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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
Preface | p. xi |
Introduction | p. 1 |
General and historical remarks | p. 1 |
Description of the content | p. 3 |
1 Fundamentals | p. 5 |
1.1 Equations for acoustic and electromagnetic waves | p. 5 |
1.1.1 Acoustic waves | p. 5 |
1.1.2 Electromagnetic waves | p. 7 |
1.2 Boundary conditions | p. 9 |
1.3 Edge and radiation conditions | p. 11 |
1.3.1 Vicinity of the edge and Meixner's condition | p. 11 |
1.3.2 On the behavior of solutions to the Helmholtz equation in the angular domain as r → 0 | p. 12 |
1.3.3 Radiation conditions: Formulation of the problem | p. 13 |
1.3.4 The limiting-absorption principle | p. 14 |
1.4 Integral transformations | p. 16 |
1.4.1 Fourier transform and the convolution theorem | p. 16 |
1.4.2 The Sommerfeld integral | p. 17 |
1.4.3 Malyuzhinets's theorem: Sommerfeld-Malyuzhinets (SM) transform | p. 18 |
1.4.4 Kontorovich-Lebedev (KL) transform and its connection with the Sommerfeld integral | p. 22 |
1.4.5 Watson-Bessel integral | p. 24 |
1.5 Malyuzhinets's solution for the impedance wedge diffraction problem | p. 26 |
1.5.1 Functional equations for the Malyuzhinets problem | p. 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 properties | p. 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 solution | p. 35 |
1.6 Theory of Malyuzhinets functional equations for one unknown function | p. 38 |
1.6.1 General Malyuzhinets equations | p. 38 |
1.6.2 Solution to the homogeneous Malyuzhinets equations | p. 39 |
1.6.3 Solution to the inhomogeneous Malyuzhinets equations | p. 40 |
1.6.4 Modified Fourier transform and S-integrals | p. 41 |
1.6.5 The direct application of S-integrals | p. 42 |
2 Diffraction of a skew-incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces | p. 45 |
2.1 Introduction | p. 45 |
2.2 Statement of the problem and uniqueness | p. 46 |
2.2.1 Statement of the problem | p. 46 |
2.2.2 On uniqueness of a solution | p. 48 |
2.3 Sommerfeld integral and functional equations | p. 51 |
2.4 A functional difference equation of higher order | p. 53 |
2.4.1 A difference equation for one spectrum | p. 53 |
2.4.2 The generalized Malyuzhinets function X¿(¿) | p. 54 |
2.4.3 Simplifying the functional difference equation of higher order | p. 55 |
2.5 Second-order functional difference equation and Fredholm integral equation of the second kind | p. 56 |
2.5.1 An integral equivalent to the difference equation | p. 56 |
2.5.2 Determining the constants C 1l ± | p. 57 |
2.5.3 Fredholm integral equation of the second kind | p. 58 |
2.6 Uniform asymptotic solution | p. 58 |
2.6.1 Poles and residues | p. 58 |
2.6.2 First-order uniform asymptotics | p. 60 |
2.7 Numerical results | p. 62 |
2.7.1 Numerical computation of the spectra | p. 62 |
2.7.2 Examples | p. 63 |
2.8 Appendix: Computation of the generalized Malyuzhinets function | p. 64 |
2.8.1 Numerical integration | p. 64 |
2.8.2 Series representation | p. 67 |
3 Scattering of waves from an electric dipole over an impedance wedge | p. 69 |
3.1 Formulation of the problem and plane-wave expansion of the incident field | p. 69 |
3.1.1 Statement of the problem | p. 69 |
3.1.2 The Hertz vector and plane-wave expansion of the incident field | p. 71 |
3.2 The integral representation of the total field | p. 74 |
3.2.1 Integral formulation | p. 74 |
3.2.2 Formulation of the problem for U(r, ¿, ¿, ß) | p. 74 |
3.2.3 Representation for the spectral functions | p. 76 |
3.3 Deformation of the contours of integration and the geometrical-optics (GO) field | p. 77 |
3.3.1 Saddle points, polar singularities, and residues | p. 77 |
3.3.2 Branch cuts for auxiliary angles | p. 79 |
3.3.3 The geometrical-optics field | p. 80 |
3.4 The diffracted wave from the edge of the wedge | p. 82 |
3.4.1 Nonuniform expression | p. 82 |
3.4.2 The UAT formulation | p. 83 |
3.5 Expressions for surface waves | p. 85 |
3.5.1 Surface waves excited directly by the dipole | p. 85 |
3.5.2 Surface waves excited at the edge by an incident space wave | p. 86 |
3.6 Numerical results | p. 87 |
3.7 Appendices | p. 89 |
3.7.1 Appendix A. Multidimensional saddle-point method | p. 89 |
3.7.2 Appendix B. The reciprocity principle | p. 92 |
4 Diffraction of a TM surface wave by an angular break of an impedance sheet | p. 95 |
4.1 Formulation of the problem | p. 95 |
4.2 Functional equations and reduction to integral equations | p. 97 |
4.2.1 Reduction to a second-order functional equation | p. 98 |
4.2.2 An integral equation of the second kind | p. 100 |
4.3 Analytic continuation of the spectral functions and scattering diagram | p. 101 |
4.3.1 Scattering diagram | p. 102 |
4.3.2 Reflected and transmitted surface waves | p. 103 |
4.4 Discussion of uniqueness | p. 104 |
5 Acoustic scattering of a plane wave by a circular impedance cone | p. 109 |
5.1 Formulation of the problem and uniqueness | p. 109 |
5.1.1 Formulation of the problem | p. 109 |
5.1.2 On uniqueness of the classical solution | p. 111 |
5.2 Kontorovich-Lebedev (KL) transform and incomplete separation of variables | p. 113 |
5.2.1 Integral representation of the solution | p. 113 |
5.2.2 Formulation of the problem for the spectral function u v | p. 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 cone | p. 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 cone | p. 122 |
5.4.1 Problems for the leading terms and for the first corrections | p. 124 |
5.4.2 Calculation of V 1 and B 2j | p. 126 |
5.4.3 Basic formula for the diffraction coefficient of the spherical wave from the vertex of a narrow cone | p. 128 |
5.5 Numerical calculation of the diffraction coefficient in the oasis M | p. 130 |
5.5.1 Numerical aspects | p. 130 |
5.5.2 A perturbation series for |¿| " 1 | p. 131 |
5.5.3 Examples | p. 132 |
5.6 Sommerfeld-Malyuzhinets transform and analytic continuation | p. 133 |
5.6.1 Analytic properties of ¿(¿,¿, ¿ 0 ) and ¿(¿,¿, ¿ 0 ) | p. 135 |
5.6.2 Problems for the Sommerfeld transformants | p. 136 |
5.6.3 The singularity corresponding to the wave reflected from the conical surface | p. 137 |
5.7 The reflected wave | p. 139 |
5.8 Scattering diagram of the spherical wave from the vertex | p. 140 |
5.9 Surface wave at axial incidence | p. 142 |
5.9.1 Ray solution for the surface wave | p. 142 |
5.9.2 Singularities of the Sommerfeld transformanis corresponding to the surface wave | p. 143 |
5.9.3 Asymptotic evaluation of the surface wave | p. 145 |
5.10 Uniform asymptotics of the far field and the parabolic cylinder functions | p. 147 |
5.11 Appendices | p. 150 |
5.11.1 Appendix A | p. 150 |
5.11.2 Appendix B. Reduction of integrals | p. 151 |
5.11.3 Appendix C. Derivation of the constant C 0 | p. 153 |
6 Electromagnetic wave scattering by a circular impedance cone | p. 155 |
6.1 Formulation and reduction to the problem for the Debye potentials | p. 155 |
6.1.1 The far-field pattern | p. 158 |
6.1.2 The Debye potentials | p. 159 |
6.1.3 Boundary conditions for the Debye potentials | p. 161 |
6.2 Kontorovich-Lebedev (KL) integrals and spectral functions | p. 161 |
6.2.1 KL integral representations | p. 161 |
6.2.2 Properties of the spectral functions | p. 163 |
6.2.3 Boundary conditions for the spectral functions | p. 164 |
6.2.4 Verification of the boundary and other conditions | p. 166 |
6.2.5 Diffraction coefficients | p. 168 |
6.3 Separation of angular variables and reduction to functional-difference (FD) equations | p. 170 |
6.4 Fredholm integral equations for the Fourier coefficients | p. 172 |
6.4.1 Reduction to integral equations | p. 172 |
6.4.2 Comments on the Fredholm property and unique solvability of the integral equations | p. 175 |
6.5 Electromagnetic diffraction coefficients in M' and numerical results | p. 175 |
6.5.1 Numerical Solution | p. 176 |
6.5.2 Numerical examples | p. 177 |
6.6 Sommerfeld and Watson-Bessel (WB) integral representations | p. 179 |
6.6.1 Sommerfeld integral representations | p. 180 |
6.6.2 Regularity domains for the Sommerfeld transformants | p. 182 |
6.6.3 Diffraction coefficients and Sommerefeld transformants | p. 183 |
6.7 The diffraction coefficients outside the oasis as ¿ ∈ M'' | p. 185 |
6.8 Problems for the Sommerfeld transformants and some complex singularities | p. 187 |
6.8.1 Problems for the Sommerfeld transformants | p. 187 |
6.8.2 Local behavior of the Sommerfeld transformants near complex singularities | p. 189 |
6.9 Asymptotics of the Sommerfeld integrals and the electromagnetic surface waves | p. 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 observation | p. 195 |
7 Epilogue | p. 197 |
References | p. 199 |
Index | p. 213 |