Stationary Diffraction by Wedges: Method of Automorphic Functions on Complex Characteristics

157 Pages·2019·2.757 MB·English

by  Alexander Komech, Anatoli Merzon

Lecture Notes in Mathematics 2249
Alexander Komech
Anatoli Merzon
Stationary
Diffraction
by Wedges
Method of Automorphic Functions on
Complex Characteristics
Lecture Notes in Mathematics 2249
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryEditors:
KarinBaur,Leeds
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
AnnetteHuber,Freiburg
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Cambridge
AngelaKunoth,Cologne
ArianeMézard,Paris
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
GabrieleVezzosi,Firenze
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Alexander Komech (cid:129) Anatoli Merzon
Stationary Diffraction
by Wedges
Method of Automorphic Functions
on Complex Characteristics
123
AlexanderKomech AnatoliMerzon
FacultyofMathematics InstitutodeFisicayMatematicas
UniversityofVienna UniversidadMichoacanadeSanNicolasde
Vienna,Austria Hidalgo
Morelia
Michoacán,Mexico
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-030-26698-1 ISBN978-3-030-26699-8 (eBook)
https://doi.org/10.1007/978-3-030-26699-8
MathematicsSubjectClassification(2010):Primary:35J25,78A45;Secondary:35Q60
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To theblessed memoryof NinaIlina
Preface
We present a complete solution to the classical problem of stationary diffraction
bywedgeswithgeneralboundaryconditions(b.c.).FortheDirichletandNeumann
b.c.,thesolutionwasfoundbySommerfeldin1896andfortheimpedanceb.c.(or
LeontovichandRobinb.c.)byMalyuzhinetzin1958.Ourapproachreliesonanovel
“methodofautomorphicfunctions(MAF)oncomplexcharacteristics”whichgives
allsolutionstoboundaryproblemsforsecond-orderellipticoperatorswithgeneral
boundaryconditions.Thismethodisalsoapplicabletotheproblemsofguidedwater
wavesonaslopingbeach,scatteringofseismicwaves, etc.
This methodwas introducedby one of the authorsin 1973 for angles Φ < π,
andwasextendedin1992–2007toanglesΦ > π bybothauthorsincollaboration.
ThemethodreliesonthecomplexFourier-Laplacetransformintwovariableswhich
inturnleadstoasystemofalgebraicequationsontheRiemannsurfaceofcomplex
characteristics.Thissystemisreducedtoonealgebraicequationwithtwounknown
functionsontheRiemannsurface.Wereducethisundeterminedalgebraicequation
to the Riemann–Hilbert problem on the Riemann surface applying Malyshev’s
method of automorphicfunctions. This reduction is the key step of our approach.
Finally, the Riemann–Hilbertproblemis solved in quadratures.This method gen-
eralizestheMalyuzhinetzapproachintroducedintheframeworkoftheimpedance
boundarycondition.
Ourpresentationcontainsmanyimportantdetailsandresults,whichwepublish
hereforthefirsttime.Allproofsandconstructionsareconsiderablystreamlinedand
simplified.
We also outline the creation of the diffraction theory by Fresnel, Kirchhoff,
Poincaré,andSommerfeldandsurveysubsequentresultsbySobolev,Malyuzhinetz,
Keller, Maz’ya, Grisvard, and others on the diffraction by wedges and on related
problemsinangulardomains.
Vienna,Austria AlexanderKomech
Morelia,Mexico AnatoliMerzon
vii
Contents
1 Introduction................................................................. 1
1.1 EarlyTheoryofDiffraction.......................................... 1
1.2 DiffractionbyWedges ............................................... 2
1.3 Method of Automorphic Functions on Complex
Characteristics(MAF) ............................................... 3
1.4 ApplicationsoftheMAFMethod................................... 4
1.5 GeneralSchemeoftheMAFMethod............................... 5
1.6 DevelopmentoftheMAFMethod .................................. 9
1.7 Comments ............................................................ 10
1.8 PlanoftheBook...................................................... 10
PartI SurveyofDiffractionTheory
2 TheEarlyTheoryofDiffraction.......................................... 15
2.1 TheGrimaldiObservations.......................................... 15
2.2 TheHuygensPrinciple............................................... 16
2.3 TheYoungTheoryoftheInterferenceandDiffraction............. 16
3 Fresnel–KirchhoffDiffractionTheory................................... 19
3.1 FresnelTheoryofDiffraction........................................ 19
3.2 TheKirchhoffTheoryofDiffraction................................ 22
3.2.1 TheKirchhoffApproximation............................... 24
3.2.2 TheFraunhoferDiffraction.................................. 25
3.3 DiffractionbyaHalf-PlaneintheFresnel–KirchhoffTheory ..... 25
3.4 TheFraunhofer/FresnelLimit:DiffractionofPlaneWaves........ 30
3.5 GeometricalOpticsandDiffraction................................. 33
4 StationaryandTime-DependentDiffraction............................ 37
4.1 Time-DependentDiffractionandLimitingAmplitude
Principle .............................................................. 37
4.2 StationaryDiffractionTheory ....................................... 40
4.2.1 TheLimitingAbsorptionPrinciple ......................... 41
4.2.2 TheSommerfeldRadiationCondition...................... 41
ix
x Contents
5 TheSommerfeldTheoryofDiffractionbyHalf-Plane................. 43
5.1 StationaryDiffractionbytheHalf-Plane............................ 43
5.2 ReflectionsontheRiemannSurface................................. 45
5.3 IntegralRepresentationforBranchingSolutions................... 45
5.3.1 TheMaxwellRepresentationforSphericalFunctions..... 46
5.3.2 TheSommerfeldLimitofSphericalFunctions............. 47
5.3.3 TheSommerfeldIntegralRepresentation................... 48
5.4 InstructiveExamples................................................. 50
5.4.1 RationalDensity ............................................. 50
5.4.2 BranchingDensities.......................................... 52
5.5 DiffractionbytheHalf-Plane........................................ 52
5.6 TheDiffractedWave ................................................. 54
5.7 ExpressionviaFresnelIntegrals..................................... 57
5.8 AgreementwiththeFresnel-KirchhoffApproximation............ 58
5.9 SelectionRulesfortheSommerfeldSolution....................... 61
6 DiffractionbyWedgeAfterSommerfeld’sArticle...................... 63
6.1 StationaryProblemsinAngles ...................................... 63
6.2 Time-DependentDiffractionbyWedges............................ 66
PartII Method of Automorphic Functions on Complex
Characteristics
7 StationaryBoundaryValueProblemsinConvexAngles............... 71
8 ExtensiontothePlane...................................................... 77
8.1 RegularSolutions .................................................... 77
8.2 DistributionalSolutions.............................................. 79
9 BoundaryConditionsviatheCauchyData.............................. 85
10 ConnectionEquationontheRiemannSurface.......................... 89
11 OnEquivalenceoftheReduction......................................... 93
11.1 DistributionalSolutions.............................................. 93
11.2 RegularSolutions .................................................... 93
12 UndeterminedAlgebraicEquationsontheRiemannSurface......... 97
13 AutomorphicFunctionsontheRiemannSurface....................... 99
14 FunctionalEquationwithaShift ......................................... 101
15 LiftingtotheUniversalCovering......................................... 105
16 TheRiemann-HilbertProblemontheRiemannSurface .............. 111
17 TheFactorization........................................................... 117
17.1 EllipticityandBoundBelow......................................... 117
17.2 TheEdge-PointValues............................................... 118
17.3 EquatingEdge-PointValues......................................... 119
Contents xi
17.4 UnwindingtheSymbol .............................................. 120
17.5 AsymptoticsoftheFactorization.................................... 122
18 TheSaltusProblemandFinalFormula.................................. 125
18.1 TheSaltusProblem .................................................. 125
18.2 TheFinalFormula.................................................... 126
19 TheReconstructionofSolutionandtheFredholmness ................ 129
19.1 ReconstructionofDistributionalSolutions ......................... 129
19.2 ReconstructionofRegularSolutions................................ 131
20 ExtensionoftheMethodtoNon-convexAngle.......................... 139
20.1 ConnectionEquationforNon-convexAngle ....................... 140
20.2 IntegralConnectionEquationontheRiemannSurface ............ 141
20.3 LiftingontotheUniversalCovering................................. 144
20.4 TheCauchyKernelontheRiemannSurface ....................... 146
20.5 ReductiontotheRiemann–HilbertProblem........................ 148
21 Comments................................................................... 149
A SobolevSpacesontheHalf-Line.......................................... 151
References......................................................................... 153
Index............................................................................... 161