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

263 Pages·1994·14.28 MB·English
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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 ~INTERSCIENCE AJOHNWILEY&SONS, INC.,PUBLICATION ANOTETOTHEREADER Thisbookhasbeen electronicallyreproducedfrom digitalinformationstoredatJohnWiley&Sons)Inc. Wearepleasedthattheuseofthisnewtechnology willenableustokeepworksofenduringscholarly valueinprintaslongasthereisareasonabledemand forthem. Thecontentofthisbookisidenticalto previousprintings. EditorialBoard WilliamPerkins,EditorinChief R.S.Blicq 1.D.Irwin J.M.F.Moura M.Eden S.V.Kartalopoulos I.Peden D.M.Etter P.Laplante E.Sanchez-Sinencio G.F.Hoffnagle A.J.Laub L.Shaw R.F.Hoyt M.Lightner D.J.Wells DudleyR.Kay,DirectorofBookPublishing CarrieBriggs,AdministrativeAssistant LisaSmey-Mizrahi,ReviewCoordinator ValerieZaborski,ProductionEditor IEEEAntennasandPropagationSociety,Sponsor AP-SLiaisontoIEEEPRESS RobertJ.Mailloux RomeLaboratory/ERI ©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

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