Mathematical Foundations for Electromagnetic Theory (IEEE Press Series on Electromagnetic Wave Theory)

263 Pages·1994·14.28 MB·English

by  Donald G. Dudley

Mathematical Foundations
for Electromagnetic Theory
IEEE SeriesonElectromagnetic WaveTheory
TheIEEE SeriesonElectromagneticWaveTheoryconsistsofnewtitlesas
wellasreprintingsandrevisionsofrecognizedclassicsthatmaintainlong-term
archivalsignificanceinelectromagneticwavesandapplications.
Series Editor
DonaldG.Dudley
UniversityofArizona
Advisory Board
RobertE.Collin
CaseWesternReserveUniversity
AkiraIshimaru
UniversityofWashington
D.S.Jones
UniversityofDundee
AssociateEditors
ElectromagneticTheory,Scattering,andDiffraction
EhudHeyman
Tel-AvivUniversity
DifferentialEquationMethods
AndreasC.Cangellaris
UniversityofArizona
IntegralEquationMethods
DonaldR.Wilton
UniversityofHouston
Antennas,Propagation,andMicrowaves
DavidR.Jackson
UniversityofHouston
SeriesBooksPublished
Chew,W.C.,WavesandFieldsinlnhomogenousMedia
Christopoulos,C.,TheTransmission-LineModelingMethod: TLM
Collin,R.E.,FieldTheoryofGuidedWaves, SecondEdition
Dudley,D.G.,MathematicalFoundationsforElectromagneticTheory
Elliott,R.S.,Electromagnetics: History, Theory, andApplications
Felsen,L.B.,andMarcuvitz,N.,RadiationandScatteringofWaves
Harrington,R.F.,FieldComputationbyMomentMethods
Jones,D.S.,MethodsinElectromagneticWavePropagation,SecondEdition
Lindell,I.V.,MethodsforElectromagneticFieldAnalysis
Tai,C.T.,GeneralizedVectorandDyadicAnalysis:
AppliedMathematicsinFieldTheory
Tai,C.T.,DyadicGreenFunctionsinElectromagneticTheory, SecondEdition
VanBladel,J.,SingularElectromagneticFieldsandSources
Wait,J.,ElectromagneticWavesinStratifiedMedia
Mathematical Foundations
for Electromagnetic Theory
Donald G. Dudley
University of Arizona, Tucson
.IEEE
TheInstituteofElectricalandElectronicsEngineers,Inc.,NewYork
ffiWILEY
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©1994bytheInstituteofElectricalandElectronicsEngineers,Inc.
345East47thStreet,NewYork,NY10017-2394
Allrightsreserved. Nopartofthisbookmaybereproducedinanyform,
normayitbestoredinaretrievalsystemortransmittedinanyform,
withoutwrittenpermissionfromthepublisher.
IEEEISBN0·7803·1022·5
or
Library CongressCataloging-in-PublicationData
Dudley,DonaldG.
Mathematicalfoundationsforelectromagnetictheory/ DonaldG.
Dudley.
p. em.- (IEEEelectromagneticwavesseries)
"IEEEAntennasandPropagationSociety,sponsor."
Includesbibliographicalreferencesandindex.
ISBN0-7803-1022-5
I. Electromagnetictheory-Mathematics. I. IEEEAntennasand
PropagationSociety." II. Title. III. Series.
QC670.D79 1994 94-3159
530.I'41'0151--dc20 CIP
S.
Dedicated to Professor Robert Elliott
Contents
Preface ix
1 Linear Analysis 1
1.1 Introduction 1
1.2 LinearSpace 1
1.3 Inner ProductSpace 7
1.4 Normed LinearSpace 10
1.5 Hilbert Space 15
1.6 BestApproximation 19
1.7 Operators in Hilbert Space 24
1.8 Method of Moments 33
A.l Appendix-ProofofProjectionTheorem 36
Problems 38
References 43
2 The Green's Function Method 45
2.1 Introduction 45
2.2 Delta Function 45
2.3 Sturm-LiouvilleOperatorTheory 50
2.4 Sturm-LiouvilleProblem of theFirst Kind 53
2.5 Sturm-LiouvilleProblem of theSecond Kind 68
2.6 Sturm-LiouvilleProblem of theThird Kind 77
Problems 94
References 97
vii
viii Contents
3 The Spectral Representation Method 99
3.1 Introduction 99
3.2 EigenfunctionsandEigenvalues 99
3.3 SpectralRepresentationsforSLPI andSLP2 106
3.4 SpectralRepresentationsforSLP3 111
3.5 Green's FunctionsandSpectralRepresentations 134
Problems 135
References 138
4 Electromagnetic Sources 139
4.1 Introduction 139
4.2 DeltaFunctionTransformations 139
4.3 Time-HarmonicRepresentations 143
4.4 TheElectromagneticModel 144
4.5 TheSheetCurrentSource 147
4.6 TheLineSource 153
4.7 TheCylindricalShellSource 166
4.8 TheRingSource 168
4.9 ThePointSource 172
Problems 178
References 179
5 Electromagnetic Boundary Value Problems 181
5.1 Introduction 181
5.2 SLPI ExtensiontoThreeDimensions 182
5.3 SLPI inTwoDimensions 191
5.4 SLP2andSLP3ExtensiontoThreeDimensions 194
5.5 TheParallelPlateWaveguide 198
5.6 IrisinParallelPlateWaveguide 206
5.7 ApertureDiffraction 216
5.8 ScatteringbyaPerfectlyConductingCylinder 226
5.9 PerfectlyConductingCircularCylinder 233
5.10 DyadicGreen's Functions 242
Problems 242
References 244
Index 246
Preface
Thisbookiswrittenfortheseriousstudentofelectromagnetictheory.
Itisaprincipal product ofmyexperienceoverthepast25yearsinteracting
withgraduatestudents inelectromagneticsandapplied mathematicsatthe
University of Arizona.
A large volume of literature has appeared since the latter days of
World War II, written by researchers expanding the basic principles of
electromagnetic theory and applying the electromagnetic model to many
important practical problems. In spite of widespread and continuing in
terest in electromagnetics, the underlying mathematical principles used
freely throughoutgraduate electromagnetictextshavenotbeen systemati
callypresentedinthetextsaspreambles. This isincontrasttothesituation
regarding undergraduateelectromagnetictexts,mostofwhichcontain pre
liminary treatments offundamental applied mathematicalprinciples, such
as vector analysis, complex arithmetic, and phasors. It is my beliefthat
there should be a graduate electromagnetic theory text with linear spaces,
Green's functions, and spectral expansions as mathematical cornerstones.
Suchatextshould allowthereader access tothemathematicsandtheelec
tromagneticapplicationswithout thenecessity forconsultingawiderange
ofmathematicalbooks written atavarietyoflevels. This book isaneffort
tobringthepowerofthemathematicstobearonelectromagneticproblems
in asingle text.
Since the mastery of the foundations for electromagnetics provided
in this book can involve a considerable investment of time, I should like
to indicate some ofthe potential rewards. When the student firstbegins a
lx
x Preface
studyofelectromagnetictheory atthe graduate level,he/she isconfronted
with a large array of series expansions and transforms with which to re
duce the differential equations and boundary conditions in a wide variety
ofcanonical problems inCartesian, cylindrical,andspherical coordinates.
Often, itseems tothestudent thatexperienceistheonly way todetermine
specifically which expansions or transforms to use in agivenproblem. In
addition, convergence properties seem quite mysterious. These issues can
be approached on a firmmathematical base through the foundations pro
vided in this book. Indeed, the reader will find that different differential
operators withtheirassociatedboundary conditionsleadtospecificexpan
sionsandtransforms thatare"natural"inaconcretemathematicalsensefor
the problem being considered. My experience with graduate students has
beenthatmasteryofthefoundations allowsthemtoappreciate whycertain
expansions and transforms are used in the study of canonical problems.
Then, what is potentially more important, the foundations allow them to
begin the more difficulttask offormulating and solving problems on their
own.
Ifirstbecame interestedinGreen'sfunctions andspectral representa
tionsduring mygraduate studies atUCLA inthe 1960s. Iwasparticularly
influenced by the treatment of the spectral representations of the delta
function by Bernard Friedman [1], whose book at that time formed the
cornerstoneof theApplied MathematicsProgram atUCLA intheCollege
ofEngineering. Subsequently, examples ofspectral representationsbegan
appearing in texts on electromagnetic theory, such as [2]-[4], and, more
recently, [5]. However, no text specifically devoted to Green's functions
and spectral expansions andtheir application toelectromagneticproblems
hasbeen forthcoming.
Thematerial inthisbookforms atwo-semestersequenceforgraduate
students at the University of Arizona. The firstthree chapters contain the
mathematical foundations, and are covered in acourse offered every year
to electrical engineering and applied mathematics graduate students with
a wide range of interests. Indeed, the first three chapters in this book
couldbestudiedbyappliedmathematicians,physicists,andengineerswith
no particular interest in the electromagnetic applications. The fourth and
fifth chapters are concerned with the electromagnetics, and are covered
in a course on advanced electromagnetic theory, offered biennially. In
this book, I have presumed that the reader has a working knowledge of
complex variables. In addition, in the last two chapters, I have assumed
thatthereader hasstudied anintroductorytreatmentofelectromagneticsat
thegraduate level,ascanbefound, forexample, inthetextsbyHarrington