139,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The book has a dual purpose. The first is to expose a general methodology to solve problems of electromagnetism in geometries constituted of angular regions. The second is to bring the solutions of some canonical problems of fundamental importance in modern electromagnetic engineering with the use of the Wiener-Hopf technique. In particular, the general mathematical methodology is very ingenious and original. It is based on sophisticated and attractive procedures exploiting simple and advanced properties of analytical functions. Once the reader has acquired the methodology, she/he can easily obtain the solution of the canonical problems reported in the book. The book can be appealing also to readers who are not directly interested in the detailed mathematical methodology and/ or in electromagnetics. In fact the same methodology can be extended to acoustics and elasticity problems. Moreover, the proposed practical problems with their solutions constitute a list of reference solutions and can be of interests in engineering production in the field of radio propagations, electromagnetic compatibility and radar technologies.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 212
Veröffentlichungsjahr: 2020
Ähnliche
Table of Contents
Cover
Title page
Copyright
Preface
Introduction
4 Exact Solutions for Electromagnetic Impedance Wedges
4.1. Introduction
4.2. A list of the impedance wedge problems amenable to exact WH solutions
4.3. Cases involving classical WH equations
4.4. Exact solutions for impedance wedge problems with the GWHE form of section 3.5 – form #1
4.5. Exact solutions for the impedance wedge problems with the GWHEs written in an alternative form – form #2
4.6. A general form of the GWHEs to study the arbitrary face impedance wedges – form #3
Appendix 4.A. Some important formulas of decomposition for wedge problems
5 Fredholm Factorization Solutions of GWHEs for the Electromagnetic Impedance Wedges Surrounded by an Isotropic Medium
5.1. Introduction
5.2. Generalized Wiener-Hopf equations for the impenetrable wedge scattering problem of an electromagnetic plane wave at skew incidence
5.3. Fredholm factorization solution in the η plane of GWHEs
5.4. Fredholm factorization solution in the w plane of GWHEs
5.5. Approximate solution of FIEs derived from GWHEs
5.6. Analytic continuation of approximate solutions of GWHEs
5.7. Far-field computation
5.8. Criteria for the examples
5.9. Example 1: Symmetric isotropic impedance wedge at normal incidence with Ez polarization
5.10. Example 2: Non-symmetric isotropic impedance wedge at normal incidence with Hz polarization and surface wave contribution
5.11. Example 3: PEC wedge at skew incidence
5.12. Example 4: Arbitrary impedance half-plane at skew incidence
5.13. Example 5: Arbitrary impedance wedge at skew incidence
5.14. Example 6: Arbitrary impedance concave wedge at skew incidence
5.15. Discussion
Appendix 5.A. Fredholm properties of the integral equation (5.3.1)
6 Diffraction by Penetrable Wedges
6.1. Introduction
6.2. GWHEs for the dielectric wedge at normal incidence (Ez-polarization)
6.3. Reduction of the GWHEs for the dielectric wedge at Ez-polarization to Fredholm integral equations
6.4. Analytic continuation for the solution of the dielectric wedge at Ez-polarization
6.5. Some remarks on the Fredholm integral equations (6.3.24), (6.3.26) and numerical solutions
6.6. Field evaluation in any point of the space
6.7. The dielectric wedge at skew incidence
6.8. Criteria for examples of the scattering by a dielectric wedge at normal incidence (Ez-polarization)
6.9. Example: the scattering by a dielectric wedge at normal incidence (Ez-polarization)
6.10. Discussion
Appendix 6.A. Fredholm factorization applied to (6.3.2)–(6.3.5)
Appendix 6.B. Source term η
References
Index
Summary of Volume 1
End User License Agreement
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Introduction
Begin Reading
References
Index
Summary of Volume 1
End User License Agreement
Pages
v
ii
iii
iv
ix
x
xi
xiii
xiv
xv
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
205
206
207
209
210
211
213
214
215
216
217
218
219
To my beloved grandchildren Marianna and Roberto V.G.D.
To my family Stefania, Marina and Alessandro for their patience and support, this book is affectionately dedicated G.L.
Waves and Scattering Set
coordinated by
Jean-Michel L. Bernard
Volume 2
Scattering and Diffraction by Wedges 2
The Wiener-Hopf Solution – Advanced Applications
Vito G. Daniele
Guido Lombardi
First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2020
The rights of Vito G. Daniele and Guido Lombardi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2020936444
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-664-7
Preface
The theory of wave diffraction by wedges constitutes one of the fundamental problems in mathematical physics. Beyond the obvious applications to engineering and physics, this topic has generated – and continues to generate – new progress in the techniques of applied mathematics (e.g. the Sommerfeld–Malyuzhinets technique and the Kontorovich–Lebedev method), and applied physics (e.g. the geometrical theory of diffraction and the physical theory of diffraction).
The scattering of wedge structures is of interest in the study of electromagnetic response/interaction of more complex, multiscale computational electromagnetic problems, using diffraction coefficients for modeling subdomains where high-frequency methods hold promise.
This book, which is divided into two volumes, presents a general flexible, novel and powerful methodology for solving problems of electromagnetism in regions containing arbitrary wedge objects; this technique can be extended, as suggested, to different physics.
The proposed technique is based on a generalization of the Wiener-Hopf (WH) technique that allows the study of complex canonical scattering problems.
Research carried out during the past two decades has ordered and systematized the procedure, to obtain spectral equations and integral representations for complex problems to avoid redundancy.
The purpose of this book is to present the application of the so-called generalized Wiener-Hopf technique (GWHT) to wedge scattering problems. This method often adopts very sophisticated mathematical methods; however, since the aim of the authors is to favor the acquisition of fundamental concepts in view of possible applications, they prefer to omit rigorous mathematical proofs for theory such as theorems. For all unproven mathematics, we refer to papers and books that are easily available.
Due to the electrical engineering background of the authors, the two volumes are focused on electromagnetic applications. However all of the material is presented in such a way that extension to other physics and applied mathematical fields is straightforward; from applications in all areas of diffraction problems, such as acoustics, elasticity, aerodynamics, hydrodynamics and so on.
Analogies between some resolved problems in electromagnetics and other areas of physics are reported. In particular, for example, the diffraction of a half-infinite crack in an elastic solid medium is described in detail.
Before dealing with the applications to wedge problems, we present some useful remarks about the WH technique. First, the classical and consolidated method of solution of WH equations is founded on a deep knowledge of the properties of complex functions, which is based on the following classical steps: the multiplicative factorization of the kernel, the additive factorization (decomposition) of functions and the application of Liouville’s theorem.
Second, in general, no WH problem is simple to study and solve. The formulation can be cumbersome to obtain and very specialized techniques are needed to implement the factorization procedure. Moreover, these methods frequently do not allow small variations of the problem itself. In this context, it is fundamental to adopt spectral domains using spatial Laplace transforms where all the unknown field representations are analytic functions.
Furthermore, even though formal solutions can be obtained in particular cases, a long and difficult elaboration may be required to make them effective from the physical and engineering points of view. For this purpose, tools such as the saddle point method are necessary for the asymptotic evaluation of integrals.
In this book, we aim to solve these problems systematically by helping the reader to implement solutions using the WH method with precise, simple steps and reducing the restrictions on generalization.
In particular, we present one of the most relevant recent advances in the WH technique, i.e. the transformation of WH equations into integral equations for their effective solution via simple discretization.
This method, known as the Fredholm factorization, is an approximate technique that extends the applicability of the WH method to a new variety of problems, consisting of different geometries and materials.
The Fredholm factorization is a semi-analytical solution method that preserves the spectral properties of the problem through its structural singularities and singularities related to the sources. Approximations are present on correction terms that are only related to diffractive properties. Thus, using a suitable analytic continuation, the method allows the reader to look at the physical properties of the problems, as done in analytical closed-form solution methods.
The success of the Fredholm factorization, in solving complex diffraction problems involving isotropic media, has also encouraged the authors to extend the WH technique developed in this book to wedge problems involving anisotropic or, more general, bi-anisotropic media. Moreover, in these cases, the Fredholm factorization is conceptually possible. However, in most cases, it yields complex kernels that involve cumbersome multivalued functions. Much effort is needed to complete the general WH theory for solving wedge problems involving arbitrary linear media. In particular, difficulties increase considerably when it is not possible to take advantage of the presence of Helmholtz equations. In fact, in these cases, it is very difficult and cumbersome to implement the indispensable analytic continuation procedures.
However, to successfully develop the mathematical theory and applications of this book, it is useful to have a technical computing software, particularly in analytical and numerical manipulations of the equations.
Finally, we admit that the authors are strongly influenced by the fundamental work of Felsen and Marcuvitz on radiation and scattering by waves, and therefore the adopted notations follow their studies. In particular, equivalent network representations of equations are provided to systemize the mathematical procedure. The use of different notations in applied physics, applied mathematics and engineering communities should not present a major difficulty in reading this book.
Vito G. DANIELE
Guido LOMBARDI
May 2020
This book consists of two volumes; any references to Chapters 1, 2 or 3 can be found in the first volume: Scattering and Diffraction by Wedges 1.
Introduction
Diffraction by objects presenting edges is a fundamental problem in the theory of diffraction. Its importance has been demonstrated by the vast number of papers and books that deal with this topic. These works concern mathematical and physical aspects, with applications to many branches of physics and engineering such as electromagnetism, acoustics, elasticity, aerodynamics and hydrodynamics. To provide a complete theory of the effects of the edges present in a body, we study wedge problems, i.e. canonical problems where the diffracting body is composed of homogeneous wedges. Wedge problems can be classified in different ways. For example, in electromagnetism, we introduce impenetrable and penetrable wedges. Impenetrable wedges arise from the introduction of approximate boundary conditions on the surfaces of wedges. For example, near a metallic face, at microwave frequencies, we can assume that the tangential electrical field vanishes. The introduction of impenetrable wedges simplifies the study of wedge problems considerably since the external problem (i.e. the evaluation of the electromagnetic field outside the wedge) is decoupled from the internal problem (i.e. the evaluation of the electromagnetic field inside the wedge). Considerable advances have been made to obtain more realistic approximate boundary conditions. Unfortunately, the introduction of sophisticated boundary conditions is not sufficient to study penetrable wedges. In particular, the scattering by a penetrable wedge (e.g. the dielectric wedge in electromagnetics) is a very difficult problem that has arisen for a long time and drawn the interest of many scientists.
Nevertheless, even for the simplest cases, no closed-form solution for this problem is known.
The solutions of wedge problems have a long history, which have been well documented by numerous books and papers. This history also documents the importance of the use of spectral representations to obtain the solutions of wedge problems. One of the aims of these two volumes is to claim the power of the representations of the field spectra through Laplace transforms in radial direction. In particular, the use of these representations allows the extension of the powerful WH technique to formulate and efficiently solve many wedge problems.
This book is divided into two volumes that are consistent with each other. The first volume introduces the general theory with simple examples, while the second volume reports more advanced applications. Each volume consists of three chapters and their cross-references.
In each volume, sections, equations and figures are numbered as (chapter.section. item) in order for the content to be easily accessible.
Chapter 1 starts by recalling some fundamental properties of the mathematical theory of the WH technique. The separation of the plus and minus Laplace transforms is fundamental in introducing the concept of the classical factorization, an ingenious idea forming the basis of this technique. This chapter presents a few cases where exact factorizations are available. A novel aspect presented in this chapter is the introduction of the solution procedure for the Fredholm factorization, as an alternative to the classical WH solution procedure. It presents the fundamental advantage of its applicability in general cases where the classical factorizations are not available.
To discuss the importance of the mathematical tools described in Chapter 1, the solution of the most simple diffraction problem is presented in Chapter 2: the electromagnetic scattering of a plane wave by a perfectly electrically conducting (PEC) half-plane. Both the classical factorization procedure and the Fredholm factorization method are applied to obtain the solution of this problem. The comparison of these two different factorization techniques illustrates their advantages and disadvantages. To provide a general methodology for studying basic problems in other diffraction areas, Chapter 2 presents the complete theory of the diffraction of a half-infinite crack located in an indefinite elastic solid.
Chapter 3 deals with the application of the WH technique to wedge problems. This application comes from the introduction of generalized Wiener-Hopf equations (GWHEs) and from the mapping that reduces GWHEs to classical WH equations (CWHEs).
In order to compare the WH technique with the Sommerfeld–Malyuzhinets technique, this chapter introduces two angular complex planes called w and bar w. In this framework, we present a new tool called rotating waves to study angular regions. Moreover, the introduction of angular complex planes is also useful in developing the application of the Fredholm factorization method.
Chapter 4 (Volume 2) deals with the solution of impenetrable wedges immersed in free space. Exact solutions obtained with the classical factorization theory are compared with those obtained by the Sommerfeld–Malyuzhinets method.
Chapter 5 (Volume 2) is particularly important for the development of immediate engineering applications. In fact, it presents the application of the Fredholm factorization method to the solution of the general scattering problem by an arbitrary impedance wedge illuminated by plane waves at skew incidence.
Finally, Chapter 6 (Volume 2) deals with the use of the WH technique to solve an arbitrary homogeneous penetrable wedge immersed in free space. In particular, this chapter studies, in depth, the diffraction by a dielectric wedge illuminated by plane waves at normal and skew incidence. The WH solution of the dielectric wedge problem presented in this chapter is the first one that the authors of this book have developed in the literature. It contains very sophisticated mathematical methods to overcome difficult analytical problems related to the presence of two media. Recently, many improvements have been made to the WH method to study the diffraction of complex problems consisting of multiple wedges and layers (see 2016–2019 papers by the authors of this book). We expect that these improvements will allow a more simple mathematical procedure to solve the diffraction by a dielectric wedge.
This book consists of two volumes; any references to Chapters 1, 2 or 3 can be found in the first volume: Scattering and Diffraction by Wedges 1.
