Canonical Problems in Scattering and Potential Theory Part II: Acoustic and Electromagnetic Diffraction by Canonical Structures

516 Pages·2002·8.04 MB·\516

by  S.S. Vinogradov (Author), P. D. Smith (Author), E.D. Vinogradova (Author)

CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 127
CANONICAL PROBLEMS
IN SCATTERING AND
POTENTIAL THEORY
PART II:
Acoustic and Electromagnetic
Diffraction by Canonical Structures
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 127
CANONICAL PROBLEMS
IN SCATTERING AND
POTENTIAL THEORY
PART II:
Acoustic and Electromagnetic
Diffraction by Canonical Structures
S. S. VINOGRADOV
P. D. SMITH
E. D. VINOGRADOVA
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
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.— (Monographs 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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To our children

Contents
Preface xi
1 Mathematical Aspects of Wave Scattering. 1
1.1 The Equations of Acoustic and Electromagnetic Waves. . . 2
1.2 Solution of the HelmholtzEquation: Separation of Variables. 19
1.3 Fields of Elementary Sources. Green’s Functions. . . . . . . 28
1.4 Representations of Incident Electromagnetic Waves. . . . . 37
1.5 Formulationof Wave Scattering Theory for Structures with
Edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.6 Single- or Double-Layer Surface Potentials and Dual Series
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.7 A Survey of Methods for Scattering. . . . . . . . . . . . . . 62
2 AcousticDi(cid:11)ractionfrom a CircularHole in a ThinSpher-
ical Shell. 65
2.1 Plane Wave Di(cid:11)raction froma Soft or Hard Spherical Cap. 67
2.2 Rigorous Theory of the Spherical Helmholtz Resonator. . . 79
2.3 Quasi-Eigenoscillations:SpectrumoftheOpenSphericalShell. 84
2.4 Total and Sonar Cross-Sections. . . . . . . . . . . . . . . . . 90
2.4.1 RayleighScattering. . . . . . . . . . . . . . . . . . . 93
2.4.2 Resonance Region. . . . . . . . . . . . . . . . . . . . 98
2.4.3 High Frequency Regime. . . . . . . . . . . . . . . . . 101
viii Contents
2.5 The Mechanical Force Factor. . . . . . . . . . . . . . . . . . 107
2.6 The Focal Region of a Spherical Re(cid:13)ector Antenna.. . . . . 112
2.7 The TransmittingSpherical Re(cid:13)ector Antenna. . . . . . . . 123
2.7.1 ComplexPoint Source. . . . . . . . . . . . . . . . . . 124
2.7.2 Regularised Solution and Far-Field Characteristics. . 125
2.7.3 NumericalResults. . . . . . . . . . . . . . . . . . . . 128
3 Acoustic Di(cid:11)raction from Various Spherical Cavities. 135
3.1 The Hard Spherical Barrel and Soft Slotted Spherical Shell. 136
3.2 The Soft Spherical Barrel and Hard Slotted Spherical Shell. 142
3.3 HelmholtzResonators: Barrelled or Slotted Spherical Shells. 148
3.4 Quasi-Eigenoscillationsof the Spherical Cavity. . . . . . . . 156
3.5 Totaland Sonar Cross-Sections; Mechanical Force Factor. . 163
4 ElectromagneticDi(cid:11)ractionfroma MetallicSphericalCav-
ity. 181
4.1 Electric or Magnetic Dipole Excitation. . . . . . . . . . . . 182
4.1.1 The Vertical Electric Dipole (TM Case). . . . . . . . 184
4.1.2 The Vertical Magnetic Dipole (TE Case) . . . . . . 194
4.2 PlaneWaveDi(cid:11)ractionfromaCircularHoleinaThinMetal-
lic Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.3 Re(cid:13)ectivity of Open Spherical Shells. . . . . . . . . . . . . . 215
4.4 The FocalRegion ofa Receiving Spherical Re(cid:13)ector Antenna.220
4.5 The TransmittingSpherical Re(cid:13)ector Antenna. . . . . . . . 225
4.5.1 TheComplex-PointHuygens Source:DebyePotentials.226
4.5.2 Excitation of the Re(cid:13)ector by a CPHS. . . . . . . . 232
4.5.3 NumericalResults . . . . . . . . . . . . . . . . . . . 236
5 Electromagnetic Di(cid:11)raction from Various Spherical Cavi-
ties. 243
5.1 EM Plane Wave Scattering by Two Concentric Spherical
Shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
5.2 Dipole Excitation of a Slotted Sphere. . . . . . . . . . . . . 255
5.2.1 The Vertical Electric Dipole. . . . . . . . . . . . . . 256
5.2.2 The Vertical Magnetic Dipole . . . . . . . . . . . . . 263
5.3 Dipole Excitation of Doubly-Connected Spherical Shells. . . 266
5.4 PlaneWaveDi(cid:11)raction:PerfectlyConductingSlottedSpher-
ical Shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.5 Magnetic Dipole Excitation of an Open Spherical Resonator. 284
5.6 Open Resonators Composed of Spherical and Disc Mirrors. 288
6 Spherical Cavities with Spherical Dielectric Inclusions. 297
6.1 Resonant CavityHeating of a SmallLossy Dielectric Sphere. 298
Contents ix
6.2 Re(cid:13)ectivity of a PartiallyScreened Dielectric Sphere. . . . . 308
6.3 The Luneberg Lens Re(cid:13)ector. . . . . . . . . . . . . . . . . . 320
7 Di(cid:11)raction from Spheroidal Cavities. 335
7.1 Regularisation in Spheroidal Coordinates. . . . . . . . . . . 338
7.2 Acoustic Scattering by a RigidSpheroidal Shell with a Cir-
cular Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
7.3 Rigorous Theory of the Spheroidal HelmholtzResonator.. . 347
7.4 Axial Electric Dipole Excitation of a Metallic Spheroidal
Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
7.5 Axial Magnetic Dipole Excitation of a Metallic Spheroidal
Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
7.6 Axial Electric Dipole Excitation of a Spheroidal Barrel. . . 367
7.6.1 The Series Equations with Odd Index CoeÆcients. . 371
7.6.2 The Series Equations with Even Index CoeÆcients. . 373
7.6.3 Numerical Results. . . . . . . . . . . . . . . . . . . . 374
7.7 Impedance Loading of the Spheroidal Barrel. . . . . . . . . 379
7.8 Spheroid Embedded in a Spheroidal Cavitywith Two Holes. 386
8 Wave Scattering Problems for Selected Structures. 401
8.1 Plane Wave Di(cid:11)raction fromIn(cid:12)nitely Long Strips. . . . . . 402
8.2 AxiallySlotted In(cid:12)nitely Long Circular Cylinders. . . . . . 413
8.3 AxiallySlotted Cylinders of Arbitrary Pro(cid:12)le. . . . . . . . . 424
8.4 Di(cid:11)raction fromCircular Discs. . . . . . . . . . . . . . . . . 434
8.5 Di(cid:11)raction fromElliptic Plates. . . . . . . . . . . . . . . . . 444
8.6 Wave Scattering Problems for Hollow Finite Cylinders. . . . 455
8.7 Wave Scattering Problems for Periodic Structures. . . . . . 469
8.7.1 Periodic Linear Array of Strips. . . . . . . . . . . . . 469
8.7.2 Periodic Linear Array of Hollow Finite Cylinders.. . 475
8.8 Shielded Microstrip Lines. . . . . . . . . . . . . . . . . . . . 478
References 485
Index 497