Dyadic Green Functions in Electromagnetic Theroy.pdf

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00 Preface

01 General Theorems and Formulas

02 Scalar Green Functions

03 Electromagnetic Theory

04 Dyadic Green Functions

05 Rectangular Waveguides

06 Cylindrical Waveguides

07 Circular Cylinder in Free Space

08 Perfectly Conducting Elliptical Cylinder

09 Perfectly Conducting Wedge and the Half Sheet

10 Spheres and Perfectly Conducting Cones

11 Planar Stratified Media

12 Ingomogeneous Media and Moving Medium

13 Name Index

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 electromagnetic waves and applications. Donald G. Dudley Editor University of Arizona Advisory Board Robert E. Collin Case Western University Akira Ishimaru University of Washington Electromagnetic Theory, Scattering, and Diffraction Associate Editors Ehud Heyrnan Tel-Aviv University Differential Equation Methods Andreas C. Cangellaris University of Arizona Integral Equation Methods Donald R. Wilton University of Houston Antennas, Propagation, and Microwaves David R. Jackson University of Houston Series Books Published Collin, R. E., Field Theory of Guided Waves, 2d. rev. ed., 1991 Tai, C. T., Generalized kctor and Dyadic Analysis: Applied Mathematics in FieM Theory, 1991 Dyadic Green Functions in Electromagnetic Theory Second Edition Chen-To Tai Professor Emeritus Radiation Laboratory Department of Electrical Engineering and Computer Science University of Michigan IEEE PRESS Series on Electromagnetic Waves G. Dudley, Series Editor 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 G. F. Hoffnagle R. F. Hoyt J. D. Irwin S. V. Kartalopoulos P. Laplante 1. Peden M. Lightner L. Shaw E. K. Miller M. Simaan J. M. F. Moura D. J. Wells R. S. Blicq M. Eden D. M. Etter J. J. Farrell I11 L. E. Frenzel Dudley R. Kay, Director of Book Publishing Carrie Briggs, Administrative Assistant Karen G. Miller, production Editor IEEE Antennas and Propagation Society, Co-sponsor 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 Dedicated to Professor Chih Kung Jen (An Inspiring Teacher of Science and Humanity) Nicolaos G. Alexopoulos Edmund K. Miller University of California at Los Angeles Robert E. Collin University Los Alamos National Laboratory Case Western Reserve 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-Publication Data Dyadic green functions in electromagnetic theory Tai Chen-To (date) by Chen-to Tai.-2nd cm. p. ed. Sponsors : IEEE Antennas and Propagation Society and IEEE Microwave The0 and Techniques Society. ~ n c l u z s Biblio aphical references and index. ISBN 0-7803-&-7 1. Electroma etic theory-Mathematics. 2. Green's functions. I. IEEE Antennas and Propagation 3. Boundary v a E problems. Society. 11. IEEE Microwave Theory and Techniques Society. Ill. Title 93-24201 CIP

Contents PREFACE ACKNOWLEDGMENTS 1 GENERAL THEOREMS AND FORMULAS 1-1 Vector Notations and the Coordinate Systems 1-2 Vector Analysis 1-3 Dyadic Analysis 1-4 Fourier Transform and Hankel Transform 1-5 Saddle-Point Method of Integration and Semi-infinite 4 6 12 1 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 2-4 Free-Space Green Function of the Three-Dimensional 33 Scalar Wave Equation 35 3 ELECTROMAGNETIC THEORY 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 3-3 Boundary Conditions 3-4 Monochromatically Oscillating Fields 42 in Free Space 47 3-5 Method of Potentials 49 41 xi xiii 1 38 vii

viii 4 DYADIC GREEN FUNCTIONS 4-1 Maxwell's Equations in Dyadic Form and Dyadic Green Functions of the Electric and Magnetic Trpe 4-2 Free-Space Dyadic Green Functions 4-3 Classification of Dyadic Green Functions 4-4 Symmetrical Properties of Dyadic Green Functions 4-5 Reciprocity Theorems 4-6 Transmission Line Model of the Complementary 62 85 59 Reciprocity Theorems 90 4-7 Dyadic Green Functions for a Half Space Bounded by a Plane Conducting Surface 92 5 RECTANGULAR WAVEGUIDES 5-1 Rectangular Vector Wave Functions 5-2 The Method of Em 5-3 The Method of ??, 5-4 The Method of EA 5-5 Parallel Plate Waveguide 5-6 Rectangular Waveguide Filled 103 110 114 115 with Two Dielectrics 5-7 Rectangular Cavity 5-8 The Origin of the Isolated Singular Term in F, 118 124 128 6 CYLINDRICAL WAVEGUIDES 6-1 Cylindrical Wave Functions with Discrete Eigenvalues 133 6-2 Cylindrical Waveguide 6-3 Cylindrical Cavity 6-4 Coaxial Line 143 142 140 7 CIRCULAR CYLINDER IN FREE SPACE 7-1 Cylindrical Vector Wave Functions with Continuous Eigenvalues 149 7-2 Eigenfunction Expansion of the Free-Space Dyadic Green Functions 152 7-3 Conducting Cylinder, Dielectric Cylinder, and Coated Cylinder 154 7-4 Asymptotic Expression 159 8 PERFECTLY CONDUCTING ELLIPTICAL CYLINDER 8-1 Vector Wave Functions in an Elliptical Cylinder Coordinate System 161 8-2 The Electric Dyadic Green Function of the First Kind 166 9 PERFECTLY CONDUCTING WEDGE AND THE HALF SHEET 9-1 Dyadic Green Functions for a Perfectly Conducting Wedge 169 9-2 The Half Sheet 173 Contents Contents 55 55 74 9-3 Radiation from Electric Dipoles in the Presence 174 of a Half Sheet 9-3.1 Longitudinal Electrical Dipole 9-3.2 Horizontal Electrical Dipole 9-3.3 Vertical ~lectric Dopole 178 174 176 9-4 Radiation from Magnetic Dipoles in the Presence of a Half Sheet 179 9-5 Slots Cut in a Half Sheet 9-5.1 Longititudinal Slot 9-5.2 Horizontal Slot 184 182 183 96 9-6 Diffraction of a Plane Wave by a Half Sheet 9-7 Circular Cylinder and Half Sheet 196 187 10 SPHERES AND PERFECTLY CONDUCTING CONES 10-1 Eigenfunction Expansion of Free-Space Dyadic 10-2 An Algebraic Method of Finding E,, without the Green Functions 198 Singular Term 204 10-3 Perfectly Conducting and Dielectric Spheres 10-4 Spherical Cavity 10-5 Perfectly Conducting Conical Structures 10-6 Cone with a Spherical Sector 218 223 220 210 1 1 PLANAR STRATIFIED MEDIA 11-1 Flat Earth 11-2 Radition from Electric Dipoles in the Presence 225 of a Flat Earth and Sommerfeld's Theory 228 11-3 Dielectric Layer on a Conducting Plane 11-4 Reciprocity Theorems for Stratified Media 11-5 Eigenfunction Expansions 11-6 A Dielectric Slab in Air 11-7 Two-Dimensional Fourier Transform of the Dyadic 244 249 233 237 Green Functions 251 12 169 INHOMOGENEOUS MEDIA AND MOVING MEDIUM 12-1 Vector Wave Functions for Plane 255 12-2 Vector Wave Functions for Spherically Stratified Media Stratified Media 255 259 12-3 Inhomogeneous Spherical Lenses 12-4 Monochromatically Oscillating Fields in a Moving 260 Isotropic Medium 270 12-5 Time-Dependent Field in a Moving Medium 12-6 Rectangular Waveguide with a Moving Medium 12-7 Cylindrical Waveguide with a Moving Medium 12-8 Infinite Conducting Cylinder 293 in a Moving Medium 277 286 291

APPENDIX A MATHEMATICAL FORMULAS A-1 Gradient, Divergence, and Curl 296 in Orthogonal Systems A-2 Vector Identities A-3 Dyadic Identities A-4 Integral Theorems APPENDIX B VECTOR WAVE FUNCTIONS 298 298 299 A N D THEIR MUTUAL RELATIONS B-1 Rectangular Vector Wave Functions B-2 Cylindrical Vector Wave Functions with Discrete 302 Eigenvalues 304 B-3 Spherical Vector Wave Functions B-4 Conical Vector Wave Functions APPENDIX C EXERCISES 305 306 REFERENCES NAME INDEX SUBJECT INDEX Contents 296 Preface 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

xii Preface to uncover the symmetrical relations of the magnetic dyadic Green functions not derivable from the Rayleigh-Carson theorem. Various dyadic Green functions for problems involving plain layered media have been derived, including a two-dimensional Fourier-integral representation of these functions. In the area of moving media, the problem of transient radi- ation is formulated with the aid of an affine transformation which enables us to solve the Maxwell-Minkowski equation in a relatively simple manner. Many new exercises have been added to this edition to help the reader bet- ter understand the materials covered in the book. Answers for some exercises are given, and sufficient hints are provided for many others so that the book may be used not only as a reference but also as a text for a graduate course in electromagnetic theory. Acknowledgments I am very grateful to Professor Per-Olof Brundell of the University of Lund, Sweden, who, in 1972, called my attention to the incompleteness of the eigen- function expansion of the electric dyadic Green function in the original edition of this book. My discussion with Dr. Olov Einarsson, then a faculty member of the same institution, on the dependence of the integral of the electric dyadic Green function on the shape of the cell in the source region was very valuable, particularly, on the aspect ratio of a cylindrical cell. The works of Prof. Robert E. Collin consolidate our understanding of the singularity behavior of the dyadic Green functions. His many communications with me on this subject were very valuable prior to the publication of a book in this field by Prof. J. Van Blade1 [1991]. I am also very grateful to Prof. Donald G. Dudley and Dr. William A. Johnson for their very careful review of my original manuscript. Section 5-8 of Chapter 5 was written as a result of their thoughtful comments. During the preparation of this manuscript I received the most valuable help from Ms. Bonnie Kidd. Her expertise in typing this manuscript was invaluable. The assistance of Dr. Leland Pierce and Ms. Patricia Wolfe are also very much appreciated. I would also like to express my sincere thanks to Prof. Fawwaz T. Ulaby, Director of the Radiation Laboratory at the University of Michigan, for his con- stant encouragement by providing me with the technical support necessary to complete this manuscript. Mr. Dudley Kay, Director of Book Publishing, and Ms. Karen Miller, Production Editor of IEEE Press, have proved to be most efficient and helpful during all stages of the production of this book. Chen-To Tai Ann Arbor, Michigan xiii

Dyadic Green Functions in Electromagnetic Theory

General Theorems and Formulas In this chapter we review some of the important theorems and formulas needed in the subsequent chapters. It is assumed that the reader has had an adequate course in advanced calculus, including vector analysis, Fourier series and integrals, and the theory of complex variables. Our review will contain suf- ficient material so that references to other books will be kept to a minimum. We sacrifice to some extent the mathematic rigor that may be required in a more thorough treatment. For example, we use quite freely the integral representa- tion of the delta function, assuming that an exponential function with imaginary argument is Fourier transformable. Whenever necessary, adequate references1 will be given to strengthen any plausible statement or to remove possible ambiguity. 1-1 VECTOR NOTATIONS AND THE COORDINATE SYSTEMS A vector quantity or a vector function will be denoted by F. A letter with a hat, such as P, is used to denote a unit vector in the direction of the covered letter. In most cases, these letters correspond to the variables - - in a coordinate system. The - scalar product of two vectors is denoted by A . B and the vector product by A x B. The three commonly used systems in this book are 1. Rectangular, or Cartesian, x, y, z 2. Circular cylindrical or simply cylindrical, r, 4, z 3. Spherical, R, 0, 4 '1n the citations in the text, the author's name is used as the identification. If it is a book, either the section number or the pages will be cited, if necessary.

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