inc ϑ , p⃗ i ) = − sin ϑ 0 cos , ϑ , t i ) = sin , 2 2 𝛾 𝛾 ⃗ (e ⃗ inc (e ⃗ inc φ , p⃗ i ) = − sin , φ , t i ) = − sin ϑ 0 cos , 2 2 𝛾 𝛾 (e ⃗ sϑ , t s⃗ ) = − cos φ sin , (e ⃗ sϑ , p⃗ s ) = − cos , 2 2 𝛾 𝛾 (e ⃗ sφ , t s⃗ ) = cos . (e ⃗ sφ , p⃗ s ) = − cos φ sin , 2 2

D Summary of Formulas for Diffraction by Plane Angular Sector

| 165

Formulas (D.4) also involve the following expressions: φ𝛾0 φ𝛾 sin , 2 2 φ𝛾0 φ𝛾 = g (φ𝛾, φ𝛾0 ) (cos φ𝛾 + cos φ𝛾0 ) = −2 cos cos . 2 2

󸀠󸀠 = f (φ𝛾, φ𝛾0 ) (cos φ𝛾 + cos φ𝛾0 ) = 2 sin U THβ 󸀠󸀠 U TEβ

(D.5)

Equations (D.5) describe the nominators of diffraction coefficients, i. e., their polarization component on the imaginary edge. The factors corresponding to the denominators of diffraction coefficients are determined on the real edge, they appear in the formulas via the vertex waves I j . After some transformations, we obtain more concise expressions: D ϑϑ =

∞ ∞ kR E ϑ kR H φ = inc inc e ikR E ϑ e ikR H φ

ik2 j 󸀠󸀠 󸀠󸀠 + etϑϑ (ϑ0 , φ) U TEβ I [epϑϑ (ϑ0 , φ) U THβ ], 2π ∞ ∞ kR E φ kR −H ϑ = ikR inc = ikR inc e Eφ e Hϑ =

D φφ

ik2 j 󸀠󸀠 󸀠󸀠 + etφφ (ϑ0 , φ) U TEβ I [epφφ (ϑ0 , φ) U THβ ], 2π ∞ ∞ kR E ϑ kR H φ = ikR inc = ikR inc e Eφ e Hϑ =

D ϑφ

ik2 j 󸀠󸀠 󸀠󸀠 + etϑφ (ϑ0 , φ) U TEβ I [epϑφ (ϑ0 , φ) U THβ ], 2π ∞ ∞ kR E φ kR −H ϑ = ikR inc = ikR inc e Eϑ e Hφ =

D φϑ

=

ik2 j 󸀠󸀠 󸀠󸀠 + etφϑ (ϑ0 , φ) U TEβ I [epφϑ (ϑ0 , φ) U THβ ]. 2π

The expressions involved in formulas (D.6) are: epϑϑ (ϑ0 , φ) = − cos

𝛾 2

× − sin ϑ0 cos

etϑϑ (ϑ0 , φ) = − cos φ sin epφφ (ϑ0 , φ) = − cos φ sin etφφ (ϑ0 , φ) = cos

𝛾 2

𝛾 2

𝛾

2

× sin

𝛾 2

× − sin

× − sin ϑ0 cos

𝛾 2

,

,

𝛾 2

,

𝛾 2

,

epϑφ (ϑ0 , φ) = (e ⃗ sϑ , p⃗ s ) (e ⃗ inc φ , p⃗ i ) = − cos etϑφ (ϑ0 , φ) = − cos φ sin epφϑ (ϑ0 , φ) = − cos φ sin

𝛾 2

𝛾

2

𝛾 2

× − sin ϑ0 cos × − sin ϑ0 cos

× − sin

𝛾

2

,

𝛾

2

,

𝛾 𝛾 ⃗ etφϑ (ϑ0 , φ) = (e ⃗ sφ , t s⃗ ) (e ⃗ inc × sin . ϑ , t i ) = cos 2 2

𝛾 2

,

(D.6)

166 | D Summary of Formulas for Diffraction by Plane Angular Sector These equations involve functions related to the angle of imaginary edge 𝛾/2: cos

𝛾 2

=

sin φ √sin φ + cos2 ϑ0

,

sin

2

𝛾 2

=

cos ϑ0 √sin φ + cos2 ϑ0

.

(D.7)

2

Consider one more form of coefficients D. ∞ kR E ϑ kR = ikR inc ikR e Eϑ e ∞ E kR φ kR = ikR inc = ikR e Eφ e

DH ϑϑ = DH φφ

DH ϑφ = DH φϑ =

∞ kR E ϑ e ikR E inc φ ∞ kR E φ e ikR E inc ϑ

H φ∞ H φinc

=

−H ϑ∞

ik2 j ⃗ a I (H , e ⃗ φ ) , 2π

=−

ik2 j ⃗ b I (H , e ⃗ ϑ ) , 2π

H ϑinc ∞ kR H φ ik2 j ⃗ b = ikR inc = I (H , e ⃗ φ ) , 2π e Hϑ =

(D.8)

∞ kR −H ϑ ik2 j ⃗ a = − I (H , e ⃗ ϑ ) . 2π e ikR H φinc

In the formulas for coefficients D, the scalar products involved are: (H⃗ a , e ⃗ ϑ ) = − cos β (H⃗ b , e ⃗ ϑ ) = − cos β

𝛾

−U TEβ (− sin 2 ) √1 − cos2 ϑ0 cos2 2𝛾

(Dφϑ) ,

𝛾

−U TEβ sin ϑ0 cos 2

√1 − cos2 ϑ0 cos2 2𝛾

(H⃗ a , e ⃗ φ ) = sin (θ − φ)

(Dφφ) , 𝛾

U THβ (− sin ϑ0 cos 2 ) √1 − cos2 ϑ0 cos2 2𝛾

(D.9)

𝛾

+ sin β cos (θ − φ) (H⃗ b , e ⃗ φ ) = sin (θ − φ)

−U TEβ (− sin 2 ) √1 − cos2 ϑ0 cos2 2𝛾

(Dϑϑ) ,

𝛾

U THβ (− sin 2 ) √1 − cos2 ϑ0 cos2 2𝛾 𝛾

+ sin β cos (θ − φ)

−U TEβ sin ϑ0 cos 2

√1 − cos2 ϑ0 cos2 2𝛾

(Dϑφ) .

In these formulas, expressions for the magnetic fields H⃗ a and H⃗ b corresponding to excitation modes (a) and (b) are used (see Appendix C):

D Summary of Formulas for Diffraction by Plane Angular Sector

H⃗ a = (sin θ, − cos θ, 0)

𝛾

U THβ (− sin ϑ0 cos 2 ) √1 − cos2 ϑ0 cos2 2𝛾 𝛾

+ (sin β cos θ, sin β sin θ, cos β)

−U TEβ (− sin 2 )

(a)

√1 − cos2 ϑ0 cos2 2𝛾

= (H xa , H ya , H za ) ,

(D.10)

𝛾

⃗b

H = (sin θ, − cos θ, 0)

| 167

U THβ (− sin 2 ) √1 − cos2 ϑ0 cos2 2𝛾 𝛾

+ (sin β cos θ, sin β sin θ, cos β)

−U TEβ sin ϑ0 cos 2

(b)

√1 − cos2 ϑ0 cos2 2𝛾

= (H xb , H yb , H zb ) . Formulas (D.4) describe the principle by which solution to the problem of scattering by a plane angular sector is constructed, whereas formulas (D.6) and (D.8) provide an alternative form of the solution (D.4). Additional relationships for the vectors. 1. Direction vectors (see [16]): n⃗ 󸀠 = (− sin ϑ0 , 0, − cos ϑ0 ), n󸀠󸀠⃗ = (cos φ, sin φ, 0). 2. Auxiliary vector Δ⃗ = (0, − sin φ, − cos ϑ0 ). 𝛾 𝛾 3. Vectors of inner normal n⃗ 𝛾 = (0, − cos 2 , − sin 2 ) and tangent to the “imaginary 𝛾 𝛾 󸀠 edge” ρ ⃗ 𝛾 = (0, − sin 2 , cos 2 ). 4. Diffraction cone condition for an individual edge: (Δ,⃗ ρ ⃗ 󸀠 ) = 0. Denote by β the inner apex angle. Then we have: ρ ⃗ 󸀠1 = (0, − sin n⃗ 1 = ρ ⃗ 󸀠2 ,

β β , cos ) , 2 2

ρ ⃗ 󸀠2 = (0, − sin and

β β , − cos ) , 2 2

n⃗ 2 = −ρ ⃗ 󸀠1 .

The latter two formulas are valid only for the quadrant where β = π/2. In the general case: β β β β 󵄨 󵄨2 , − sin ) , n⃗ 2 = (0, cos , − sin ) , 󵄨󵄨󵄨󵄨Δ⃗ 󵄨󵄨󵄨󵄨 = sin2 (φ) + cos2 (ϑ0 ) , 2 2 2 2 β β β β 1 2 (Δ,⃗ n⃗ ) = sin (φ) cos + cos (ϑ0 ) sin , (Δ,⃗ n⃗ ) = − sin (φ) cos + cos (ϑ0 ) sin , 2 2 2 2 β β β β (Δ,⃗ ρ ⃗ 󸀠1 ) = sin (φ) sin − cos (ϑ0 ) cos , (Δ,⃗ ρ ⃗ 󸀠2 ) = sin (φ) sin + cos (ϑ0 ) cos . 2 2 2 2 n⃗ 1 = (0, − cos

168 | D Summary of Formulas for Diffraction by Plane Angular Sector In the formulas from the above table, two angles are present: β and 𝛾. As we discussed above, angle 𝛾 substitutes the angle β whenever the imaginary edge is considered. In items 1–3, angle 𝛾 is present, while in item 4 (where the real diffraction cone is considered) angle β is involved. In other words, when geometrical parameters are required in order to obtain the solution, one can use angle 𝛾. However, if the diffraction cone is in question (which really exists in space; special experiments were even conducted to detect its location), one has to deal only with real angle β. A propos, the same follow from formulas (D.5). From the imaginary edge solution we only extract the “polarization component” of the diffraction coefficient which is substituted into (D.4), while the nominator determining the singularity (“geometrical component”) remains the same as is determined by the real edge, see (D.2) and (D.3). As we discussed above, in the physical optics solution for diffraction by a plane angular sector (Appendix C) we changed the polarization component of diffraction coefficient in the integral (Appendix A) which was also obtained in the physical optics approximation. There is no contradiction here. Indeed, the solution was modified by the factors cp𝛾 and cr𝛾 for which the explicit formulas (C.16) can be obtained by comparing the diffraction coefficients of the rigorous and physical optics solutions in the case of normal incidence on an edge. This modification was only possible because we knew beforehand the relation between rigorous and physical optics coefficients in the case of normal incidence. As for the solutions (D.4), (D.6), and (D.8) presented in this appendix, they were obtained not by modification of separate parts of the physical optics solution, but by simultaneous substitution of both polarization components factored out of the physical optics integral over the edge by the polarization components of rigorous solution. In equations (D.5), the polarization components for different polarization types are separated, but this only occurs because these solution components entering equations (D.4) and (D.6) for coefficients D correspond to normal incidence on the edge. Comparison of the solution based on rigorous formulas [30] and physical optics solution (Appendix C) with the rigorous solution [16] reveals differences between these solutions, even though in the physical optics solution some components were corrected by the coefficients cp𝛾 and cr𝛾 taking into account field perturbation near the edge. In the solution based solely on the physical optics principle, field perturbation near the edge is not taken into account. Nevertheless, we do not rule out the physical optics solution for a number of reasons. Firstly, the physical optics solution helps us to clarify the differences between the rigorous solution and its physical optics counterpart. Secondly, it may well be that for some boundary conditions the physical optics solution will describe the scattered signal better. The ratios of diffraction coefficients cp𝛾 and cr𝛾 can be used in order to refine the solution for 2D diffraction coefficient on the edge, as well as 3D solution for plane angular sector.

E Fresnel Integral and its Properties Definition of Fresnel integral y

y

0

1 1 [ F (y) = ∫ exp (iξ 2 ) dξ = ∫ exp (iξ 2 ) dξ + ∫ exp (iξ 2 ) dξ ] √πi √πi −∞ 0 ] [−∞ =

y

√πi

(E.1)

1 [ + ∫ exp (iξ 2 ) dξ ] . 2 √πi 0 [ ]

Asymptotic behavior of Fresnel integral Perform integration by parts, denoting u = exp (iξ 2 ) and v = (2iξ )−1 : d d 1 = u󸀠 v; [exp (iξ 2 )] = 2iξ exp (iξ 2 ) ; exp (iξ 2 ) = [exp (iξ 2 )] dξ dξ 2iξ 1 1 −dξ d ; ( )= dξ 2iξ 2i ξ 2 ∫ exp (iξ 2 ) dξ = ∫

1 d = ∫ u󸀠 v = uv − ∫ uv󸀠 [exp (iξ 2 )] dξ 2iξ

exp (iξ 2 ) 1 dξ + ∫ exp (iξ 2 ) 2 . 2iξ 2i ξ 1 1 1 d 1 1 = [exp (iξ 2 )] ∫ exp (iξ 2 ) 2 = ∫ ∫ u󸀠 v 3 2i 2i dξ 2i ξ 2iξ =

exp (iξ 2 ) 2dξ 1 1 + (uv − ∫ uv󸀠 ) = ∫ exp (iξ 2 ) 3 . 2 2 3 2i ξ (2i) ξ (2i) 2) exp (iξ dξ 1 + ∫ exp (iξ 2 ) 2 ∫ exp (iξ 2 ) dξ = 2iξ 2i ξ =

=

exp (iξ 2 ) exp (iξ 2 ) 1 2dξ + + ∫ exp (iξ 2 ) 3 = . . . . 2iξ ξ (2i)2 ξ 3 (2i)2

As a result, we obtain an asymptotic series valid for large values of argument ∫ exp (iξ 2 ) dξ = exp (iξ 2 ) [

1 1 + ...] + 2iξ (2i)2 ξ 3

∞

= exp (iξ 2 ) ∑ n=0

n! (2i)n+1 ξ 2n+1

(E.2)

.

To obtain the power series, it is necessary to expand the exponential function in the integrand and perform integration.

170 | E Fresnel Integral and its Properties

Limiting case at the light–shadow boundary Basic formulas Consider the field scattered by a perfectly conducting wedge.

v (w0 ) ≅ ∑ m=1,2

×

ŵ sm /n ŵ sm − w0

2iP (ŵ sm ) r0 √2πikr z0 r z0 + r0 (E.3)

√S c (ŵ sm )−S c (w0 )

√ S c (ŵ sm ) − S c (w0 ) exp [iS c (ŵ sm ) − iS c (w0 )]

∫

exp (iq2 ) dq.

∞√S c (ŵ sm )−S c (w0 )

Depending on the problem geometry, we substitute into equation (E.3) one of the following (see also (1.46) and (1.51)): S c (ŵ sm ) − S c (w0 ) = k (r z0 + r0 ) − k√(r z0 + r0 )2 − 2r z0 r0 [1 + cos (φ z0 − φ0 )], S c (ŵ sm ) − S c (w0 ) = k (r z2 + h − r z1 ) , S c (ŵ sm ) − S c (w0 ) = k (r z1 + h − r z2 ) , (E.4) φ0 + π exp (i ) w s2 w s1 n − = w s1 − w0 w s2 − w0 exp (i φ0 + π ) − exp (i φ z ) n n φ0 − π π exp (i ) i sin n n − π φ z − φ0 . φ0 − π φz = exp (i ) − exp (i ) cos − cos n n n n

Scattered field near the shadow boundary In the vicinity of saddle points nφ cw − φ0 = ±π, we have S c (w sm ) − S c (w0 ) →

φ z0 − φ0 kr z0 r0 2 cos2 . r z0 + r0 2

When φ z0 → φ0 ± (π − x), where x is a small quantity, we have cos cos

x x φ z0 − φ0 → sin ≈ , 2 2 2 φ z0 − φ0 ∓π ± x = cos n n ∓π ∓x ∓π ∓x π π x = cos cos + sin sin ≅ cos + sin sin . n n n n n n n

E Fresnel Integral and its Properties |

Therefore,

171

π π i sin 1 n n n π x ≅ ix . x = π φ z − φ0 ≅ − sin sin i sin cos − cos n n n n n i sin

√

y

Since ∫∞y exp (iq2 ) dq󳨀󳨀󳨀󳨀→ ± 2πi , we obtain for the scattered field at the shadow y→0 boundary v (w0 ) ≅

1 ix

2iP (ŵ sm ) kr r x ±√πi ±P (ŵ sm ) √ 2 z0 0 = . r z0 + r0 2 2 2 r0 √2πikr z0 r z0 + r0

(E.5)

Equation (E.5) means that on the light–shadow boundary the scattered field is equal to one half of the incident geometrical optics field not only in the case of plane field incident on a half-plane, but also in the case of spherical wave incident on a wedge. This fact can be useful for construction of heuristic solutions.

F Generalized Fresnel Integral and Its Properties The Generalized Fresnel Integral (GFE) arises in many problems of mathematical physics. GFE is a special function having, unlike the ordinary Fresnel integral, two arguments, rather than one. There are several GFE formulations, and each form of GFE corresponds to a particular form of the ordinary Fresnel integral. In this appendix, formulas for the generalized Fresnel integral taken from different sources are summarized; the sources are denoted by the following acronyms: Karatygin–Rozov [58] (KR), Clemmow–Senior [55] (CS), Tischenko–Khestanov [20] (TK), and Borovikov–Kinber [43] (BK). The expressions given below summarize the formulations from the four sources: (KR), (CS), (TK), and (BK).

Fresnel integral ∞

K1 (z) = ∫ e iξ z

dξ √ξ

(KR),

∞

F (a) = e

ia2

∫ e−iλ dλ 2

(CS),

a ∞

F (p) = e

−ip2

2

∫ e iξ dξ

(TK), transformed from (CS),

p y

F (y) =

2 1 ∫ e iξ dξ √iπ

(BK).

−∞

Generalized Fresnel integral ∞

G1 (z, α) = α ∫ z

exp [i (t2 + α2 )] dt t2 + α2 ∞ 2

G (a, b) = be ia ∫ a

e−iλ dλ 2 λ + b2

∞

G (p, q) = qe−ip ∫ 2

p

(KR),

2

(CS),

2

e iξ dξ ξ 2 + q2

(TK),

∞

exp [i (ξ 2 + y2 )] dξ y G (x, y) = ∫ 2π ξ 2 + y2 x

(BK).

174 | F Generalized Fresnel Integral and Its Properties

The (KR) and (BK) formulations of Fresnel integral are related by K1 (z) = 2√πiF [− sign (z) √|z|] ,

K1 (α2 ) = 2√πi [F (− |α|)] ,

√π −iπ/4 K1 (α2 ) = π [F (− |α|)] , e 2

G1 = 2πG.

Sum with the opposite sign of the argument

G1 (−z, α) = 2G1 (0, α) − G1 (z, α) G (−a, b) = 2√πe

1 4 iπ

2

e ia F (b) − G (a, b) + δ

G (x, y) + G (−x, y) = sign yF (− |y|)

(KR), (CS), (BK).

The limiting case in the vicinity of zero argument √π −iπ/4 K1 (α2 ) e 2 1 G (0, b) = ±√πe 4 iπ F (±b) , Re b ⋛ 0 1 G (0, y) = sign yF (− |y|) 2

G 1 (0, α) =

(KR), (CS), (BK).

Sum with transposition of arguments. i G1 (z, α) + G1 (α, z) = (− ) K1 (z2 ) K1 (α2 ) 2 G (a, b) + G (b, a) = 2iF (a) F (b) + δ G (x, y) + G (y, x) = F (−x) F (−y)

(KR), (CS), (BK).

Asymptotic behavior near zero

lim

z→0,α→0

π z − arctan 2 α 1 x 1 G (x, y) → − arctan + sign y 2π y 4

G1 (z, α) =

(KR), (BK).

F Generalized Fresnel Integral and Its Properties |

175

Asymptotic formulas For the Fresnel integral used in this book: X

2

∫ exp (iq2 ) dq ≈ ∞⋅X

e ikX for |X| ≫ 1; 2iX

−|x|

F (− |x|) =

2

2

2 1 i e ikx 1 −e ikx =√ for x ≫ 1, ∫ e iξ dξ ≅ π 2 |x| √iπ √iπ 2i |x|

−∞

where the Fresnel integral is taken from (BK); Formulas for the generalized Fresnel integral: G (a, b) ≅

a2

b 3 1 󵄨 󵄨 F (a) for large 󵄨󵄨󵄨󵄨a2 + b2 󵄨󵄨󵄨󵄨 and − π < a < π 2 4 4 +b

(CS),

2

G (x, y) ≅

e iy √ i F (−x) for x, y → ∞ 2y π

G (x, y) ≅

2 2 iy e i(x +y ) , 4πx (x2 + y2 )

y≫1

x≫1,y≫1

(BK),

this formula from (BK) agrees better with the expression from (CS), rather than with the previous formula.

G Electromagnetic Wave Diffraction by Semi-transparent Plate in the Physical Optics Approximation In this appendix, theoretical derivation of the diffraction solution for semi-transparent plates in the physical optics approximation is presented. The theory relies on the reflection and transmission coefficients for plane scatterers, as well as involves the surface integral over the plane scatterer for which a rigorous expression was obtained in the previous Appendix. The formulas are based on the approach suggested in [62]. Consider the solution in the physical optics approximation for diffraction by a plane semi-transparent scatterer [32, 62]. Formulation of this problem corresponds to equations (3.1)–(3.4). The field calculated from equation (3.1) satisfies homogeneous Maxwell equations rot E⃗ = ik H,⃗

rot H⃗ = −ik E.⃗

(G.1)

In the component form, the Maxwell equations (G.1) are written as ∂E z ∂E y − = ikH x , ∂y ∂z ∂E x ∂E z − = ikH y , ∂z ∂x ∂E y ∂E x − = ikH z , ∂x ∂y

∂H z ∂H y − = −ikE x , ∂y ∂z ∂H x ∂H z − = −ikE y , ∂z ∂x ∂H y ∂H x − = −ikE z . ∂x ∂y

(G.2)

In Fig. G.1, elastic wave incidence on a plane polygonal scatterer from [62] is sketched. Let the polarization of a plane wave incident on the semi-transparent plate be such that its vector E⃗ (i) is parallel to the plane of the plate ( E⃗ (i) ⊥ YOZ). In this case, the direction vector of the wave and position vectors are, respectively, n⃗ 󸀠 = (0, sin 𝛾, cos 𝛾) and r ⃗ = (0, y, z). We then have (i)

⃗ = E0x exp {ik (z cos 𝛾 + y sin 𝛾)} , E x = E0x exp {ik (n⃗ 󸀠 , r)}

(i)

H x = 0.

(G.3)

An infinite semi-transparent screen is located in plane XOY, the normal to plate n⃗ is directed opposite to axis OZ, n⃗ 󸀠 is the direction vector of the plane wave (i. e., this vector is normal to the wave front), r ⃗ is the position vector of the observation point, i is imaginary unit, k = 2π/λ is the wave number. Electric field in the incident wave E0x is related to the source power by E0x ∼ √ P0 G/r0 , where P0 is the source power, G is the source antenna pattern, r0 is the source-to-screen distance. Components of the reflected and transmitted waves can be described by using the reflection (r e ) and transmission (t e ) coefficients, respectively: (r)

E x = r e E0x exp {ik (−z cos 𝛾 + y sin 𝛾)} ,

(G.4)

(t) Ex

(G.5)

= t e E0x exp {ik (z cos 𝛾 + y sin 𝛾)} .

178 | G Electromagnetic Wave Diffraction by Semi-transparent Plate

y n⃗ʹ

n⃗ S

γ y

Fig. G.1: To the problem of electromagnetic wave diffraction in the physical optics approximation.

Magnetic field components are determined from the Maxwell equations (G.1) and (G.2), subject to condition ∂/∂x = 0: H x = 0,

Hy =

1 ∂E x , ik ∂z

Hz = −

1 ∂E x . ik ∂y

(G.6)

It follows from equations (G.3)–(G.5) and (G.6) for the incident, reflected, and transmitted waves (i)

H y = E0x cos 𝛾 exp {ik (z cos 𝛾 + y sin 𝛾)} , (r)

H y = −r e E0x cos 𝛾 exp {ik (−z cos 𝛾 + y sin 𝛾)} , (t) Hy

(G.7)

= t e E0x cos 𝛾 exp {ik (z cos 𝛾 + y sin 𝛾)} .

As a result, the fields on the illuminated (z = −0) and shadow (z = +0) sides of the plane are (−) E x = E0x (1 + r e ) exp {iky sin 𝛾} , (−)

H y = E0x (1 − r e ) cos 𝛾 exp {iky sin 𝛾} , (−) Ey

and

=

(−) Hx

(G.8)

= 0,

(+)

E x = t e E0x exp {iky sin 𝛾} , (+)

H y = t e E0x cos 𝛾 exp {iky sin 𝛾} , (+) Ey

=

(+) Hx

(G.9)

= 0.

Integrating the incident field over the finite-dimension plate surface area, we obtain, in the physical optics approximation, the reflected field: exp {ikr} 1 A⃗ e = [n,⃗ H⃗ (−) − H⃗ (+) ] ds, ∬ 4π r S

exp {ikr} 1 A =− [n,⃗ E⃗ (−) − E⃗ (+) ] ds, ∬ 4π r ⃗m

S

r = √(x − ξ )2 + (y − η)2 + z2 .

(G.10)

G Electromagnetic Wave Diffraction by Semi-transparent Plate | 179

In equations (G.10), integration is performed over one side of the plate; the direction of normal n⃗ is shown in the figure. With reference to equations (G.8) and (G.9), we obtain A ex =

exp {ikr} 1 E0x (1 − r e − t e ) cos 𝛾 ∬ exp {ikη sin 𝛾} ds, 4π r S

exp {ikr} 1 E0x (1 + r e − t e ) ∬ exp {ikη sin 𝛾} ds, Am y = 4π r

(G.11)

S e,m Az

=

A ey

=

Am x

= 0.

Consider now the case of H-polarized wave (i)

⃗ = H0x exp {ik (z cos 𝛾 + y sin 𝛾)} , H x = H0x exp {ik (n⃗ 󸀠 , r)}

(i)

E x = 0.

(G.12)

incident on a semi-transparent plane z = 0. The reflected and transmitted plane waves can be represented in the form (r)

H x = r h H0x exp {ik (−z cos 𝛾 + y sin 𝛾)} , (t)

H x = t h H0x exp {ik (z cos 𝛾 + y sin 𝛾)} .

(G.13)

Electric field components are determined from the Maxwell equations in the following way: 1 ∂H x 1 ∂H x , Ez = . (G.14) E x = 0, E y = − ik ∂z ik ∂y The electric field components for the incident, reflected, and transmitted waves are obtained from the Maxwell equations, similar to equation (G.7): (i)

E y = −H0x cos 𝛾 exp {ik (z cos 𝛾 + y sin 𝛾)} , (r)

E y = r h H0x cos 𝛾 exp {ik (−z cos 𝛾 + y sin 𝛾)} , (t) Ey

(G.15)

= −t h H0x cos 𝛾 exp {ik (z cos 𝛾 + y sin 𝛾)} .

The fields on the illuminated (z = −0) and shadow (z = +0) sides of the plane are (−)

H x = H0x (1 + r h ) exp {iky sin 𝛾} , (−)

E y = −H0x (1 − r h ) cos 𝛾 exp {iky sin 𝛾} , (−)

(G.16)

(−)

H y = E x = 0, and

(+)

H x = t h H0x exp {iky sin 𝛾} , (+)

E y = −t h H0x cos 𝛾 exp {iky sin 𝛾} , (+)

(+)

H y = E x = 0.

(G.17)

180 | G Electromagnetic Wave Diffraction by Semi-transparent Plate

Integrating the incident field over the finite-dimension plate surface area, we obtain, in the physical optics approximation, the field reflected by the plate: A ey = −

exp {ikr} 1 H0x (1 + r h − t h ) ∬ exp {ikη sin 𝛾} ds, 4π r S

exp {ikr} 1 H0x (1 − r h − t h ) cos 𝛾 ∬ exp {ikη sin 𝛾} ds, Am x = 4π r

(G.18)

S e,m Az

=

A ex

=

Am y

= 0.

Note that the coefficients r e,h and t e,h can be either real or complex-valued, they can depend on the screen material and incident wave direction. It can be shown that in the far zone (R ≫ ka2 , where a is the maximum dimension of the plate) the condition r ≈ R − (n⃗ 󸀠󸀠 , ρ)⃗ is satisfied, in which case equations (G.11) and (G.18) take the form (with reference to expression for I from (4.3) and η sin 𝛾 = ⃗ (n⃗ 󸀠 , ρ)): 1 exp {ikR} E0x (1 − r e − t e ) cos 𝛾 ⋅ I, A ex = 4π R 1 exp {ikR} Am E0x (1 + r e − t e ) I, y = 4π R (G.19) 1 exp {ikR} e Ay = − H0x (1 + r h − t h ) I, 4π R 1 exp {ikR} Am H0x (1 − r h − t h ) cos 𝛾 ⋅ I. x = 4π R To obtain the final relationships, it is sufficient to apply the well-known transformation formulas for the physical coordinates of vector potential A ϑ = A x cos ϑ cos φ + A y cos ϑ sin φ,

A φ = −A x sin φ + A y cos φ,

(G.20)

after which the formulas for far zone fields take the form corresponding to equation (3.4).

H Generalized Diffraction Coefficient and its Application to Diffraction Problems In this appendix, we derive generalized diffraction coefficients for diffraction of a field of arbitrary physical nature by a scatterer with plane or wedge-shaped edges. The rigorous solution for the scattered field upon diffraction of a plane electromagnetic wave of unit amplitude by a 2D perfectly conducting wedge with the external apex angle πn is well known (see, e. g., [1]): sin πn 2 v(r, ψ) = − n cos π − cos n

ψ n

ψ e−ikr cos ψ cos 2 √πi

√2kr cos(ψ/2)

exp (iq2 ) dq,

∫

(H.1)

∞ cos(ψ/2)

where ψ = φ ∓ φ0 , i is the imaginary unit, k = 2π/λ is the wave number, r is the distance from the edge to the observation point, φ0 is the source angle, φ is the observation point angle. Function v (r, ψ) is continuous and periodic on the two-sheet domain. Depending on the type of boundary conditions, the scattered field E is constructed either as a difference, or as a sum: E = v(r, φ − φ0 ) − v(r, φ + φ0 )

or

E = v(r, φ − φ0 ) + v(r, φ + φ0 )

(H.2)

for TH and TE modes (having either H or E vectors normal to the edge), respectively. Taking into account the asymptotic formulas −p

∫ e

∞ iq2

2

2

dq = ∫ e iq dq ≈ −

−∞

p

e ip , 2ip

(H.3)

we obtain far away from the light–shadow boundary ψ e−ikr cos ψ cos 2 √πi

√2kr cos(ψ/2)

∫

exp (iq2 ) dq ≈

∞

e ikr , 2i√2πikr

(H.4)

and, therefore, v(r, ψ) = √

sin πn i exp (ikr) 1 2π √kr n cos π − cos n

ψ n

.

(H.5)

From equations (H.2) and (H.5), we obtain the field components (

i exp (ikr) f Ez ( ). )=√ 2π √kr g Eφ

(H.6)

182 | H Generalized Diffraction Coefficient and its Application to Diffraction Problems

Here, f and g are the diffraction coefficients of rigorous solution for TH and TE modes, respectively: f (φ, φ0 ) =

sin πn ( n cos

sin πn g (φ, φ0 ) = ( n cos

1 π n

− cos

φ−φ0 n

1 π n

− cos

φ−φ0 n

− +

1 cos

π n

− cos

φ+φ0 n

1 cos

π n

− cos

φ+φ0 n

)

(TH), (H.7)

)

(TE).

In the case of half-plane, n = 2 and, consequently, 1 1 1 − ( φ+φ0 ) 0 2 − cos φ−φ − cos n n

(TH),

1 1 1 + g (φ, φ0 ) = ( ) φ−φ 0 0 2 − cos n − cos φ+φ n

(TE).

f (φ, φ0 ) =

(H.8)

Solution to the 2D diffraction problem for a perfectly conducting half-plane obtained in the physical optics approximation far away from the shadow boundary (kr ≫ 1), where k is the wave number, is very similar to the rigorous solution: (

i exp (ikr) f 0 Ez ( 0) , )=√ 2π √kr Eφ g

(H.9)

where E z and E φ are TH and TE modes, while f 0 and g0 are the diffraction coefficients of the physical optics solution for TH and TE modes: f 0 (φ, φ0 ) =

0 0 sin φ+φ sin φ−φ 1 n n − ( ) 0 0 2 − cos φ−φ − cos φ+φ n n

0 0 sin φ+φ sin φ−φ 1 n n + g (φ, φ0 ) = ( ) 0 0 2 − cos φ−φ − cos φ+φ n n

0

(TH), (H.10) (TE).

Rigorous solution to 2D problem (H.6) and its physical optics counterpart are related by the formula from [3] which states that 2D solution in the physical optics approximation v0 (r, ψ) for the scattered field, continuous on the two-sheet Riemann surface, takes the form which is very similar to that of the rigorous solution v(r, ψ): v0 (r, ψ) = sin

ψ v (r, ψ) , 2

where ψ = φ ∓ φ0 .

(H.11)

Equation (H.11) gives relationship between the diffraction coefficients (H.8) from rigorous solution and physical optics ones for diffraction by half-plane; in particular, it describes the relationship between asymptotic forms of the physical optics and rigorous solutions at large distances from the light-shadow boundary. As for the 3D case, direct verification of relationship (H.11) is impossible because there is no rigorous solution in this case. However, we can expect that equation (H.11) holds true in the 3D case, and it is this assumption on which we base the derivation of heuristic solution.

H Generalized Diffraction Coefficient and its Application to Diffraction Problems |

183

Geometrical optics reflection and transmission coefficients As was discussed in Section 2.3.2, equation (2.40) for the field scattered by a perfectly conducting wedge can be factored into a number of multiplicands. One of these factors is the diffraction coefficient characterizing one of two boundary condition types upon incidence of an electromagnetic wave with one of two polarization types on the surface of a perfectly conducting scatterer. Orientation of the polarization vector can be considered either with respect to the normal to scatterer, or with respect to the edge. If the wave is incident normally with respect to the edge, polarizations are divided equally, while in the case of oblique incidence they are divided differently. Compare now the diffraction coefficients of rigorous (H.8) and physical optics (H.10) polarizations upon normal incidence on the edge. Each diffraction coefficient consists of two terms. Denominators of these terms are the same, they vanish on the light–shadow boundary. Therefore, this part of diffraction coefficients can be regarded as the one related to problem geometry (“geometrical” part). Nominators of these terms are different. Since the polarizations of rigorous and physical optics solutions are divided differently, the nominators can be regarded as the “polarization” part of diffraction coefficient. The nominators not only affect the field amplitude, but also determine the interaction of incident wave with the scatterer surface. The first terms of diffraction coefficients are identical and related to the shadow boundary of the incident field; the second terms are different, they are related to the shadow boundary of reflected field. It is quite natural to assume that these terms involve additional unit factors related to the surface properties. In the first chapters of this book describing the method of generalized eikonal, we considered the field scattered by a perfectly conducting wedge and half-plane on the two-sheet Riemann surface. Having obtained by MGE the known solution for diffraction by a wedge (1.57), we can compare the behavior of this solution on two-sheet domain with expressions (H.7) for diffraction coefficients. It becomes evident that the first term of the diffraction coefficient is connected with the upper half-plane, i. e., with the sheet related to the incident wave, whereas the second one is connected with the lower half-plane, i. e., with the sheet related to the reflected wave. These considerations allow us to derive a heuristic formula for generalized diffraction coefficient which takes into account non-ideal boundary conditions on the scatterer surface. To this end, we use the reflection coefficient, R, and transmission coefficient, T, describing wave interaction with an infinite plane surface. The coefficients R and T can be either constant or variable, but anyway it is much easier to derive them than to solve with non-ideal boundary conditions a 3D scattering problem, or a 2D problem of diffraction by an infinite edge upon normal incidence, or “2.5D” diffraction problem for oblique incidence on an infinite edge. In comparison with these problems, finding out R and T is a 1D problem. Expressions (H.8) are valid far away from the light–shadow boundary; they correspond to diffraction by a perfectly conducting half-plane. Consider now the case of

184 | H Generalized Diffraction Coefficient and its Application to Diffraction Problems

non-ideal boundary conditions. Let a field interact with an infinite plane surface with such boundary conditions that the incident wave be reflected from the surface with the reflection coefficient R and pass through the surface leaving it on the shadow side with the transmission coefficient T. Our purpose is to pose the same boundary condition on a half-plane and to obtain a solution which would take into account the scatterer shape. Such an approach allows us to obtain the final formulas much easier than by finding out two-dimensional solution for diffraction by a half-plane with non-ideal boundary conditions. The proposed approach to taking into account the boundary conditions is of heuristic type, it is based on the method of generalized eikonal (MGE). MGE allows one to obtain solutions not only for a wedge or half-plane, but also for two-dimensional scatterers of complex shapes. We, however, will limit ourselves to consideration of scatterers having simple shapes. According to MGE, solution to the diffraction problem for a two-dimensional perfectly conducting scatterer with edge of complex shape can be represented in the auxiliary domain. This domain consists of the upper and lower half-planes, the boundary between these corresponds to the scatterer surface. Such representation allows one to give physical interpretation to the solution. We consider the upper half-plane to be corresponding to the sheet related to incident field, the lower one corresponds to the reflected field. Solution for the scattered field is continuous and periodic on the two-sheet domain. The diffraction coefficient formula also consists of two parts which can be related to the two sheets of the auxiliary domain. We relate the first terms in the diffraction coefficients (H.8) with the incident wave sheet, while the second ones with the reflected wave sheet.

Derivation of heuristic solution Multiplying the solution part related to the scattered field and located in the lower half-plane by the reflection coefficient R we obtain the input into the scattered part of the reflected field. Multiplying the solution part related to the transmitted field and located in the upper half-plane by (1 − T), we obtain the input into the scattered part of the transmitted field. The factor (1 − T) has such a form because we subtract from the unity amplitude of the “shadow” field the field with amplitude T which passes through the scatterer and, therefore, changes the scattered field amplitude. Applying the above method, we obtain from (H.8) the generalized diffraction coefficient for semi-transparent half-plane: fg (φ, φ0 , R, T) =

1 R 1−T + ( φ+φ ) . 0 2 − cos φ−φ − cos 2 0 2

(H.12)

H Generalized Diffraction Coefficient and its Application to Diffraction Problems |

185

In the case T = 0 and R = ±1, equation (H.12) coincides with one of expressions (H.8). Thus, depending on the value of R, equation (H.12) describes both electromagnetic wave polarizations. Derive now a heuristic formula for the solution of three-dimensional diffraction problem in the case of normal incidence of a wave on an edge of a plane polygonal scatterer under the far zone conditions. For the scattered field, we obtain far away from the edge: E=

1 ka j exp (ikR0 ) exp {iΦ j } fg (φ, φ0 , R, T) . 2π kR0

(H.13)

Here, E is the scattered field, fg (φ, φ0 , R, T) is the generalized diffraction coefficient, a j is the length of j-th side of the scatterer, i is imaginary unit, R0 is the distance from the center of scatterer to the observation point, Φ j is the phase at the end point of the edge. On the diffraction cone (including the normal incidence case) the phase is the same in all points of the j-th side. Polarization of the scattered field is dependent on the physical problem specifics. As was mentioned above, equation (H.6), with (H.12) taken into account, is only valid far away from the light–shadow boundary. By analogy with (H.12), one can derive a more general formula for wedge scatterer with non-ideal boundary conditions, valid, in particular, near the light–shadow boundary: E = vK [(1 − T) , r, φ − φ0 ] + vK [R, r, φ + φ0 ] ,

(H.14)

where vK (K, r, ψ) = K

sin πn −2 n cos π − cos n

ψ n

cos

ψ e−ikr cos ψ 2 √πi (H.15)

√2kr cos(ψ/2)

×

∫

exp (iq ) dq, 2

where ψ = φ ∓ φ0 .

∞ cos(ψ/2)

Far away from the light–shadow boundary, it is easy to obtain from equations (H.14) and (H.15) expression (H.6) with diffraction coefficients (H.12): fg (φ, φ0 , R, T, n) =

sin πn 1−T ( π n cos n − cos

φ−φ0 n

+

R cos

π n

− cos

φ+φ0 n

).

(H.16)

For a half-plane, we have n = 2 and equation (H.16) is reduced to (H.12). Note that the following formula is true: fg (φ, φ0 , R, T, n) = (1 − T) fg (φ, φ0 , R1 , 0, n) ,

where R1 = R/ (1 − T) .

(H.17)

Equation (H.17) means that, from the formal point of view, heuristic formulas for a semi-transparent scatterer can be based on an appropriate model for an opaque scatterer for which T = 0.

186 | H Generalized Diffraction Coefficient and its Application to Diffraction Problems

Verification of heuristic solution For any heuristic solution, verification is of paramount importance. It can be carried out by numerical or analytical methods. One can say that availability of a good verification solution is a pre-requisite for beginning of work on derivation of heuristic formulas. As we discussed earlier, it is quite difficult to obtain analytical solutions for semi-infinite scatterers with non-ideal boundary conditions. Besides, the analytical solutions obtained not always have the form which render them suitable for verification of heuristic formulas. Therefore, an analytical solution suitable for verification purposes can only be found for some kinds of boundary conditions. For other boundary conditions, numerical verification can be performed. One can say that we first “guess” the formulas, after which we check them. Results of this check either let use state that the formulas obtained really give some solution, or they have to be modified, or a different solution must be searched. Verification can be analytical, numerical, or experimental. Depending on the purposes and resources available, it is possible to check 2D or 3D formulas. If a heuristic solution derived with GDC coincides with the verification solution, we have reached our goal and built a simple enough and computationally efficient analytical solution. However, if the heuristic solution deviates from the verification one, such a comparison will also be very useful because the discrepancy can be attributed to additional physical factors; thus, we can better understand the process physics. In order to verify equation (H.12) in the electromagnetic case, parameters were chosen according to the particular analytic solution found in the literature. In work [100], detailed verification of a solution was performed which is similar to (H.14) with functions vK (H.15), into which the reflection coefficient R for an impedance wedge and T = 0 are substituted. It was shown that results obtained by formula (H.14) agree quite well with the rigorous solution [99]. The formulas studied in [100] are somewhat more complex than (H.14) because they allow non-compensated singularities occurring at some angles for diffraction by a wedge to be avoided. In the book [1] on which the formula (H.14) is based, singularities in equation (H.1) are eliminated by taking more complex form of formula (H.2). The way in which non-compensated singularities are eliminated is a purely technical question which does not affect the essence of the solution. If we assume that the singularities have already been eliminated, one can show that expressions from [100] coincide exactly with formula (H.14) and, thus, verification performed in [100] equally applies to formula (H.14). It follows from equation (H.17) that the diffraction coefficient of semi-transparent half-plane can be expressed via that of an impedance wedge. It is fair to admit, though, that in reality these models can differ more significantly. But we will show below that expressing the boundary conditions in terms of coefficients R and T for infinite plane surface is a rightful approach, applicable to diffraction processes not only in electrodynamics, but in other areas of physics too.

H Generalized Diffraction Coefficient and its Application to Diffraction Problems | 187

Accuracy of obtained formulas and their application to 3D diffraction problems In the general case, the proposed approach is not rigorous and can be subject to systematic errors. The solution will be continuous and periodic in the two-sheet domain only in the case where R and T are constant quantities equal to unity, i. e., for a scatterer with ideal boundary conditions. Impedance boundary conditions can lead to solution discontinuity by a constant value on the illuminated surface of the scatterer, while on the shadow side this discontinuity will be variable. Therefore, we obtain quite an accurate solution on the illuminated side and on the light–shadow boundary, but on the shadow side the heuristic solution will deviate somewhat from the given boundary conditions. Nevertheless, this drawback is offset by an important advantage consisting in the simplicity of the analytical formulas obtained. The new approach can be applied to three-dimensional problems. For diffraction by a plane angular sector with ideal boundary conditions (Chapter 4), a technique was offered by which solutions obtained by the equivalent edge current method (EECM) can be refined. An intrinsic error of EECM is that field perturbation at the edge ends is not taken into account. This drawback is similar to that of the physical optics (PO) method where field perturbation at the edge of half-plane is neglected. It was shown in Chapter 4 that EECM solution can be refined significantly by introducing heuristic correction factors containing the ratio of physical optics and rigorous diffraction coefficients. Rigorous three-dimensional solutions are very scarce, they are much fewer in number than 2D solutions which, in turn, are also quite rare. Therefore, the proposed heuristic approach (and other heuristic approaches as well) is valuable because it provides 2D or even 3D analytical formulas in the problems where no rigorous solutions have yet been found. The approach is as universal as PO. Using MGE, one can additionally make the edge shape more complex by changing the diffraction coefficients (H.12), (H.16), and (H.17). Of course, one has to keep in mind that verification of heuristic formulas is necessary; in the absence of rigorous analytical solutions this can be performed numerically. If we change in formula (H.12) not only the coefficients R and T, but also the angle φ0 , we can derive heuristic formulas for a wave which has different geometrical optics light–shadow boundaries, namely, reflected in the direction different from the mirror one and, passing through the scatterer, leaves it on the shadow side in the direction not aligned with that of the incident excitation wave. Such solutions are relevant to waves of the non-electromagnetic types, e. g., elastic waves. Now, by analogy with equations (H.14) for the scattered field, we obtain relations for the total field. Let a plane wave with the phase factor U = exp (−ikS) = exp [−ik (x cos α + y sin α)]

(H.18)

188 | H Generalized Diffraction Coefficient and its Application to Diffraction Problems be incident on a perfectly conducting screen occupying the half-plane y = 0, x > 0. A solution for the total field is available in [30], see also Appendix B: U TH0 = E z = (πi)−1/2 exp (ikr) {G (u) − G (v)} , U TE0 = H z = (πi)−1/2 exp (ikr) {G (u) + G (v)} , where u = −√2kr cos

θ−α , 2

v = −√2kr cos

θ+α . 2

(H.19)

(H.20)

In these equations, G (x) = exp (−ix2 ) F (x) , ∞

(H.21) ∞

x

F (x) = ∫ exp (iq2 ) dq = ∫ exp (iq2 ) dq − ∫ exp (iq2 ) dq x

0

=

0

√πi 2

x

(H.22)

− ∫ exp (iq ) dq. 2

0

For the “straight forward” scattering, the corresponding angles are θ = π+α; θ−α = π; u = 0. For “mirror reflection” scattering, the angles are θ = π − α; θ + α = π; v = 0. The position vector of current filament in Cartesian coordinates is r 0⃗ = (x0 , y0 ) = r0 (cos α, sin α, ), that of the observation point is r ⃗ = (x, y) = r (cos θ, sin θ). Formulas (H.19) describes the total field. If θ ± α < π, then u < 0 and v < 0 (the observation point is located on the illuminated side both for transmitted and reflected waves). Following the method of generalized eikonal, consider the field on a two-sheet domain, each sheet being a half-plane. The upper half-plane, containing the source, corresponds to the incident field sheet, the lower half-plane corresponds to the reflected field sheet. Field behavior on the scatterer boundary is determined by the particular boundary conditions posed. If the boundary conditions are ideal, as in the case of a perfectly conducting half-plane, the field on the scatterer boundary is continuous, with T = 0 and R = ±1. Non-ideal boundary conditions are characterized by the coefficients T and R different from those given above. For example, if T = 0 but |R| ≠ 1, we can take into account the boundary conditions approximately, by multiplying the reflected field part by the factor R. In this case we obtain U R = (πi)−1/2 exp (ikr) {G (u) + RG (v)} .

(H.23)

If we substitute into (H.23) the reflection coefficient R = ±1, equation (H.23) will coincide with one of expressions (H.19), while the scalar function U R will describe the amplitude of total field scattered by a half-plane with ideal boundary conditions upon normal incidence of an electromagnetic wave with TH or TE polarization. The first term in the curly brackets in (H.23) describes the input of incident field, the second one is due to the input of reflected field. If, additionally, T ≠ 0, then these

H Generalized Diffraction Coefficient and its Application to Diffraction Problems | 189

two terms will be supplemented by a third one describing the input of passing wave. This input has the same shadow boundary as the incident field, but differs from it by the opposite positions of the light and shadow domains. Therefore, we take the argument of function G with the opposite sign: U TR = (πi)−1/2 exp (ikr) {G (u) + TG (−u) + RG (v)} .

(H.24)

If we consider the scattered field instead of the total one, then, in the case of diffraction by a semi-transparent half-plane, one can eliminate the geometrical optics input and obtain for the diffraction coefficients relations (H.12) which are valid for the scattered field far away from the light–shadow boundaries. For formulas (H.20) to be applicable not only to a plane wave, but also to the current filament located at a distance r0 from the edge, it is necessary to substitute rr0 . Then, instead of (H.20) we obtain the variable r in expressions for u and v by r+r 0 u = −√2k

rr0 θ−α cos , r + r0 2

v = −√2k

rr0 θ+α cos . r + r0 2

(H.25)

rr0 → r. By setting r0 equal In the case of plane wave we have r0 → ∞ and, therefore, r+r 0 to a large value, one obtains formulas for a plane wave. Compare now our generalized diffraction coefficients with the physical optics solution (Appendix G) determined by formulas (G.19) together with (3.4). An important feature of formula (G.19) in that the reflection, r, and transmission, t, coefficients enter it as a single expression multiplied by the surface integral I. Solution behavior on the light–shadow boundaries is determined by the integral I: this integral must provide for the existence of light–shadow boundaries on both sides of the scatterer plane, these boundaries must be located symmetrically with respect to the scatterer plane, and also the condition r = 1 − t must be satisfied. Only in this case expressions (G.19) give a correct result. Formulas (H.14)–(H.17), (H.24) are more general than (G.19) because coefficients R and T can take arbitrary values, and the formulas themselves are derived in the two-sheet domain and can allow for asymmetric orientation at given angles of the light–shadow boundaries of reflected and transmitted waves. This is important for derivation of heuristic formulas for complex diffraction problems, for example, for diffraction of elastic waves. If the boundary conditions are such that the geometrical optics field “cut-outs” (Fig. H.1 (b)) corresponding to the reflected and transmitted waves are directed arbitrarily, our approach allows one to refine the physical optics solution and obtain independent solutions for each above-mentioned spatial “cutouts”. In Fig. H.1 (a), the scheme of incident wave interaction with the edge of a plane semi-infinite scatterer is shown (“two-dimensional” case). This picture is similar to elastic wave interaction with plane infinite surface presented in Fig. 6.1 (“onedimensional” case). The difference between 1D and 2D scattering is that in the onedimensional case waves are scattered only in the selected spatial directions. In the

190 | H Generalized Diffraction Coefficient and its Application to Diffraction Problems

(a)

(b)

Fig. H.1: To the derivation of generalized diffraction coefficient. Scattering by half-plane with nonideal boundary conditions (a) and spatial “cut-outs” upon diffraction by a finite-size object (b).

two-dimensional case, these directions correspond to the light–shadow boundary, while scattering occurs in all directions. The approach proposed in this book and based on the use of generalized diffraction coefficients (GDC) allows the simpler (onedimensional) solution to be applied to more complex (two-dimensional) problems. In Fig. H.1 (b), the scheme of incident wave interaction with two-dimensional scatterer of finite size (strip) is shown. Upon plane wave reflection by the strip and passage through it, the geometrical optics approximation gives several “cut-outs” of the plane wave. Similar “cut-outs” can be obtained in the three-dimensional case. Each “cut-out” can be considered separately, which provides flexibility in the derivation of heuristic solutions for the given boundary conditions. Approaches described in this book allow one to derive in three-dimensional space heuristic formulas for the signal generated by each separate “cut-out”. Accuracy of the heuristic solutions is comparable with that of rigorous solutions, while simplicity of the formulas is similar to that of physical optics solutions. Let’s say a few words on the expected accuracy of the solutions based on the generalized diffraction coefficient concept. The accuracy will be determined by how accurately the conditions of boundary-value problem are satisfied (see also Section 1.1.1). Mathematical formulation of the problem includes i) wave equation; ii) initial, and iii) boundary conditions. Uniqueness of the solution also requires iv) condition at infinity, and v) condition on the edge. Many of the approaches considered in this book consist in the refinement of physical optics approximation. For diffraction by a perfectly conducting half-plane, one can prove that PO approximation satisfies all boundary-value problem conditions, except the Meixner condition on the edge. For diffraction by a perfectly conducting wedge, there are already two conditions which PO solution does not satisfy: the edge condition and boundary conditions on the shadow side of the wedge.

H Generalized Diffraction Coefficient and its Application to Diffraction Problems | 191

Solution with generalized diffraction coefficients for semi-transparent half-plane is based on the rigorous solution for a perfectly conducting half-plane. Therefore, conditions on the edge will be satisfied. As for the boundary conditions, the degree to which they are fulfilled depends on how accurately the reflection and transmission coefficients R and T set by us correspond to the real situation on the surface of the scatterer. In the formulas for GDC, we take R and T from the problem on wave interaction with a plane infinite surface and assume that they depend on the angle φ0 , but are independent of angle φ. Thus, R and T determine the geometrical optics filed on the light–shadow boundaries, whereas field behavior far away from these boundaries in the heuristic solution is the same as in the problem of diffraction by a perfectly conducting scatterer. In this case, one can expect that boundary conditions will be satisfied more accurately on the illuminated side of the scatterer, and less accurately on the shadow side. This is because the wave incident on the illuminated side of the surface originates from the surface located at the angle φ0 with respect to the edge, while the wave incident on the shadow side originates on the edge which is not available at all for plane infinite surface. One can say that insufficiently accurate account for boundary conditions on the shadow side of the scatterer is a systematic error inherent in our approach. As a matter of fact, we can take into account the presence of edge and calculate multiple reflections between the illuminated and shadow sides. Such an approach was followed in the book [44] where multiple reflections were treated in the physical optics approximation; the solution was obtained as a series converging to the rigorous solution. However, taking multiple reflections into account makes the solution quite cumbersome, while simplicity is one of the advantages of heuristic solutions. Besides, it may well be that verification of simple formulas will demonstrate their good agreement with the rigorous solution. The approach based on GDC can prove useful not only for researchers wishing to obtain accurate enough solutions by simple means. The author would like to convince specialists in numerical simulations to obtain, in addition to 2D or 3D solutions, the “one-dimensional” reflection and transmission coefficients R and T. Then, on the basis of such numerical simulations, heuristic formulas could be derived and compared with rigorous solutions, allowing for better understanding of physics of the phenomenon. Comprehensive research on the diffraction properties of a scatterer with non-ideal boundary conditions is contingent on the solution of several key problems. In addition to diffraction by a plane polygon, which is of high applied value, it is also required to obtain solution for the edge, as well as to obtain a set of solutions for a plane angular sector. This solution set would allow proper validation and tuning of heuristic analytical solution to this problem.

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Index Analytical solutions 1 Angular coordinates 48, 151, 159 Applicability limits of heuristic approaches 106 Asymptotic solutions 2, 8, 78 Auxiliary domain 16, 18, 20, 29 Binomial expansion formula 33 Boundary conditions 6, 10, 12, 16, 20, 23, 55, 87, 96, 127, 129, 132, 181 Boundary-value problem 2, 9, 11, 15, 17, 20, 23, 24, 27, 69, 112, 125, 190 Cauchy residue theorem 11, 18, 55 Coefficient – reflection 115, 121, 131, 177, 183, 191 – transmission 121, 177, 183, 191 Complexity/accuracy ratio 8 Conformal mapping 16, 23, 28, 36, 37, 42, 44, 59, 62 Contour integrals 24 Curvature 41, 43, 54, 60, 62 Curve r d0 17, 22, 25, 28, 29, 35, 39, 40, 43, 44, 48, 51, 57, 60, 66 Cylindrical wave 27, 34, 47, 77 Deterministic approach 109 Diffraction – by edges 6, 113 – by half-plate 36 – by polygons and polyhedrons 1 – by semi-transparent plate 177 – by truncated wedge 56, 67 – by vertices 6 Diffraction coefficients 35, 78, 81, 87, 90, 92, 95, 98, 101, 113, 125, 129, 130, 151 Diffraction cone 9, 10, 73, 77, 78, 81, 83, 90, 94, 98, 113, 114, 143, 157, 167 Directrix 73, 75, 76, 81, 94 Dirichlet condition 20 E and H-polarizations 54 Edge generatrix 73 EECM approximation 96, 101, 104, 163 Elastic medium 125, 127 Elastic wave diffraction 126

Elementary integration strips 92 Energy relationships 117, 121 Enforced far zone condition 89, 163 Engineering methods 5 Equivalent edge current method (EECM) 4, 81, 85 Equivalent wedge 40–42, 60, 65, 66, 70 Far zone condition 8, 76, 77, 89, 119, 126, 147 Fourier transformation 129, 131 Frequency domain 129, 130, 132 Fresnel integral 60, 150, 169 Fresnel zone 113, 117, 119 Generalized diffraction coefficient (GDC) 131, 181 Generalized eikonal function 24, 29, 34, 35 Generalized Fresnel integral 64, 173 Generalized geometrical optics function 11, 16, 30 Geometrical theory of diffraction (GTD) 4, 10, 78, 108 Half-plate of finite thickness 11, 35, 43 Hankel function 28, 76, 93 Helmholtz equation 11, 17, 23, 24, 27, 29, 31, 132 Heuristic methods 3, 4, 6, 108 Huygens–Kirchhoff principle 81 Ideal boundary conditions 7, 11, 187 Imaginary edge 12, 85, 95, 98, 103, 154, 157, 166 Incident and scattered waves 16, 85, 92, 95, 99, 154 Incident wave and observation point direction vectors 139 Incremental length diffraction coefficients (ILDC) 4, 82 Inhomogeneity in the medium 13 Integral over a closed contour 11, 22, 81 Integral representation 10, 15, 18, 64 Integrals over elementary strips 73 Key problems 6, 8

200 | Index

Laplace equation 18, 23, 29 Light-shadow boundary 8, 11, 61, 84, 106, 151, 170 Light—shadow boundary 190 Linear slip conditions 132 Linear source 15, 52 Longitudinal and transverse polarizations 125, 127, 133 Maxwell equations 94, 149, 177 Meixner’s edge condition 16 Method – of edge waves (MEW) 10, 73, 78 – of generalized diffraction coefficients 10, 129, 131, 181, 189 – of generalized eikonal (MGE) 11, 15, 35 – of geometrical optics (GO) 9, 41 – of physical optics (PO) 9, 106, 109 – of stationary phase 24 – of successive diffractions (MSD) 46, 48, 62 Mirror reflection direction 7, 9, 88, 114, 118 Modified EECM 101, 163 Multi-beam signal propagation 112 Multiple reflections 8, 191 Multiple-sheet Riemann surface 55 Mutual coupling between two antennas 115 Neumann condition 20 Non-ideal boundary conditions 4, 6, 10, 115, 183, 188 Numerical methods 1, 5, 109 Oblique incidence 77, 87, 89, 98, 104, 149, 152, 154, 183 Observation point 16, 17, 21, 27, 29, 37, 41, 48, 54, 57, 74, 82, 86, 87, 93, 95, 101, 106, 114, 126, 127, 139, 151, 157, 177, 181 Pair of combined wedges 56 Perfectly conducting angular sector 85 Phase function 7, 94, 147 Phenomenological coefficients 63, 64, 69 Physical coordinates of the vector 74 Physical optics approximation 9, 74, 81, 87, 89, 92, 106, 163 Physical optics solution for diffraction by a plane scatterer 81 Physical theory of diffraction (PTD) 4, 10, 81, 108

Plane angular sector 6, 85, 89, 95, 96, 143, 151, 158, 163 Plane wave 16, 20, 21, 27, 31 Polarization components of diffraction coefficients 12, 98, 164, 168 Polygonal line 73, 79, 81 Position vector 74, 82, 87, 128, 133, 139, 141, 145, 177 Power normalization 42, 115 Problem geometry 2, 7, 85, 95, 126, 128, 129, 144, 170, 183 Propagation mechanisms 109, 111 Radar cross section (RCS) 4, 13 Radiation condition 16, 23 Radio wave propagation in urban environment 111, 121 Ray tracing 113, 121 Rayleigh scattering theory 111 Receiver point 113, 114 Reciprocity principle 93, 147 Regularity condition 16 Relationship between the surface and contour integrals 139 Ricker wavelet 127, 129, 132 Rigorous analytical solutions 2, 6 Saddle point 18, 25, 26, 32, 47, 58, 60, 61, 170 Scatterer – perfectly conducting 15, 35, 81, 125 – plane semi-transparent 177 – polygonal 7, 83, 125, 177 – polyhedral 7, 106 – semi-infinite 11, 16, 35, 55, 59, 115, 118, 151, 153, 186, 189 Schwarz–Christoffel integral 59, 63 Size parameter 3, 10, 23, 35, 37, 39, 51, 57, 67, 70, 103 Soft input surface 4 Solutions – 1D 7, 102, 183 – 2.5D 7, 183 – 2D 7, 102, 183 – 3D 7, 102, 183 Sommerfeld integral 10, 22, 135 Source point 114 Specular reflection 18, 113, 122 Sphero-conal coordinate system 2 Stationary phase point groups 7

Index |

Statistical models 109 Stealth objects 4, 94 Stokes theorem 82, 139 Straight through (straight forward) direction 8, 9, 114, 118 Surface currents 73, 74, 87, 95, 158 Symmetrization of solution 59–61, 66 TE and TH-polarizations 98 Thick screen 56 Time domain 129, 130, 132 Total current diffraction coefficients 84 Truncated wedge 56, 63, 64, 67 Uncompensated singularities 61, 62 Urban environment 85, 109, 111, 114, 119, 122

201

Variable wave number 16, 31, 59, 69 Vector potentials 73, 74, 159 Verification 6, 62, 71, 103, 125, 132, 135, 182, 186 Vertex waves 90, 144 Vertically and horizontally polarized waves 116 Wave equation 16, 22, 35, 57, 69, 149, 190 Wave length 15, 82, 106, 116, 117, 126, 131 Wave number 15, 17, 35, 61, 69, 127 Wiener–Hopf method 2 Zone of stationary phase (ZSP) 120 Zone significant for radio wave propagation (ZSWP) 112, 120

De Gruyter Studies in Mathematical Physics

Volume 28 Carlo Rovelli (Ed.) General Relativity: The most beautiful of theories, 2015 ISBN 978-3-11-034042-6, e-ISBN 978-3-11-034330-4, e-ISBN (ePUB) 978-3-11-038364-5 Set-ISBN 978-3-11-034331-1 Volume 27 Ivan A. Lukovsky, Peter V. Malyshev Nonlinear Dynamics, 2015 ISBN 978-3-11-031655-1, e-ISBN 978-3-11-031657-5, e-ISBN (ePUB) 978-3-11-038973-9 Set-ISBN 978-3-11-031658-2 Volume 26 Vincent Laude Phononic crystals, 2015 ISBN 978-3-11-030265-3, e-ISBN 978-3-11-030266-0, e-ISBN (ePUB) 978-3-11-038791-9 Set-ISBN 978-3-11-030267-7 Volume 25 Sergey D. Algazin, Igor A. Kijko Aeroelastic vibrations and stability of plates and shells, 2015 ISBN 978-3-11-033836-2, e-ISBN 978-3-11-033837-9, e-ISBN (ePUB) 978-3-11-038945-6 Set-ISBN 978-3-11-040491-3 Volume 24 Igor Olegovich Cherednikov, Tom Mertens, Frederik F. van der Veken Wilson Lines in Quantum Field Theory, 2014 ISBN 978-3-11-030910-2, e-ISBN 978-3-11-030921-8, e-ISBN (ePUB) 978-3-11-038293-8, Set-ISBN 978-3-11-030922-5

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