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Title:
Scattering of waves by wedges and cones with impedance boundary conditions
Personal Author:
Series:
The Mario Boella series on electromagnetism in information & communication

ISMB series

Mario Boella series on electromagnetism in information & communication
Publication Information:
Edison, N.J. : Scitech Publishing, 2013
Physical Description:
xiii, 215 p. : ill. (black and white) ; 26 cm.
ISBN:
9781613530030
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30000010319041 QC665.S3 L93 2013 Open Access Book Book
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33000000009143 QC665.S3 L93 2013 Open Access Book Book
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Summary

Summary

This book is a systematic and detailed exposition of different analytical techniques used in studying two of the canonical problems, the wave scattering by wedges or cones with impedance boundary conditions. It is the first reference on novel, highly efficient analytical-numerical approaches for wave diffraction by impedance wedges or cones. This text includes calculations of the diffraction or excitation coefficients, including their uniform versions, for the diffracted waves from the edge of the wedge or from the vertex of the cone; study of the far-field behavior in diffraction by impedance wedges or cones, reflected waves, space waves from the singular points of the boundary (from edges or tips), and surface waves; and the applicability of the reported solution procedures and formulae to existing software packages designed for solving real-world high-frequency problems encountered in antenna, wave propagation, and radar cross section. This book is for researchers in wave phenomena physics, radio, optics and acoustics engineers, applied mathematicians and specialists in mathematical physics and specialists in quantum scattering of many particles.

s, radio, optics and acoustics engineers, applied mathematicians and specialists in mathematical physics and specialists in quantum scattering of many particles.


Author Notes

Mikhail A. Lyalinov is a Professor in the Department of Mathematics and Mathematical Physics at Saint Petersburg University, Russia. He has published more than 50 research papers on different mathematical aspects of diffraction theory and is co-author of two monographs. He is a principal organizer of the annual international "Days on Diffraction" seminars.
Ning Yan Zhu is a Privatdozent at the Institute of Radio Frequency Technology, University of Stuttgart, Germany. His research includes rigorous techniques and their applications to antennas and radio wave propagation in complex environments. He has published 25 journal articles and co-author one monograph in these fields. He is also an editorial advisor of the Alpha Science Series on Wave Phenomena (Oxford, UK).


Table of Contents

Prefacep. xi
Introductionp. 1
General and historical remarksp. 1
Description of the contentp. 3
1 Fundamentalsp. 5
1.1 Equations for acoustic and electromagnetic wavesp. 5
1.1.1 Acoustic wavesp. 5
1.1.2 Electromagnetic wavesp. 7
1.2 Boundary conditionsp. 9
1.3 Edge and radiation conditionsp. 11
1.3.1 Vicinity of the edge and Meixner's conditionp. 11
1.3.2 On the behavior of solutions to the Helmholtz equation in the angular domain as r → 0p. 12
1.3.3 Radiation conditions: Formulation of the problemp. 13
1.3.4 The limiting-absorption principlep. 14
1.4 Integral transformationsp. 16
1.4.1 Fourier transform and the convolution theoremp. 16
1.4.2 The Sommerfeld integralp. 17
1.4.3 Malyuzhinets's theorem: Sommerfeld-Malyuzhinets (SM) transformp. 18
1.4.4 Kontorovich-Lebedev (KL) transform and its connection with the Sommerfeld integralp. 22
1.4.5 Watson-Bessel integralp. 24
1.5 Malyuzhinets's solution for the impedance wedge diffraction problemp. 26
1.5.1 Functional equations for the Malyuzhinets problemp. 26
1.5.2 The multiplication principle and the auxiliary solution ¿ 0 (z) to the functional equations (1.104)p. 28
1.5.3 The Malyuzhinets function ¿ ¿ (z)and its basic propertiesp. 30
1.5.4 Examination of (d/dz) In ¿ ¿ (z)p. 30
1.5.5 The Malyuzhinets function ¿ 0 (z)p. 32
1.5.6 Completion of the construction of ¿ 0 (z) and of s(z)p. 34
1.5.7 Far-field analysis of the exact solutionp. 35
1.6 Theory of Malyuzhinets functional equations for one unknown functionp. 38
1.6.1 General Malyuzhinets equationsp. 38
1.6.2 Solution to the homogeneous Malyuzhinets equationsp. 39
1.6.3 Solution to the inhomogeneous Malyuzhinets equationsp. 40
1.6.4 Modified Fourier transform and S-integralsp. 41
1.6.5 The direct application of S-integralsp. 42
2 Diffraction of a skew-incident plane electromagnetic wave by a wedge with axially anisotropic impedance facesp. 45
2.1 Introductionp. 45
2.2 Statement of the problem and uniquenessp. 46
2.2.1 Statement of the problemp. 46
2.2.2 On uniqueness of a solutionp. 48
2.3 Sommerfeld integral and functional equationsp. 51
2.4 A functional difference equation of higher orderp. 53
2.4.1 A difference equation for one spectrump. 53
2.4.2 The generalized Malyuzhinets function X¿(¿)p. 54
2.4.3 Simplifying the functional difference equation of higher orderp. 55
2.5 Second-order functional difference equation and Fredholm integral equation of the second kindp. 56
2.5.1 An integral equivalent to the difference equationp. 56
2.5.2 Determining the constants C 1l ±p. 57
2.5.3 Fredholm integral equation of the second kindp. 58
2.6 Uniform asymptotic solutionp. 58
2.6.1 Poles and residuesp. 58
2.6.2 First-order uniform asymptoticsp. 60
2.7 Numerical resultsp. 62
2.7.1 Numerical computation of the spectrap. 62
2.7.2 Examplesp. 63
2.8 Appendix: Computation of the generalized Malyuzhinets functionp. 64
2.8.1 Numerical integrationp. 64
2.8.2 Series representationp. 67
3 Scattering of waves from an electric dipole over an impedance wedgep. 69
3.1 Formulation of the problem and plane-wave expansion of the incident fieldp. 69
3.1.1 Statement of the problemp. 69
3.1.2 The Hertz vector and plane-wave expansion of the incident fieldp. 71
3.2 The integral representation of the total fieldp. 74
3.2.1 Integral formulationp. 74
3.2.2 Formulation of the problem for U(r, ¿, ¿, ß)p. 74
3.2.3 Representation for the spectral functionsp. 76
3.3 Deformation of the contours of integration and the geometrical-optics (GO) fieldp. 77
3.3.1 Saddle points, polar singularities, and residuesp. 77
3.3.2 Branch cuts for auxiliary anglesp. 79
3.3.3 The geometrical-optics fieldp. 80
3.4 The diffracted wave from the edge of the wedgep. 82
3.4.1 Nonuniform expressionp. 82
3.4.2 The UAT formulationp. 83
3.5 Expressions for surface wavesp. 85
3.5.1 Surface waves excited directly by the dipolep. 85
3.5.2 Surface waves excited at the edge by an incident space wavep. 86
3.6 Numerical resultsp. 87
3.7 Appendicesp. 89
3.7.1 Appendix A. Multidimensional saddle-point methodp. 89
3.7.2 Appendix B. The reciprocity principlep. 92
4 Diffraction of a TM surface wave by an angular break of an impedance sheetp. 95
4.1 Formulation of the problemp. 95
4.2 Functional equations and reduction to integral equationsp. 97
4.2.1 Reduction to a second-order functional equationp. 98
4.2.2 An integral equation of the second kindp. 100
4.3 Analytic continuation of the spectral functions and scattering diagramp. 101
4.3.1 Scattering diagramp. 102
4.3.2 Reflected and transmitted surface wavesp. 103
4.4 Discussion of uniquenessp. 104
5 Acoustic scattering of a plane wave by a circular impedance conep. 109
5.1 Formulation of the problem and uniquenessp. 109
5.1.1 Formulation of the problemp. 109
5.1.2 On uniqueness of the classical solutionp. 111
5.2 Kontorovich-Lebedev (KL) transform and incomplete separation of variablesp. 113
5.2.1 Integral representation of the solutionp. 113
5.2.2 Formulation of the problem for the spectral function u vp. 114
5.3 The boundary value problem for the spectral function u v (¿, ¿ 0 )p. 117
5.3.1 Separation of the angular variables for the circular conep. 119
5.3.2 Study of the integral equation for R(v, n)p. 120
5.4 Diffraction coefficient in the oasis M for a narrow conep. 122
5.4.1 Problems for the leading terms and for the first correctionsp. 124
5.4.2 Calculation of V 1 and B 2jp. 126
5.4.3 Basic formula for the diffraction coefficient of the spherical wave from the vertex of a narrow conep. 128
5.5 Numerical calculation of the diffraction coefficient in the oasis Mp. 130
5.5.1 Numerical aspectsp. 130
5.5.2 A perturbation series for |¿| " 1p. 131
5.5.3 Examplesp. 132
5.6 Sommerfeld-Malyuzhinets transform and analytic continuationp. 133
5.6.1 Analytic properties of ¿(¿,¿, ¿ 0 ) and ¿(¿,¿, ¿ 0 )p. 135
5.6.2 Problems for the Sommerfeld transformantsp. 136
5.6.3 The singularity corresponding to the wave reflected from the conical surfacep. 137
5.7 The reflected wavep. 139
5.8 Scattering diagram of the spherical wave from the vertexp. 140
5.9 Surface wave at axial incidencep. 142
5.9.1 Ray solution for the surface wavep. 142
5.9.2 Singularities of the Sommerfeld transformanis corresponding to the surface wavep. 143
5.9.3 Asymptotic evaluation of the surface wavep. 145
5.10 Uniform asymptotics of the far field and the parabolic cylinder functionsp. 147
5.11 Appendicesp. 150
5.11.1 Appendix Ap. 150
5.11.2 Appendix B. Reduction of integralsp. 151
5.11.3 Appendix C. Derivation of the constant C 0p. 153
6 Electromagnetic wave scattering by a circular impedance conep. 155
6.1 Formulation and reduction to the problem for the Debye potentialsp. 155
6.1.1 The far-field patternp. 158
6.1.2 The Debye potentialsp. 159
6.1.3 Boundary conditions for the Debye potentialsp. 161
6.2 Kontorovich-Lebedev (KL) integrals and spectral functionsp. 161
6.2.1 KL integral representationsp. 161
6.2.2 Properties of the spectral functionsp. 163
6.2.3 Boundary conditions for the spectral functionsp. 164
6.2.4 Verification of the boundary and other conditionsp. 166
6.2.5 Diffraction coefficientsp. 168
6.3 Separation of angular variables and reduction to functional-difference (FD) equationsp. 170
6.4 Fredholm integral equations for the Fourier coefficientsp. 172
6.4.1 Reduction to integral equationsp. 172
6.4.2 Comments on the Fredholm property and unique solvability of the integral equationsp. 175
6.5 Electromagnetic diffraction coefficients in M' and numerical resultsp. 175
6.5.1 Numerical Solutionp. 176
6.5.2 Numerical examplesp. 177
6.6 Sommerfeld and Watson-Bessel (WB) integral representationsp. 179
6.6.1 Sommerfeld integral representationsp. 180
6.6.2 Regularity domains for the Sommerfeld transformantsp. 182
6.6.3 Diffraction coefficients and Sommerefeld transformantsp. 183
6.7 The diffraction coefficients outside the oasis as ¿ ∈ M''p. 185
6.8 Problems for the Sommerfeld transformants and some complex singularitiesp. 187
6.8.1 Problems for the Sommerfeld transformantsp. 187
6.8.2 Local behavior of the Sommerfeld transformants near complex singularitiesp. 189
6.9 Asymptotics of the Sommerfeld integrals and the electromagnetic surface wavesp. 191
6.9.1 Derivation of the functionals C 0u (n)p. 193
6.9.2 Some comments on the asymptotics uniform with respect to the direction of observationp. 195
7 Epiloguep. 197
Referencesp. 199
Indexp. 213