Scattering of Waves by Wedges and Cones with Impedance Boundary Conditions: ISMB Series

232 Pages·2013·7.398 MB·English

by  Mikhail A. Lyalinov, Ning Yan Zhu

This book is a systematic and detailed exposition of different analytical techniques
used in studying two of the canonical problems, the wave scattering by wedges or
cones with impedance boundary conditions. It is the first reference on novel, highly
efficient analytical-numerical approaches for wave diffraction by impedance wedges
or cones.
KEY FEATURES
• Development of new approaches which lead to exact (but not explicit) solutions of
key canonical problems like diffraction by an impedance wedge or cone.
• Calculations of the diffraction or excitation coefficients, including their uniform
versions, for the diffracted waves from the edge of the wedge or from the vertex of
the cone.
• Study of the far-field behavior in diffraction by impedance wedges or cones, reflected
waves, space waves from the singular points of the boundary (from edges or tips),
and surface waves.
• Applicability of the reported solution procedures and formulae to existing software
packages designed for solving real-world high-frequency problems encountered in
antenna, wave propagation, and radar cross section.
AUDIENCE
• Researchers in wave phenomena physics.
• Radio, optics and acoustics engineers.
• Applied mathematicians and specialists in mathematical physics.
• Specialists in quantum scattering of many particles.
ABOUT THE AUTHORS
Mikhail A. Lyalinov is a Professor in the Department of Mathematics and Mathematical
Physics at Saint Petersburg University, Russia. He has published more than 50 research
papers on different mathematical aspects of diffraction theory and is co-author of
two monographs. He is a principal organizer of the annual international “Days on
Diffraction” seminars.
Ning Yan Zhu is a Privatdozent at the Institute of Radio Frequency Technology, University
of Stuttgart, Germany. His research includes rigorous numerical techniques and their
applications to antennas and radio wave propagation in complex environments. He
has published 25 journal articles and co-authored one monograph in these fields.
He is also an editorial advisor of the Alpha Science Series on Wave Phenomena
(Oxford, UK).
Lyalinov-5220034 lyal5220034˙fm October18,2012 15:54
Scattering of Waves by Wedges
and Cones with Impedance
Boundary Conditions
Lyalinov-5220034 lyal5220034˙fm October18,2012 15:54
The Mario Boella Series on Electromagnetism in
Information & Communication
TheMarioBoellaseriesofferstextbooksandmonographsinallareasofradioscience,witha
specialemphasisontheapplicationsofelectromagnetismtoinformationandcommunication
technologies. The series is scientifically and financially sponsored by the Istituto Superiore
MarioBoellaaffiliatedwiththePolitecnicodiTorino,Italy,andisscientificallyco-sponsored
by the International Union of Radio Science (URSI). It is named to honor the memory of
ProfessorMarioBoellaofthePolitecnicodiTorino,whowasapioneerinthedevelopmentof
electronicsandtelecommunicationsinItalyforhalfacentury,andaVicePresidentofURSI
from1966to1969.
ScatteringofWavesbyWedgesandConeswithImpedanceBoundaryConditions,
ISMBSeries
MikhailA.LyalinovandNingYanZhu
Thismonographonscatteringofelectromagneticwavesbyimpedancewedgesandconesis
the most comprehensive treatise in existence on a specialized topic that is of great interest
to electrical engineers and mathematical physicists. Written by two international experts on
scattering and diffraction, it constitutes a fundamental and lasting contribution to the field
of applied mathematics. Even though its obvious applications are to radar and to mobile
communications,themathematicalmethodsemployedhereinarealsoapplicabletoproblems
involvingscatteringofacousticandelasticwaves.Thus,thismonographwillbeofinterestto
allresearchersonwavepropagationandwillremainastandardreferencefortheforeseeable
future.
PiergiorgioL.E.Uslenghi–SeriesEditor
Chicago,July2012
Other titles in this series
FundamentalsofWavePhenomena,SecondEdition
byAkiraHiroseandKarlE.Lonngren
Forthcoming titles in this series
TheWiener-HopfMethodinElectromagnetics
byVitoG.DanieleandRodolfoS.Zich
HigherOrderNumericalSolutionTechniquesinElectromagnetics
byRobertoD.GragliaandAndrewF.Peterson
Lyalinov-5220034 lyal5220034˙fm October18,2012 15:54
Scattering of Waves by Wedges
and Cones with Impedance
Boundary Conditions
ISMB Series
Mikhail A. Lyalinov
InstituteofPhysics
St.PetersburgUniversity
St.Petersburg,Russia
Ning Yan Zhu
Institutfu¨rHochfrequenztechnik
Universita¨tStuttgart
Stuttgart,Germany
Edison,NJ
scitechpub.com
Lyalinov-5220034 lyal5220034˙fm October18,2012 15:54
PublishedbySciTechPublishing,animprintoftheIET.
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10 9 8 7 6 5 4 3 2 1
ISBN978-1-61353-003-0(hardback)
ISBN978-1-61353-025-2(PDF)
TypesetinIndiabyMPSLtd
PrintedintheUSAbySheridanBooks,Inc.
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Contents
Preface xi
Introduction 1
Generalandhistoricalremarks 1
Descriptionofthecontent 3
1 Fundamentals 5
1.1 Equationsforacousticandelectromagneticwaves 5
1.1.1 Acousticwaves 5
1.1.2 Electromagneticwaves 7
1.2 Boundaryconditions 9
1.3 Edgeandradiationconditions 11
1.3.1 VicinityoftheedgeandMeixner’scondition 11
1.3.2 OnthebehaviorofsolutionstotheHelmholtzequationinthe
angulardomainasr→0 12
1.3.3 Radiationconditions:Formulationoftheproblem 13
1.3.4 Thelimiting-absorptionprinciple 14
1.4 Integraltransformations 16
1.4.1 Fouriertransformandtheconvolutiontheorem 16
1.4.2 TheSommerfeldintegral 17
1.4.3 Malyuzhinets’stheorem:Sommerfeld–Malyuzhinets(SM)transform 18
1.4.4 Kontorovich–Lebedev(KL)transformandits
connectionwiththeSommerfeldintegral 22
1.4.5 Watson–Besselintegral 24
1.5 Malyuzhinets’ssolutionfortheimpedancewedgediffractionproblem 26
1.5.1 FunctionalequationsfortheMalyuzhinetsproblem 26
1.5.2 ThemultiplicationprincipleandtheauxiliarysolutionΨ (z)
0
tothefunctionalequations(1.104) 28
v
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vi Contents
1.5.3 TheMalyuzhinetsfunctionψΦ(z)anditsbasicproperties 30
1.5.4 Examinationof(d/dz)lnψΦ(z) 30
1.5.5 TheMalyuzhinetsfunctionψΦ(z) 32
1.5.6 CompletionoftheconstructionofΨ (z)andofs(z) 34
0
1.5.7 Far-fieldanalysisoftheexactsolution 35
1.6 TheoryofMalyuzhinetsfunctionalequations
foroneunknownfunction 38
1.6.1 GeneralMalyuzhinetsequations 38
1.6.2 SolutiontothehomogeneousMalyuzhinetsequations 39
1.6.3 SolutiontotheinhomogeneousMalyuzhinetsequations 40
1.6.4 ModifiedFouriertransformandS-integrals 41
1.6.5 ThedirectapplicationofS-integrals 42
2 Diffraction of a skew-incident plane electromagnetic
wave by a wedge with axially anisotropic impedance faces 45
2.1 Introduction 45
2.2 Statementoftheproblemanduniqueness 46
2.2.1 Statementoftheproblem 46
2.2.2 Onuniquenessofasolution 48
2.3 Sommerfeldintegralandfunctionalequations 51
2.4 Afunctionaldifferenceequationofhigherorder 53
2.4.1 Adifferenceequationforonespectrum 53
2.4.2 ThegeneralizedMalyuzhinetsfunctionχΦ(α) 54
2.4.3 Simplifyingthefunctionaldifferenceequationofhigherorder 55
2.5 Second-orderfunctionaldifferenceequationandFredholm
integralequationofthesecondkind 56
2.5.1 Anintegralequivalenttothedifferenceequation 56
±
2.5.2 DeterminingtheconstantsC 57
1(cid:6)
2.5.3 Fredholmintegralequationofthesecondkind 58
2.6 Uniformasymptoticsolution 58
2.6.1 Polesandresidues 58
2.6.2 First-orderuniformasymptotics 60
2.7 Numericalresults 62
2.7.1 Numericalcomputationofthespectra 62
2.7.2 Examples 63
2.8 Appendix:ComputationofthegeneralizedMalyuzhinetsfunction 64
2.8.1 Numericalintegration 64
2.8.2 Seriesrepresentation 67
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Contents vii
3 Scattering of waves from an electric dipole
over an impedance wedge 69
3.1 Formulationoftheproblemandplane-waveexpansion
oftheincidentfield 69
3.1.1 Statementoftheproblem 69
3.1.2 TheHertzvectorandplane-waveexpansionoftheincidentfield 71
3.2 Theintegralrepresentationofthetotalfield 74
3.2.1 Integralformulation 74
3.2.2 FormulationoftheproblemforU(r,ϕ,α,β) 74
3.2.3 Representationforthespectralfunctions 76
3.3 Deformationofthecontoursofintegrationandthe
geometrical-optics(GO)field 77
3.3.1 Saddlepoints,polarsingularities,andresidues 77
3.3.2 Branchcutsforauxiliaryangles 79
3.3.3 Thegeometrical-opticsfield 80
3.4 Thediffractedwavefromtheedgeofthewedge 82
3.4.1 Nonuniformexpression 82
3.4.2 TheUATformulation 83
3.5 Expressionsforsurfacewaves 85
3.5.1 Surfacewavesexciteddirectlybythedipole 85
3.5.2 Surfacewavesexcitedattheedgebyanincidentspacewave 86
3.6 Numericalresults 87
3.7 Appendices 89
3.7.1 AppendixA.Multidimensionalsaddle-pointmethod 89
3.7.2 AppendixB.Thereciprocityprinciple 92
4 Diffraction of a TM surface wave by an angular
break of an impedance sheet 95
4.1 Formulationoftheproblem 95
4.2 Functionalequationsandreductiontointegralequations 97
4.2.1 Reductiontoasecond-orderfunctionalequation 98
4.2.2 Anintegralequationofthesecondkind 100
4.3 Analyticcontinuationofthespectralfunctionsand
scatteringdiagram 101
4.3.1 Scatteringdiagram 102
4.3.2 Reflectedandtransmittedsurfacewaves 103
4.4 Discussionofuniqueness 104
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viii Contents
5 Acoustic scattering of a plane wave by a circular
impedance cone 109
5.1 Formulationoftheproblemanduniqueness 109
5.1.1 Formulationoftheproblem 109
5.1.2 Onuniquenessoftheclassicalsolution 111
5.2 Kontorovich–Lebedev(KL)transformandincomplete
separationofvariables 113
5.2.1 Integralrepresentationofthesolution 113
5.2.2 Formulationoftheproblemforthespectralfunctionuν 114
5.3 Theboundaryvalueproblemforthespectralfunctionu (ω,ω ) 117
ν 0
5.3.1 Separationoftheangularvariablesforthecircularcone 119
5.3.2 Studyoftheintegralequationfor R(ν,n) 120
5.4 DiffractioncoefficientintheoasisMforanarrowcone 122
5.4.1 Problemsfortheleadingtermsandforthefirstcorrections 124
5.4.2 CalculationofV and B 126
1 2j
5.4.3 Basicformulaforthediffractioncoefficientofthe
sphericalwavefromthevertexofanarrowcone 128
5.5 NumericalcalculationofthediffractioncoefficientintheoasisM 130
5.5.1 Numericalaspects 130
5.5.2 Aperturbationseriesfor|η|(cid:3)1 131
5.5.3 Examples 132
5.6 Sommerfeld–Malyuzhinetstransformandanalyticcontinuation 133
5.6.1 AnalyticpropertiesofΦ˜(α,ω,ω )andΦ(α,ω,ω ) 135
0 0
5.6.2 ProblemsfortheSommerfeldtransformants 136
5.6.3 Thesingularitycorrespondingtothewavereflected
fromtheconicalsurface 137
5.7 Thereflectedwave 139
5.8 Scatteringdiagramofthesphericalwavefromthevertex 140
5.9 Surfacewaveataxialincidence 142
5.9.1 Raysolutionforthesurfacewave 142
5.9.2 SingularitiesoftheSommerfeldtransformants
correspondingtothesurfacewave 143
5.9.3 Asymptoticevaluationofthesurfacewave 145
5.10 Uniformasymptoticsofthefarfieldandthe
paraboliccylinderfunctions 147
5.11 Appendices 150
5.11.1 AppendixA 150
5.11.2 AppendixB.Reductionofintegrals 151
5.11.3 AppendixC.DerivationoftheconstantC 153
0
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Contents ix
6 Electromagnetic wave scattering by a circular impedance cone 155
6.1 FormulationandreductiontotheproblemfortheDebyepotentials 155
6.1.1 Thefar-fieldpattern 158
6.1.2 TheDebyepotentials 159
6.1.3 BoundaryconditionsfortheDebyepotentials 161
6.2 Kontorovich–Lebedev(KL)integralsandspectralfunctions 161
6.2.1 KLintegralrepresentations 161
6.2.2 Propertiesofthespectralfunctions 163
6.2.3 Boundaryconditionsforthespectralfunctions 164
6.2.4 Verificationoftheboundaryandotherconditions 166
6.2.5 Diffractioncoefficients 168
6.3 Separationofangularvariablesandreductionto
functional-difference(FD)equations 170
6.4 FredholmintegralequationsfortheFouriercoefficients 172
6.4.1 Reductiontointegralequations 172
6.4.2 CommentsontheFredholmpropertyandunique
solvabilityoftheintegralequations 175
6.5 ElectromagneticdiffractioncoefficientsinM’andnumericalresults 175
6.5.1 NumericalSolution 176
6.5.2 Numericalexamples 177
6.6 SommerfeldandWatson–Bessel(WB)integralrepresentations 179
6.6.1 Sommerfeldintegralrepresentations 180
6.6.2 RegularitydomainsfortheSommerfeldtransformants 182
6.6.3 DiffractioncoefficientsandSommerefeldtransformants 183
6.7 Thediffractioncoefficientsoutsidetheoasisasω ∈ M” 185
6.8 ProblemsfortheSommerfeldtransformantsand
somecomplexsingularities 187
6.8.1 ProblemsfortheSommerfeldtransformants 187
6.8.2 LocalbehavioroftheSommerfeldtransformants
nearcomplexsingularities 189
6.9 AsymptoticsoftheSommerfeldintegralsandthe
electromagneticsurfacewaves 191
6.9.1 DerivationofthefunctionalsC (n) 193
0u
6.9.2 Somecommentsontheasymptoticsuniformwithrespect
tothedirectionofobservation 195
7 Epilogue 197
References 199
Index 213