Dyadic Green Functions in Electromagnetic Theroy

【实例简介】
Dyadic Green Functions in Electromagnetic Thero
Contents PREFACE ACKNOWLEDGMENTS 1 GENERAL THEOREMS AND FORMULAS 1-1 Vector Notations and the Coordinate systems 1-2Ⅴ ector Anal ysi 1-3 Dyadic analysis 6 4 Fourier transform and hankel transform 1-5 Saddle-Point Method of Integration and Semi-infinitc 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 o(a, a by the Conventional Method and the ohm-Rayleigh Method 25 2-3 Symmetrical Propertics of Green Functions 33 2-4 Free-Space Green Function of the Three-Dimensional Scalar Wave Equati 35 3 ELECTROMAGNETIC THEORY 38 3-1 The Independent and dependent equations and the indefinite and Definite Form of Maxwells equations 38 3-2 Integral Forms of Maxwell,s Equations 3-3 Boundary Conditions 42 3-4 Monochromatically Oscillating Fields in Free Space 3-5 Method of potentials 49 Content Contents ix 4 DYADIC GREEN FUNCTIONS 55 9-3 Radiation from Electric Dipoles in the Presence 4-1 Maxwell, s Equations in Dyadic Form and Dyadic of a half Sheet 174 Green Functions of the Electric and Magnetic Type 55 9-3.1 Longitudinal Electrical Dipole 174 4-2 Free-Space Dyadic Green Functions 9-3.2 Horizontal Electrical Dipole 176 4-3 Classification of Dyadic Green Functions 9-3.3 Vertical Electric Dopole 178 4-4 Symmetrical Properties of Dyadic Green Functions 74 9-4 Radiation from Magnetic Dipoles in the Prcsence 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 Shcct 182 Reciprocity Theorems 90 9.5.1 Longititudinal Slot 183 4-7 Dyadic Green Functions for a Halt Space Bounded 9-5.2 Horizontal slot 184 by a Plane Conducting Surface y2 9-6 Diffraction of a Plane wave by a Half Sheet 187 5 RECTANGULAR WAVEGUIDES 96 g-7 Circular Cylinder and half Sheet 196 5-1 Rectangular Vector Wave Functions 96 10 SPHERES AND PERFECTLY CONDUCTING CONES t98 5-2 The Method of Gm 103 10-1 Eigenfunction Expansion of Free-Space Dyadic 5-3 The Method of Ge 110 reen functions 198 5-4 The Method of GA 114 10-2 An Algebraic Method of Finding Geo without the 5-5 Parallel Plate Waveguide 115 Singular term 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 Cavit 10-5 Perfectly Conducting Conical Structures 22 5-8 The Origin of the Isolated Singular Term in Ge 128 10-6 Cone with a spherical sector 223 6 CYLINDRICAL WAVEGUIDES 133 11 PLANAR STRATIFIED MEDIA 225 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 23 7 CIRCULAR CYLINDER IN FREE SPACE 149 11-5 Eigenfunction Expansions 244 7-1 Cylindrical Vector Wave Functions with Continuous 11-6 A Dielectric slab in air 249 Eigenvalues 149 11-7 Two-Dimensional Fourier Transform of the Dyadic 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 161 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 Gnd 166 12-5 Time-Dependent Field in a Moving Medium 277 9 PERFECTLY CONDUCTING WEDGE AND THE HALF SHEET 169 12-6 Rectangular Waveguide with a Moving Medium 286 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 Thc half Sheet 173 in a moving medium 293 APPENDIXA MATHEMATICAL FORMULAS 29 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 302 B-1 Rectangular Vector Wave F 302 B-2 Cylindrical Vector Wave Functions with Discrete Preface Eigenvalues 304 B-3 Spherical Vector Wave Functions 305 B-4 Conical Vector Wave Functions 306 APPENDIX C EXERCISES 308 REFERENCES 332 NAME INDEX 337 SUBJECT INDEX 339 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 trcatcd; 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, 1973] 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 Grcen functions. Being a solenoidal function, its eigenfunction expansion does not require the use of non solenoidal vector wave functions or Hansens L-functions Stratton, 1941. With the aid of Maxwell-Ampere ec dyadic form, one can find the eigenfung tion expansion of the electrical dyadic Green function, including the previously missing singular term. This method is uscd extensively in the present edition. Several other new features are found in this edition For example, the inte g eP s equations are now derived with the aid of the vect 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 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 Acknowledgments electromagnetic theory 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 clectric 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 Bladel [1991]. I am als ateful 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 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 references will be given to strengthen any plausible statement or to remove possible b 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 i 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, C,y, a 2. Circular cylindrical or simply cylindrical, r, p,2 3. Spherical, R, 8, o i In the citations in the text thc 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. General Theorems and Formulas Chap I ec 1-1 Vector Notations and the Coordinate Systems The spatial variables associated with these systems are shown in Fig. 1-1 Table 1-1 Relarions Between the should be pointed out that the same -variable is used for both the cylindri Unit Vectors in the Rectangular and cal and the spherical systems. The unit vectors belonging to these systems are the Cylindrical Coordinate Systems displayed in Fig 1-2 in two cross-sectional views. The relation between these unit vectors is summarized in Tables 1-1 and 1-2 之 Ds p sin 0 -8in中 cosφ0 0 tabLE 1-2 Relations Between the unit vectors the Rectangular and the Spherical Coordinate Systems y r sin e cosφ sin e sinφ Cos 6 co8cos中cos6sinφ 日 cos o 0 Likewise, the second row gives 6=cos6cos@+cosθsinp-sin日2. (12) The reader can verify for himself or herself that these tables apply equally well Fig 1-1 Three commonly uscd coordinate systems to the transform of the components of a vector; for example, Ae= cos 9 cos oAx+cos 8 sin PAy -sin 8Az (13) R Anothcr coordinate system used in this book deals with an elliptical cylinder One set of variables that can be used in this system is designated by(u, 1, 2).a cross-sectional view of a plane perpendicular to the x-axis is shown in Fi The constant u contours and the constant v contours correspond, respectively to a family of confocal ellipses and a family of confocal hyperbolas. The relations between(a, g) and(al, 17)are e osh i cos ? y=csinh i sin vi, (15 where∝>u>0,2m>υ≥0. Two alternate variables which are used some times in place of(u, v)are defined by E=cosh (16) Fig 1-2 The unit vectors in three commonly used coordinate systems 7=c0s (1.7) To express unit vector i in terms of the unit vectors in the spherical system one uses the coefficients in the first column of Table 1-2, which gives where∞>≥0,1≥n≥-1. Table1-3 contains the transformation co efficients between the unit vectors of the rectangular system and the elliptical =si6csφR+cos6c0-sinφ (1.1) General Theorems and Formulas Chap 1 Sec. 1-2 Vector Analysis Stokes theorem states that for any continuous vector function of position with continuous first derivatives on an open surface s bounded by a contour c: (xF)dS=中Fa(Soke (1.9 It is understood that the direction of the line integral and the direction of dS =霄 follows the right-hand screw rule In addition to these two important theorems, there are several more theo vector analysis, namely: Vfd fds (gradient theorem) (1.10) x t'dv ×Fds( curl theorem, Fig, 1-3 A cross-sectional view of the elliptical coordinate system where n denotes an outward unit normal to the surface S enclosing th TABLE 1-3 Relations Bctwccn the Unit Vectors in the If we let Rectangular System and the Elliptical Cylinder System F=更Vv-yV重, (1.12) y where p and l are two scalar functions of position, then in view of identities (A.11)and(A16)of Al xA sinhu cosh u sin v V.F=V2v-yV2更, (113 cosh u sin v sinh u cos v where v2nb and V2a denote, respectively, the Laplacian of v and It follows from gauss theorem that △=(09-2) (更y2-V2型)V (重Vv-vV)·dS,(1.14) which is designated as the scalar Grcen theorem of the second kind. 1-2 VECTOR ANALYSIS If we let The entire subject of vector analysis consists of three definitions, namely, the F=PxVxQ (1.15) gradient, the divergence, and the curl; a number of identities: and two theorems named after Gauss and Stokes. For convenient reference some of the identities where P and Q are two vector functions, then according to the vector identity and formulas are listed in Appendix A. We will not review here the elementary 4.13 )of Al ppendiX aspects of vector analysis but, rather, will outline the two theorems and a number V·F=(V×P)(V×可)-PVxV×2 (116) of useful lemmas that can be derived from these theorems Gauss theorem states that for any vector function of position Fwith Upon substituting it into Gauss theorem, we obtain the vector Green theorem uous first derivatives throughout a volume V and over the enclosing surface S, of the first kind (×P)·(V×Q)-P dv V·Fdv F·ds( Gauss thcorem) (18) The ring around a surface integral is to emphasize the fact that the surface is a 焦四××Q)。-=n、P×xas1m closed one. The same notation will be applied to a closed line integral where fi denotes the outward unit normal to the surface s 【实例截图】
【核心代码】