Canonical Problems in Scattering and Potential Theory. Part I

393 Pages·4.722 MB·English

by  Sergey S Vinogradov, P D Smith, Elena D Vinogradova

CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 122
CANONICAL PROBLEMS
IN SCATTERING AND
POTENTIAL THEORY
PART I:
Canonical Structures
in Potential Theory
CHAPMAN & HALL/CRC
Monographs and Surveys in Pure and Applied Mathematics
Main Editors
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W.A. Strauss, Brown University
J. van der Hoek, University of Adelaide
CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 122
CANONICAL PROBLEMS
IN SCATTERING AND
POTENTIAL THEORY
PART I:
Canonical Structures
in Potential Theory
S.S. VINOGRADOV
P.D. SMITH
E.D. VINOGRADOVA
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.
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Library of Congress Cataloging-in-Publication Data
Vinogradov, Sergey S. (Sergey Sergeyevich)
Canonical problems in scattering and potential theory / Sergey S. Vinogradov, Paul D.
Smith, Elena D. Vinogradova.
p. cm.— (Monographis and surveys in pure and applied mathematics ; 122)
Includes bibliographical references and index.
Contents: pt. 1. Canonical structures in potential theory
ISBN 1-58488-162-3 (v. 1 : alk. paper)
1. Potential theory (Mathematics) 2. Scattering (Mathematics) I. Smith, P.D. (Paul
Denis), 1955- II. Vinogradova, Elena D. (Elena Dmitrievna) III. Title. IV. Chapman &
Hall/CRC monographs and surveys in pure and applied mathematics ; 122.
QA404.7 . V56 2001
515′.9—dc21 2001028226

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Contents
1 Laplace’s Equation 1
1.1 Laplace’s equation in curvilinear coordinates . . . . . . . . 3
1.1.1 Cartesian coordinates . . . . . . . . . . . . . . . . . 7
1.1.2 Cylindricalpolar coordinates . . . . . . . . . . . . . 8
1.1.3 Spherical polar coordinates . . . . . . . . . . . . . . 8
1.1.4 Prolate spheroidal coordinates . . . . . . . . . . . . 9
1.1.5 Oblate spheroidal coordinates . . . . . . . . . . . . . 11
1.1.6 Elliptic cylinder coordinates . . . . . . . . . . . . . . 12
1.1.7 Toroidalcoordinates . . . . . . . . . . . . . . . . . . 13
1.2 Solutions of Laplace’s equation: separation of variables . . . 14
1.2.1 Cartesian coordinates . . . . . . . . . . . . . . . . . 14
1.2.2 Cylindricalpolar coordinates . . . . . . . . . . . . . 15
1.2.3 Spherical polar coordinates . . . . . . . . . . . . . . 16
1.2.4 Prolate spheroidal coordinates . . . . . . . . . . . . 17
1.2.5 Oblate spheroidal coordinates . . . . . . . . . . . . . 18
1.2.6 Elliptic cylinder coordinates . . . . . . . . . . . . . . 19
1.2.7 Toroidalcoordinates . . . . . . . . . . . . . . . . . . 19
1.3 Formulationof potential theory for structures with edges . 21
1.4 Dual equations: a classi(cid:12)cation of solution methods . . . . . 30
1.4.1 The de(cid:12)nition method . . . . . . . . . . . . . . . . . 31
1.4.2 The substitution method . . . . . . . . . . . . . . . 32
1.4.3 Noble’s multiplyingfactor method . . . . . . . . . . 33
1.4.4 The Abel integral transformmethod . . . . . . . . . 34
1.5 Abel’s integral equation and Abel integral transforms. . . . 37
viii Contents
1.6 Abel-typeintegralrepresentations ofhypergeometricfunctions 40
1.7 Dual equations and single- or double-layer surface potentials 45
2 Series and Integral Equations 55
2.1 Dual series equations involvingJacobi polynomials . . . . . 57
2.2 Dual series equations involvingtrigonometricalfunctions . . 65
2.3 Dual series equations involvingassociated Legendre functions 74
2.4 SymmetrictripleseriesequationsinvolvingJacobipolynomials 81
2.4.1 Type A triple series equations . . . . . . . . . . . . . 82
2.4.2 Type B triple series equations . . . . . . . . . . . . . 85
2.5 Relationships between series and integral equations . . . . . 86
2.6 Dual integral equations involvingBessel functions . . . . . . 96
2.7 Nonsymmetricaltriple series equations . . . . . . . . . . . . 99
2.8 Coupled series equations . . . . . . . . . . . . . . . . . . . . 104
2.9 A class of integro-series equations . . . . . . . . . . . . . . . 107
3 Electrostatic PotentialTheory for Open Spherical Shells 109
3.1 The open conducting spherical shell . . . . . . . . . . . . . 110
3.2 A symmetricalpairofopen spherical caps andthe spherical
barrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2.1 Approximateanalytical formulaefor capacitance . . 117
3.3 An asymmetricalpair of spherical caps and the asymmetric
barrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.4 The method of inversion . . . . . . . . . . . . . . . . . . . . 131
3.5 Electrostatic (cid:12)elds in a spherical electronic lens . . . . . . . 141
3.6 Frozen magnetic (cid:12)elds inside superconducting shells . . . . 144
3.7 Screening number of superconducting shells . . . . . . . . . 149
4 Electrostatic Potential Theory for Open SpheroidalShells157
4.1 Formulationofmixedboundaryvalueproblemsinspheroidal
geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.2 The prolate spheroidal conductor with one hole . . . . . . . 163
4.3 The prolate spheroidal conductor with a longitudinalslot . 173
4.4 The prolate spheroidal conductor with two circular holes . . 178
4.5 The oblate spheroidal conductor with a longitudinalslot . . 181
4.6 The oblate spheroidal conductor with two circular holes . . 185
4.7 Capacitance of spheroidal conductors . . . . . . . . . . . . . 187
4.7.1 Open spheroidal shells . . . . . . . . . . . . . . . . . 188
4.7.2 Spheroidal condensors . . . . . . . . . . . . . . . . . 191
5 Charged Toroidal Shells 195
5.1 Formulation of mixed boundary value problems in toroidal
geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.2 The open charged toroidal segment . . . . . . . . . . . . . . 198
5.3 The toroidal shell with two transversal slots . . . . . . . . . 202
Contents ix
5.4 The toroidal shell with two longitudinalslots . . . . . . . . 208
5.5 Capacitance of toroidalconductors . . . . . . . . . . . . . . 212
5.6 An open toroidal shell with azimuthalcuts . . . . . . . . . 213
5.6.1 The toroidal shell with one azimuthalcut. . . . . . . 215
5.6.2 The toroidal shell with multiplecuts . . . . . . . . . 220
5.6.3 Limitingcases . . . . . . . . . . . . . . . . . . . . . 221
6 Potential Theory for Conical Structureswith Edges 225
6.1 Non-coplanar oppositely charged in(cid:12)nite strips . . . . . . . 226
6.2 Electrostatic (cid:12)elds of a charged axisymmetric (cid:12)nite open
conical conductor . . . . . . . . . . . . . . . . . . . . . . . . 235
6.3 The slotted hollowspindle . . . . . . . . . . . . . . . . . . . 246
6.4 A spherical shell with an azimuthalslot . . . . . . . . . . . 253
7 Two-dimensional Potential Theory 257
7.1 The circular arc . . . . . . . . . . . . . . . . . . . . . . . . . 258
7.2 Axiallyslotted open circular cylinders . . . . . . . . . . . . 262
7.3 Electrostatic potential of systems of charged thin strips . . 268
7.4 Axially-slotted elliptic cylinders . . . . . . . . . . . . . . . . 276
7.5 Slotted cylinders of arbitrary pro(cid:12)le . . . . . . . . . . . . . 282
8 More Complicated Structures 291
8.1 Rigorous solution methods for charged (cid:13)at plates . . . . . . 292
8.2 The charged elliptic plate . . . . . . . . . . . . . . . . . . . 296
8.2.1 The spherically-curved elliptic plate . . . . . . . . . 299
8.3 Polygonalplates . . . . . . . . . . . . . . . . . . . . . . . . 304
8.4 The (cid:12)nite strip . . . . . . . . . . . . . . . . . . . . . . . . . 310
8.5 Coupled charged conductors: the spherical cap and circular
disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
A Notation 323
B Special Functions 325
B.1 The Gammafunction . . . . . . . . . . . . . . . . . . . . . 325
B.2 Hypergeometric functions . . . . . . . . . . . . . . . . . . . 326
B.3 Orthogonalpolynomials:Jacobipolynomials,Legendrepoly-
nomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
B.3.1 The associated Legendre polynomials. . . . . . . . . 332
B.3.2 The Legendre polynomials. . . . . . . . . . . . . . . 332
B.4 Associated Legendre functions. . . . . . . . . . . . . . . . . 333
B.4.1 Ordinary Legendre functions . . . . . . . . . . . . . 334
B.4.2 Conical functions . . . . . . . . . . . . . . . . . . . . 338
B.4.3 Associated Legendre functions of integer order . . . 338
B.5 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 341
B.5.1 Spherical Bessel functions . . . . . . . . . . . . . . . 344