Table Of ContentThe 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 eigenfunc-
tion 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 eigenfunc-
tion 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 inte-
gral 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 eigenfunc-
tion 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 eigenfunc-
tion 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 inte-
gral 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
