Table Of Content

The IEEE PRESS Series on Electromagnetic Waves consists of new titles as well as reprints and revisions of recognized classics that maintain long-term archival significance in electromagneticw aves and applications. Dyadic Green Functions Donald G. Dudley in Electromagnetic Theory Editor University of Arizona Advisory Board Second Edition Robert E. Collin Case Western University Akira Ishimaru University of Washington Associate Editors Electromagnetic Theory, Scattering, and Diffraction Ehud Heyrnan Chen-To Tai Tel-Aviv University Differential Equation Methods Professor Emeritus Andreas C. Cangellaris Radiation Laboratory University of Arizona Department of Electrical Engineering Integral Equation Methods and Computer Science Donald R. Wilton University of Michigan University of Houston Antennas, Propagation, and Microwaves David R. Jackson University of Houston Series Books Published IEEE PRESS Series on Collin, R. E., Field Theory of Guided Waves, 2d. rev. ed., 1991 Electromagnetic Waves Tai, C. T., Generalized kctor and Dyadic Analysis: G. Dudley, Series Editor Applied Mathematics in FieM Theory, 1991 Elliott, R. S., Electromagnetics: History, Theory, and Applications, 1993 Harrington, R. F., Field Computation by Moment Methoh, 1993 Tai, C. T, Dyadic Green Functions in Electromagnetic Theory, 2nd ed., 1993 Future Series Title Dudley, D. G., Mathematical Foundations of Electromagnetic Theory IEEE Antennas and Propagation Society and IEEE Microwave Theory and Techniques Society, Co-sponsors The Institute of Electrical and Electronics Engineers, Inc., New York The IEEE PRESS Series on Electromagnetic Waves consists of new titles as well as reprints and revisions of recognized classics that maintain long-term archival significance in electromagneticw aves and applications. Dyadic Green Functions Donald G. Dudley in Electromagnetic Theory Editor University of Arizona Advisory Board Second Edition Robert E. Collin Case Western University Akira Ishimaru University of Washington Associate Editors Electromagnetic Theory, Scattering, and Diffraction Ehud Heyrnan Chen-To Tai Tel-Aviv University Differential Equation Methods Professor Emeritus Andreas C. Cangellaris Radiation Laboratory University of Arizona Department of Electrical Engineering Integral Equation Methods and Computer Science Donald R. Wilton University of Michigan University of Houston Antennas, Propagation, and Microwaves David R. Jackson University of Houston Series Books Published IEEE PRESS Series on Collin, R. E., Field Theory of Guided Waves, 2d. rev. ed., 1991 Electromagnetic Waves Tai, C. T., Generalized kctor and Dyadic Analysis: G. Dudley, Series Editor Applied Mathematics in FieM Theory, 1991 Elliott, R. S., Electromagnetics: History, Theory, and Applications, 1993 Harrington, R. F., Field Computation by Moment Methoh, 1993 Tai, C. T, Dyadic Green Functions in Electromagnetic Theory, 2nd ed., 1993 Future Series Title Dudley, D. G., Mathematical Foundations of Electromagnetic Theory IEEE Antennas and Propagation Society and IEEE Microwave Theory and Techniques Society, Co-sponsors The Institute of Electrical and Electronics Engineers, Inc., New York 1993 Editorial Board William Perkins, Editor in Chief R. S. Blicq G. F. Hoffnagle P. Laplante 1. Peden M. Eden R. F. Hoyt M. Lightner L. Shaw D. M. Etter J. D. Irwin E. K. Miller M. Simaan J. J. Farrell I11 S. V. Kartalopoulos J. M. F. Moura D. J. Wells L. E. Frenzel Dedicated Dudley R. Kay, Director of Book Publishing to Carrie Briggs, AdministrativeA ssistant Karen G. Miller, production Editor Professor Chih Kung Jen IEEE Antennas and Propagation Society, Co-sponsor (An Inspiring Teacher of Science and Humanity) AP-S Liaison to IEEE PRESS Robert J. Mailloux Rome Laboratory, ERI IEEE Microwave Theory and Techniques Society, Co-sponsor M'IT-S Liaison to IEEE PRESS Kris K. Agarwal E-Systems Technical Reviewers Nicolaos G. Alexopoulos Edmund K. Miller Robert E. Collin University of California Los Alamos National Laboratory Case Western Reserve at Los Angeles University Kai Chang Texas A & M University 01994 by the Institute of Electrical and Electronics Engineers, Inc. 345 East 47th Street, New York, NY 10017-2394 01971 International Textbook Company All rights reserved. No part of this book may be reproduced in any form, nor may it be stored in a rem'eval system or transmitted in any form, without written permission from the publisher. Printed in the United States of America 1 0 9 8 7 6 5 4 3 2 1 ISBN 0-7803-0449-7 IEEE Order Number: PC0348-3 Library of Congress Cataloging-in-PublicationD ata Tai Chen-To (date) Dyadic green functions in electromagnetic theory by Chen-to Tai.-2nd ed. p. cm. Sponsors : IEEE Antennas and Propagation Society and IEEE Microwave The0 and Techniques Society. ~ncluzBs iblio aphical references and index. ISBN 0-7803-&-7 1. Electroma etic theory-Mathematics. 2. Green's functions. 3. Boundary vaEp roblems. I. IEEE Antennas and Propagation Society. 11. IEEE Microwave Theory and Techniques Society. Ill. Title 93-24201 CIP 1993 Editorial Board William Perkins, Editor in Chief R. S. Blicq G. F. Hoffnagle P. Laplante 1. Peden M. Eden R. F. Hoyt M. Lightner L. Shaw D. M. Etter J. D. Irwin E. K. Miller M. Simaan J. J. Farrell I11 S. V. Kartalopoulos J. M. F. Moura D. J. Wells L. E. Frenzel Dedicated Dudley R. Kay, Director of Book Publishing to Carrie Briggs, AdministrativeA ssistant Karen G. Miller, production Editor Professor Chih Kung Jen IEEE Antennas and Propagation Society, Co-sponsor (An Inspiring Teacher of Science and Humanity) AP-S Liaison to IEEE PRESS Robert J. Mailloux Rome Laboratory, ERI IEEE Microwave Theory and Techniques Society, Co-sponsor M'IT-S Liaison to IEEE PRESS Kris K. Agarwal E-Systems Technical Reviewers Nicolaos G. Alexopoulos Edmund K. Miller Robert E. Collin University of California Los Alamos National Laboratory Case Western Reserve at Los Angeles University Kai Chang Texas A & M University 01994 by the Institute of Electrical and Electronics Engineers, Inc. 345 East 47th Street, New York, NY 10017-2394 01971 International Textbook Company All rights reserved. No part of this book may be reproduced in any form, nor may it be stored in a rem'eval system or transmitted in any form, without written permission from the publisher. Printed in the United States of America 1 0 9 8 7 6 5 4 3 2 1 ISBN 0-7803-0449-7 IEEE Order Number: PC0348-3 Library of Congress Cataloging-in-PublicationD ata Tai Chen-To (date) Dyadic green functions in electromagnetic theory by Chen-to Tai.-2nd ed. p. cm. Sponsors : IEEE Antennas and Propagation Society and IEEE Microwave The0 and Techniques Society. ~ncluzBs iblio aphical references and index. ISBN 0-7803-&-7 1. Electroma etic theory-Mathematics. 2. Green's functions. 3. Boundary vaEp roblems. I. IEEE Antennas and Propagation Society. 11. IEEE Microwave Theory and Techniques Society. Ill. Title 93-24201 CIP Contents PREFACE xi ACKNOWLEDGMENTS xiii 1 GENERAL THEOREMS AND FORMULAS 1 1-1 Vector Notations and the Coordinate Systems 1 1-2 Vector Analysis 4 1-3 Dyadic Analysis 6 1-4 Fourier Transform and Hankel Transform 12 1-5 Saddle-Point Method of Integration and Semi-infinite Integrals of the Product of Bessel Functions 16 2 SCALAR GREEN FUNCTIONS 2-1 Scalar Green Functions of a One-Dimensional Wave Equation-Theory of Transmission Lines 21 2-2 Derivation of go(x,x ') by the Conventional Method and the Ohm-Rayleigh Method 25 2-3 Symmetrical Properties of Green Functions 33 2-4 Free-Space Green Function of the Three-Dimensional Scalar Wave Equation 35 3 ELECTROMAGNETIC THEORY 38 3-1 The Independent and Dependent Equations and the Indefinite and Definite Forms of Maxwell's Equations 38 3-2 Integral Forms of Maxwell's Equations 41 3-3 Boundary Conditions 42 3-4 Monochromatically Oscillating Fields in Free Space 47 3-5 Method of Potentials 49 vii Contents PREFACE xi ACKNOWLEDGMENTS xiii 1 GENERAL THEOREMS AND FORMULAS 1 1-1 Vector Notations and the Coordinate Systems 1 1-2 Vector Analysis 4 1-3 Dyadic Analysis 6 1-4 Fourier Transform and Hankel Transform 12 1-5 Saddle-Point Method of Integration and Semi-infinite Integrals of the Product of Bessel Functions 16 2 SCALAR GREEN FUNCTIONS 2-1 Scalar Green Functions of a One-Dimensional Wave Equation-Theory of Transmission Lines 21 2-2 Derivation of go(x,x ') by the Conventional Method and the Ohm-Rayleigh Method 25 2-3 Symmetrical Properties of Green Functions 33 2-4 Free-Space Green Function of the Three-Dimensional Scalar Wave Equation 35 3 ELECTROMAGNETIC THEORY 38 3-1 The Independent and Dependent Equations and the Indefinite and Definite Forms of Maxwell's Equations 38 3-2 Integral Forms of Maxwell's Equations 41 3-3 Boundary Conditions 42 3-4 Monochromatically Oscillating Fields in Free Space 47 3-5 Method of Potentials 49 vii viii Contents Contents 9-3 Radiation from Electric Dipoles in the Presence 4 DYADIC GREEN FUNCTIONS 55 4-1 Maxwell's Equations in Dyadic Form and Dyadic of a Half Sheet 174 Green Functions of the Electric and Magnetic Trpe 55 9-3.1 Longitudinal Electrical Dipole 174 4-2 Free-Space Dyadic Green Functions 59 9-3.2 Horizontal Electrical Dipole 176 4-3 Classification of Dyadic Green Functions 62 9-3.3 Vertical ~lectricD opole 178 4-4 Symmetrical Properties of Dyadic Green Functions 74 9-4 Radiation from Magnetic Dipoles in the Presence 4-5 Reciprocity Theorems 85 of a Half Sheet 179 4-6 Transmission Line Model of the Complementary 9-5 Slots Cut in a Half Sheet 182 Reciprocity Theorems 90 9-5.1 Longititudinal Slot 183 4-7 Dyadic Green Functions for a Half Space Bounded 9-5.2 Horizontal Slot 184 by a Plane Conducting Surface 92 9-6 Diffraction of a Plane Wave by a Half Sheet 187 9-7 Circular Cylinder and Half Sheet 196 5 RECTANGULAR WAVEGUIDES 5-1 Rectangular Vector Wave Functions 96 10 SPHERES AND PERFECTLY CONDUCTING CONES Em 5-2 The Method of 103 10-1 Eigenfunction Expansion of Free-Space Dyadic ??, 5-3 The Method of 110 Green Functions 198 5-4 The Method of EA 114 10-2 An Algebraic Method of Finding E,, without the 5-5 Parallel Plate Waveguide 115 Singular Term 204 5-6 Rectangular Waveguide Filled 10-3 Perfectly Conducting and Dielectric Spheres 210 with Two Dielectrics 118 10-4 Spherical Cavity 218 5-7 Rectangular Cavity 124 10-5 Perfectly Conducting Conical Structures 220 F, 5-8 The Origin of the Isolated Singular Term in 128 10-6 Cone with a Spherical Sector 223 6 CYLINDRICAL WAVEGUIDES 1 1 PLANAR STRATIFIED MEDIA 6-1 Cylindrical Wave Functions with Discrete 11-1 Flat Earth 225 Eigenvalues 133 11-2 Radition from Electric Dipoles in the Presence 6-2 Cylindrical Waveguide 140 of a Flat Earth and Sommerfeld's Theory 228 6-3 Cylindrical Cavity 142 11-3 Dielectric Layer on a Conducting Plane 233 6-4 Coaxial Line 143 11-4 Reciprocity Theorems for Stratified Media 237 11-5 Eigenfunction Expansions 244 7 CIRCULAR CYLINDER IN FREE SPACE 11-6 A Dielectric Slab in Air 249 7-1 Cylindrical Vector Wave Functions with Continuous 11-7 Two-Dimensional Fourier Transform of the Dyadic Eigenvalues 149 7-2 Eigenfunction Expansion of the Free-Space Dyadic Green Functions 251 Green Functions 152 12 INHOMOGENEOUS MEDIA AND MOVING MEDIUM 255 7-3 Conducting Cylinder, Dielectric Cylinder, and Coated 12-1 Vector Wave Functions for Plane Cylinder 154 Stratified Media 255 7-4 Asymptotic Expression 159 12-2 Vector Wave Functions for Spherically 8 PERFECTLY CONDUCTING ELLIPTICAL CYLINDER Stratified Media 259 8-1 Vector Wave Functions in an Elliptical Cylinder 12-3 Inhomogeneous Spherical Lenses 260 Coordinate System 161 12-4 Monochromatically Oscillating Fields in a Moving 8-2 The Electric Dyadic Green Function of the First Isotropic Medium 270 Kind 166 12-5 Time-Dependent Field in a Moving Medium 277 12-6 Rectangular Waveguide with a Moving Medium 286 9 PERFECTLY CONDUCTING WEDGE AND THE HALF SHEET 169 9-1 Dyadic Green Functions for a Perfectly 12-7 Cylindrical Waveguide with a Moving Medium 291 Conducting Wedge 169 12-8 Infinite Conducting Cylinder 9-2 The Half Sheet 173 in a Moving Medium 293 viii Contents Contents 9-3 Radiation from Electric Dipoles in the Presence 4 DYADIC GREEN FUNCTIONS 55 4-1 Maxwell's Equations in Dyadic Form and Dyadic of a Half Sheet 174 Green Functions of the Electric and Magnetic Trpe 55 9-3.1 Longitudinal Electrical Dipole 174 4-2 Free-Space Dyadic Green Functions 59 9-3.2 Horizontal Electrical Dipole 176 4-3 Classification of Dyadic Green Functions 62 9-3.3 Vertical ~lectricD opole 178 4-4 Symmetrical Properties of Dyadic Green Functions 74 9-4 Radiation from Magnetic Dipoles in the Presence 4-5 Reciprocity Theorems 85 of a Half Sheet 179 4-6 Transmission Line Model of the Complementary 9-5 Slots Cut in a Half Sheet 182 Reciprocity Theorems 90 9-5.1 Longititudinal Slot 183 4-7 Dyadic Green Functions for a Half Space Bounded 9-5.2 Horizontal Slot 184 by a Plane Conducting Surface 92 9-6 Diffraction of a Plane Wave by a Half Sheet 187 9-7 Circular Cylinder and Half Sheet 196 5 RECTANGULAR WAVEGUIDES 5-1 Rectangular Vector Wave Functions 96 10 SPHERES AND PERFECTLY CONDUCTING CONES Em 5-2 The Method of 103 10-1 Eigenfunction Expansion of Free-Space Dyadic ??, 5-3 The Method of 110 Green Functions 198 5-4 The Method of EA 114 10-2 An Algebraic Method of Finding E,, without the 5-5 Parallel Plate Waveguide 115 Singular Term 204 5-6 Rectangular Waveguide Filled 10-3 Perfectly Conducting and Dielectric Spheres 210 with Two Dielectrics 118 10-4 Spherical Cavity 218 5-7 Rectangular Cavity 124 10-5 Perfectly Conducting Conical Structures 220 F, 5-8 The Origin of the Isolated Singular Term in 128 10-6 Cone with a Spherical Sector 223 6 CYLINDRICAL WAVEGUIDES 1 1 PLANAR STRATIFIED MEDIA 6-1 Cylindrical Wave Functions with Discrete 11-1 Flat Earth 225 Eigenvalues 133 11-2 Radition from Electric Dipoles in the Presence 6-2 Cylindrical Waveguide 140 of a Flat Earth and Sommerfeld's Theory 228 6-3 Cylindrical Cavity 142 11-3 Dielectric Layer on a Conducting Plane 233 6-4 Coaxial Line 143 11-4 Reciprocity Theorems for Stratified Media 237 11-5 Eigenfunction Expansions 244 7 CIRCULAR CYLINDER IN FREE SPACE 11-6 A Dielectric Slab in Air 249 7-1 Cylindrical Vector Wave Functions with Continuous 11-7 Two-Dimensional Fourier Transform of the Dyadic Eigenvalues 149 7-2 Eigenfunction Expansion of the Free-Space Dyadic Green Functions 251 Green Functions 152 12 INHOMOGENEOUS MEDIA AND MOVING MEDIUM 255 7-3 Conducting Cylinder, Dielectric Cylinder, and Coated 12-1 Vector Wave Functions for Plane Cylinder 154 Stratified Media 255 7-4 Asymptotic Expression 159 12-2 Vector Wave Functions for Spherically 8 PERFECTLY CONDUCTING ELLIPTICAL CYLINDER Stratified Media 259 8-1 Vector Wave Functions in an Elliptical Cylinder 12-3 Inhomogeneous Spherical Lenses 260 Coordinate System 161 12-4 Monochromatically Oscillating Fields in a Moving 8-2 The Electric Dyadic Green Function of the First Isotropic Medium 270 Kind 166 12-5 Time-Dependent Field in a Moving Medium 277 12-6 Rectangular Waveguide with a Moving Medium 286 9 PERFECTLY CONDUCTING WEDGE AND THE HALF SHEET 169 9-1 Dyadic Green Functions for a Perfectly 12-7 Cylindrical Waveguide with a Moving Medium 291 Conducting Wedge 169 12-8 Infinite Conducting Cylinder 9-2 The Half Sheet 173 in a Moving Medium 293 Contents APPENDIX A MATHEMATICAL FORMULAS 296 A-1 Gradient, Divergence, and Curl in Orthogonal Systems 296 A-2 Vector Identities 298 A-3 Dyadic Identities 298 A-4 Integral Theorems 299 APPENDIX B VECTOR WAVE FUNCTIONS AND THEIR MUTUAL RELATIONS B-1 Rectangular Vector Wave Functions 302 Preface B-2 Cylindrical Vector Wave Functions with Discrete Eigenvalues 304 B-3 Spherical Vector Wave Functions 305 B-4 Conical Vector Wave Functions 306 APPENDIX C EXERCISES REFERENCES NAME INDEX SUBJECT INDEX The first edition of this book, bearing the same title, was published by Intext Edu- cation Publishers in 1971. Since then, several topics in the book have been found to have been improperly treated; in particular, a singular term in the eigenfunction expansion of the electrical dyadic Green function was inadvertently omitted, an oversight that was later amended [Tai, 19731. In the present edition, some major revisions have been made. First, Maxwell's equations have been cast in a dyadic form to facilitate the introduction of the electric and the magnetic dyadic Green functions. The magnetic dyadic Green function was not introduced in the first edition, but it was found to be a very important entity in the entire theory of dyadic Green functions. Being a solenoidal function, its eigenfunction expansion does not require the use of non- solenoidal vector wave functions or Hansen's L-functions [Stratton, 19411. With the aid of Maxwell-Ampkre equation in dyadic form, one can find the eigenfunction expansion of the electrical dyadic Green function, including the previously missing singular term. This method is used extensively in the present edition. Several other new features are found in this edition. For example, the integral solutions of Maxwell's equations are now derived with the aid of the vector- dyadic Green's theorem instead of by the vector Green's theorem as in the old treatment. By doing so, many intermediate steps can be omitted. In reviewing Maxwell's theory we have emphasized the necessity of adopting one of two alter- native postulates in stating the boundary conditions. The implication is that the boundary conditions cannot be derived from Maxwell's differential equations without a postulate. Reciprocity theorems in electromagnetic theory are dis- cussed in detail. In addition to the classical theorems due to Rayleigh, Carson, and Helmholtz, two complementary reciprocity theorems have been formulated Contents APPENDIX A MATHEMATICAL FORMULAS 296 A-1 Gradient, Divergence, and Curl in Orthogonal Systems 296 A-2 Vector Identities 298 A-3 Dyadic Identities 298 A-4 Integral Theorems 299 APPENDIX B VECTOR WAVE FUNCTIONS AND THEIR MUTUAL RELATIONS B-1 Rectangular Vector Wave Functions 302 Preface B-2 Cylindrical Vector Wave Functions with Discrete Eigenvalues 304 B-3 Spherical Vector Wave Functions 305 B-4 Conical Vector Wave Functions 306 APPENDIX C EXERCISES REFERENCES NAME INDEX SUBJECT INDEX The first edition of this book, bearing the same title, was published by Intext Edu- cation Publishers in 1971. Since then, several topics in the book have been found to have been improperly treated; in particular, a singular term in the eigenfunction expansion of the electrical dyadic Green function was inadvertently omitted, an oversight that was later amended [Tai, 19731. In the present edition, some major revisions have been made. First, Maxwell's equations have been cast in a dyadic form to facilitate the introduction of the electric and the magnetic dyadic Green functions. The magnetic dyadic Green function was not introduced in the first edition, but it was found to be a very important entity in the entire theory of dyadic Green functions. Being a solenoidal function, its eigenfunction expansion does not require the use of non- solenoidal vector wave functions or Hansen's L-functions [Stratton, 19411. With the aid of Maxwell-Ampkre equation in dyadic form, one can find the eigenfunction expansion of the electrical dyadic Green function, including the previously missing singular term. This method is used extensively in the present edition. Several other new features are found in this edition. For example, the integral solutions of Maxwell's equations are now derived with the aid of the vector- dyadic Green's theorem instead of by the vector Green's theorem as in the old treatment. By doing so, many intermediate steps can be omitted. In reviewing Maxwell's theory we have emphasized the necessity of adopting one of two alter- native postulates in stating the boundary conditions. The implication is that the boundary conditions cannot be derived from Maxwell's differential equations without a postulate. Reciprocity theorems in electromagnetic theory are dis- cussed in detail. In addition to the classical theorems due to Rayleigh, Carson, and Helmholtz, two complementary reciprocity theorems have been formulated