The interaction of waves with obstacles is an everyday phenomenon in science and engineering, arising for example in acoustics, electromagnetism, seismology and hydrodynamics. The mathematical theory and technology needed to understand the phenomenon is known as multiple scattering, and this book is the first devoted to the subject. The author covers a variety of techniques, describing first the single-obstacle methods and then extending them to the multiple-obstacle case. A key ingredient in many of these extensions is an appropriate addition theorem: a coherent, thorough exposition of these theorems is given, and computational and numerical issues around them are explored. The application of these methods to different types of problems is also explained; in particular, sound waves, electromagnetic radiation, waves in solids and water waves. A comprehensive bibliography of some 1400 items rounds off the book, which will be an essential reference on the topic for applied mathematicians, physicists and engineers.
Author(s): P. A. Martin
Series: Encyclopedia of Mathematics and its Applications
Edition: 1
Publisher: Cambridge University Press
Year: 2006
Language: English
Pages: 450
About......Page 2
Encyclopedia of Mathematics and Its Applications......Page 3
Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles......Page 4
0521865549......Page 5
Contents......Page 8
Preface......Page 12
1.1 What is ‘multiple scattering’? ......Page 14
1.2 Narrowing the scope: previous reviews and omissions ......Page 23
1.3 Acoustic scattering by N obstacles ......Page 24
1.4 Multiple scattering of electromagnetic waves ......Page 29
1.5 Multiple scattering of elastic waves ......Page 32
1.6 Multiple scattering of water waves ......Page 36
1.7 Overview of the book ......Page 40
2.1 Introduction ......Page 42
2.2 Cartesian coordinates ......Page 43
2.3 Hobson’s theorem ......Page 44
2.4 Wavefunctions ......Page 47
2.5 Addition theorems ......Page 49
2.6 The separation matrices S and \widehat{S}......Page 53
2.7 Use of rotation matrices ......Page 56
2.8 Two-centre expansions ......Page 57
2.9 Elliptical wavefunctions ......Page 59
2.10 Vector cylindrical wavefunctions ......Page 66
2.11 Multipoles for water waves ......Page 67
3.2 Spherical harmonics ......Page 75
3.3 Legendre’s addition theorem ......Page 78
3.4 Cartesian coordinates ......Page 79
3.5 Hobson’s theorem ......Page 80
3.6 Wavefunctions and the operator y^m_n......Page 82
3.7 First derivatives of spherical wavefunctions ......Page 86
3.8 Axisymmetric addition theorems ......Page 88
3.9 A useful lemma ......Page 94
3.10 Composition formula for the operator y^m_n......Page 96
3.11 Addition theorem for j_nY^m_n......Page 100
3.12 Addition theorem for h^{(1)}_nY^m_n......Page 102
3.13 The separation matrices S and \widehat{S}......Page 104
3.14 Two-centre expansions ......Page 108
3.15 Use of rotation matrices ......Page 111
3.16 Alternative expressions for S(b\hat{z})......Page 112
3.17 Vector spherical wavefunctions ......Page 119
3.18 Multipoles for water waves ......Page 120
4.2 Separation of variables for one circular cylinder ......Page 135
4.3 Notation ......Page 138
4.4 Multipole method for two circular cylinders ......Page 139
4.5 Multipole method for N circular cylinders ......Page 141
4.6 Separation of variables for one sphere ......Page 145
4.7 Multipole method for two spheres ......Page 148
4.8 Multipole method for N spheres ......Page 151
4.9 Electromagnetic waves ......Page 153
4.10 Elastic waves ......Page 154
4.11 Water waves ......Page 156
4.12 Separation of variables in other coordinate systems ......Page 161
5.1 Introduction ......Page 165
5.2 Wave sources ......Page 166
5.3 Layer potentials ......Page 167
5.4 Explicit formulae in two dimensions ......Page 172
5.5 Explicit formulae in three dimensions ......Page 175
5.6 Green’s theorem ......Page 177
5.7 Scattering and radiation problems ......Page 179
5.8 Integral equations: general remarks ......Page 181
5.9 Integral equations: indirect method ......Page 182
5.10 Integral equations: direct method ......Page 185
6.2 Transmission problems ......Page 192
6.3 Inhomogeneous obstacles ......Page 193
6.4 Electromagnetic waves ......Page 200
6.5 Elastic waves ......Page 209
6.6 Water waves ......Page 216
6.7 Cracks and other thin scatterers ......Page 219
6.9 Modified fundamental solutions ......Page 222
6.10 Combination methods ......Page 230
6.11 Augmentation methods ......Page 232
6.12 Application of exact Green’s functions ......Page 235
6.13 Twersky’s method ......Page 242
6.14 Fast multipole methods ......Page 248
7.2 Radiation problems ......Page 255
7.3 Kupradze’s method and related methods ......Page 256
7.4 Scattering problems ......Page 259
7.5 Null-field equations for radiation problems: one obstacle ......Page 260
7.6 Null-field equations for scattering problems: one obstacle ......Page 262
7.7 Infinite sets of functions ......Page 264
7.8 Solution of the null-field equations ......Page 270
7.9 The T-matrix for one obstacle ......Page 281
7.10 The T-matrix for two obstacles ......Page 287
7.11 The T-matrix for N obstacles ......Page 294
8.2 Small scatterers ......Page 297
8.3 Foldy’s method ......Page 310
8.4 Point scatterers ......Page 317
8.5 Wide-spacing approximations ......Page 319
8.6 Random arrangements of small scatterers; suspensions ......Page 325
A Legendre functions ......Page 336
B Integrating a product of three spherical harmonics; Gaunt coefficients ......Page 338
C Rotation matrices ......Page 342
D One-dimensional finite-part integrals ......Page 346
E Proof of Theorem 5.4 ......Page 352
F Two-dimensional finite-part integrals ......Page 356
G Maue’s formula ......Page 363
H Volume potentials ......Page 364
I Boundary integral equations for G^E......Page 368
References ......Page 370
Citation index ......Page 434
Subject index ......Page 446
