Rays, waves, and scattering: Topics in classical mathematical physics

606 Pages·2017·3.864 MB·English

by  Adam J.A

Rays, Waves, and Scattering
Topics in Classical Mathematical Physics
John A. Adam
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright(cid:2)c 2017byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
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ISBN978-0-691-14837-3
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Whichisthewaytotheplacewherelightisscatteredandtheeast
windisspreadacrosstheearth?
Job38:24
Contents
Preface xvii
Acknowledgments xxiii
Chapter1 Introduction 1
1.1 TheRainbowDirectory 3
1.1.1 TheMultifacetedRainbow 3
1.2 AMathematicalTasteofThingstoCome 5
1.2.1 Rays 5
1.2.2 Waves 6
1.2.3 Scattering(Classical) 7
1.2.4 Scattering(Semiclassical) 9
1.2.5 CausticsandDiffractionCatastrophes 11
PARTI. RAYS 15
Chapter2 Introductiontothe“Physics”ofRays 17
2.1 WhatIsaRay? 17
2.1.1 SomeMathematicalDefinitions 18
2.1.2 GeometricWavefronts 19
2.1.3 Fermat’sPrinciple 21
2.1.4 TheIntensityLaw 21
2.1.5 HeuristicDerivationofSnell’sLaws 23
2.1.6 Generalization 24
2.2 GeometricandOtherProofsofSnell’sLawsofReflection
andRefraction 25
2.2.1 TheLawofReflection 25
2.2.2 TheLawofRefraction 26
2.2.3 AWave-TheoreticProof 28
2.2.4 AnAlgebraicProof 29
Chapter3 IntroductiontotheMathematicsofRays 33
3.1 Background 33
3.2 TheMethodofCharacteristics 34
viii • CONTENTS
3.3 IntroductiontoHamilton-JacobiTheory 37
3.3.1 Hamilton’sPrinciple 39
3.3.2 RaysandCharacteristics 39
3.3.3 TheOpticalPathLengthRevisited 43
3.4 RayDifferentialGeometryandtheEikonalEquationAgain 46
3.4.1 TheMirageTheoremforHorizontallyStratifiedMedia 49
3.4.2 AReturntoSphericallySymmetricMedia:
n(r)Continuous 51
3.5 DispersionRelations: AWave-RayConnection 54
3.5.1 FourierTransformsandDispersionRelations 55
3.5.2 TheBottomLine 56
3.5.3 ApplicationstoAtmosphericWaves 61
3.6 GeneralSolutionoftheLinearWaveEquation:
SomeAsymptotics 64
3.6.1 StationaryPhase 64
3.6.2 AsymptoticsforOscillatorySources: Wavenumber
Surfaces 65
3.7 RaysandWavesinaSlowlyVaryingEnvironment 70
3.7.1 SomeConsequences 71
3.7.2 WavepacketsandtheGroupSpeedRevisited 75
Chapter4 RayOptics: TheClassicalRainbow 76
4.1 PhysicalFeaturesandHistoricalDetails: ASummary 76
4.2 RayTheoryoftheRainbow: ElementaryMathematical
Considerations 78
4.2.1 SomeNumericalValues 84
4.2.2 PolarizationoftheRainbow 85
4.2.3 TheDivergenceProblem 87
4.3 RelatedTopicsinMeteorologicalOptics 89
4.3.1 TheGlory 89
4.3.2 Coronas(Simplified) 92
4.3.3 RayleighScattering—aDimensionalAnalysisArgument 93
Chapter5 AnImprovementoverRayOptics: Airy’sRainbow 95
5.1 TheAiryApproximation 95
5.1.1 SomeRayPrerequisites 95
5.1.2 TheAiryWavefront 100
5.1.3 HowAreColorsDistributedintheAiryRainbow? 104
5.1.4 TheAiryWavefront: ADerivationforArbitraryp 105
Chapter6 DiffractionCatastrophes 113
6.1 BasicGeometryoftheFoldandCuspCatastrophes 114
6.1.1 TheFold 114
6.1.2 TheCusp 115
6.2 ABetterApproximation 122
6.2.1 TheFresnelIntegrals 124
CONTENTS • ix
6.3 TheFoldDiffractionCatastrophe 126
6.3.1 TheRainbowasaFoldCatastrophe 128
6.4 Caustics: TheAiryIntegralintheComplexPlane 130
6.4.1 TheNatureofAi(X) 133
Chapter7 IntroductiontotheWKB(J)Approximation: AllThingsAiry 137
7.1 Overview 137
7.1.1 EliminationoftheFirstDerivativeTerm 139
7.1.2 TheLiouvilleTransformation 141
7.1.3 TheOne-DimensionalSchrödingerEquation 143
7.1.4 PhysicalInterpretationoftheWKB(J)Approximation 144
7.1.5 TheWKB(J)ConnectionFormulas 145
7.1.6 ApplicationtoaPotentialWell 148
7.2 TechnicalDetails 149
7.3 MatchingAcrossaTurningPoint 152
7.4 ALittleMoreaboutAiryFunctions 153
7.4.1 RelationtoBesselFunctions 154
7.4.2 TheAiryIntegralandRelatedTopics 156
7.4.3 RelatedIntegrals 159
Chapter8 IslandRays 162
8.1 StraightandParallelDepthContours 163
8.1.1 PlaneWaveIncidentonaRidge 164
8.1.2 WaveTrappingonaRidge 166
8.2 CircularDepthContours 167
8.3 ConstantPhaseLines 169
8.3.1 Case1 169
8.3.2 Case2 170
8.3.3 Case3 170
8.4 WavesandCurrents 170
Chapter9 SeismicRays 173
9.1 SeismicRayEquations 173
9.2 RayPropagationinaSphericalEarth 175
9.2.1 AHorizontallyStratifiedEarth 178
9.2.2 TheWiechert-HerglotzInversion 179
9.2.3 Further Properties of X in the Horizontally Stratified
Case 181
PARTII WAVES 187
Chapter10 ElasticWaves 189
10.1 BasicNotation 190
10.2 PlaneWaveSolutions 193
x • CONTENTS
10.3 Surfacewaves 195
10.4 LoveWaves 198
Chapter11 SurfaceGravityWaves 200
11.1 TheBasicFluidEquations 201
11.2 TheDispersionRelation 203
11.2.1 DeepWaterWaves 203
11.2.2 ShallowWaterWaves 204
11.2.3 Instability 205
11.2.4 GroupSpeedAgain 210
11.2.5 Wavepackets 212
11.3 ShipWaves 214
11.3.1 HowDoesDispersionAffecttheWavePattern
ProducedbyaMovingObject? 214
11.3.2 Whitham’sShipWaveAnalysis 218
11.3.3 AGeometricApproachtoShipWavesandWakes 221
11.3.4 ShipWavesinShallowWater 227
11.4 ADiscreteApproach 229
11.4.1 LongWaves 229
11.4.2 ShortWaves 230
11.5 FurtherAnalysisforSurfaceGravityWaves 231
Chapter12 OceanAcoustics 237
12.1 OceanAcousticWaveguides 237
12.1.1 TheGoverningEquation 237
12.1.2 LowVelocityCentralLayer 239
12.1.3 LeakyModes 240
12.2 One-DimensionalWavesinanInhomogeneousMedium 241
12.2.1 AnEigenfunctionExpansion 242
12.2.2 Poles 245
12.3 ModelforaStratifiedFluid: CylindricalGeometry 247
12.4 TheSech-SquaredPotentialWell 250
12.4.1 PositiveEnergyStates 250
12.4.2 BoundStates 253
Chapter13 Tsunamis 255
13.1 MathematicalModelofTsunamiPropagation(TransientWaves) 255
13.2 TheBoundary-ValueProblem 257
13.3 SpecialCaseI:TsunamiGenerationbyaDisplacement
oftheFreeSurface 258
13.3.1 ADigression: SurfaceWavesonDeepWater(Again) 263
13.3.2 HowFastDoestheWaveEnergyPropagate? 265
13.3.3 KinematicsAgain 267
13.4 LeadingWavesDuetoaTransientDisturbance 268
13.5 SpecialCase2: TsunamiGenerationbyaDisplacement
oftheSeafloor 270
CONTENTS • xi
Chapter14 AtmosphericWaves 273
14.1 GoverningLinearizedEquations 274
14.2 AMathematicalModelofLee/MountainWavesover
anIsolatedMountainRidge 285
14.2.1 BasicEquationsandSolutions 286
14.2.2 AnIsolatedRidge 288
14.2.3 TrappedLeeWaves 290
14.3 BillowClouds,WindShear,andHoward’sSemicircleTheorem 292
14.4 TheTaylor-GoldsteinEquation 296
PARTIII CLASSICALSCATTERING 299
Chapter15 TheClassicalConnection 301
15.1 Lagrangians,Action,andHamiltonians 301
15.2 TheClassicalWaveEquation 304
15.3 ClassicalScattering: ScatteringAnglesandCrossSections 308
15.3.1 Overview 308
15.3.2 TheClassicalInverseScatteringProblem 313
Chapter16 GravitationalScattering 316
16.1 PlanetaryOrbits: ScatteringbyaGravitationalField 317
16.1.1 RepulsiveCase: k >0 318
16.1.2 AttractiveCase: k <0 319
16.1.3 TheOrbits 319
16.2 TheHamilton-JacobiEquationforaCentralPotential 325
16.2.1 TheKeplerProblemRevisited 326
16.2.2 Generalizations 327
16.2.3 HardSphereScattering 328
16.2.4 RutherfordScattering 329
Chapter17 ScatteringofSurfaceGravityWavesbyIslands,Reefs,
andBarriers 332
17.1 TrappedWaves 333
17.2 TheScatteringMatrixS(α) 334
17.3 TrappedModes: ImaginaryPolesofS(α) 337
17.4 PropertiesofS(α)forα ∈R 338
17.5 SubmergedCircularIslands 340
17.6 EdgeWavesonaSlopingBeach 342
17.6.1 One-DimensionalEdgeWavesonaConstantSlope 345
17.6.2 WaveAmplicationbyaSlopingBeach 345
xii • CONTENTS
Chapter18 AcousticScattering 348
18.1 ScatteringbyaCylinder 350
18.2 Time-AveragedEnergyFlux: ALittleBitofPhysics 352
18.3 TheImpenetrableSphere 354
18.3.1 Introduction: SphericallySymmetricGeometry 354
18.3.2 TheScatteringAmplitudeRevisited 356
18.3.3 TheOpticalTheorem 358
18.3.4 TheSommerfeldRadiationCondition 358
18.4 RigidSphere: SmallkaApproximation 359
18.5 AcousticRadiationfromaRigidPulsatingSphere 361
18.6 TheSoundofMountainStreams 364
18.6.1 BubbleCollapse 367
18.6.2 PlayingwithMathematicalBubbles 369
Chapter19 ElectromagneticScattering: TheMieSolution 371
19.1 Maxwell’sEquationsofElectromagneticTheory 378
19.2 TheVectorHelmholtzEquationforElectromagneticWaves 379
19.3 TheLorentz-Miesolution 383
19.3.1 ConstructionoftheSolution 386
19.3.2 TheRayleighScatteringLimit: ACondensedDerivation 392
19.3.3 TheRadiationFieldGeneratedbyaHertzianDipole 394
Chapter20 DiffractionofPlaneElectromagneticWavesbyaCylinder 397
20.1 ElectricPolarization 398
20.2 MoreaboutClassicalDiffraction 406
20.2.1 Huygen’sPrinciple 406
20.2.2 TheKirchhoff-HuygensDiffractionIntegral 406
20.2.3 DerivationoftheGeneralizedAiryDiffractionPattern 409
PARTIV SEMICLASSICALSCATTERING 413
Chapter21 TheClassical-to-SemiclassicalConnection 415
21.1 Introduction: ClassicalandSemiclassicalDomains 415
21.2 Introduction: TheSemiclassicalFormulation 416
21.2.1 TheTotalScatteringCrossSection 418
21.2.2 ClassicalWaveConnections 419
21.3 TheScalarWaveEquation 420
21.3.1 SeparationofVariables 420
21.3.2 Bauer’sExpansionAgain 422
21.4 TheRadialEquation: FurtherDetails 423
21.5 SomeExamples 426
21.5.1 ScatteringbyaOne-DimensionalPotentialBarrier 426
21.5.2 TheRadiallySymmetricProblem: PhaseShifts
andthePotentialWell 428
CONTENTS • xiii
Chapter22 TheWKB(J)ApproximationRevisited 434
22.1 TheConnectionFormulasrevisited: AnAlternativeApproach 435
22.2 Tunneling: APhysicalDiscussion 437
22.3 ATriangularBarrier 438
22.4 MoreNutsandBolts 440
22.4.1 ThePhaseShift 445
22.4.2 SomeCommentsonConvergence 445
22.4.3 TheTransitiontoClassicalScattering 446
22.5 CoulombScattering: TheAsymptoticSolution 448
22.5.1 ParabolicCylindricalCoordinates(ξ,η,φ) 449
22.5.2 AsymptoticFormof F (−iμ,1;ikξ) 450
1 1
22.5.3 TheSphericalCoordinateSystemRevisited 451
22.6 CoulombScattering: TheWKB(J)Approximation 453
22.6.1 CoulombPhases 453
22.6.2 FormalWKB(J)SolutionsfortheTIRSE 454
22.6.3 TheLangerTransformation: FurtherJustification 456
Chapter23 ASturm-LiouvilleEquation: TheTime-Independent
One-DimensionalSchrödingerEquation 459
23.1 VariousTheorems 460
23.2 BoundStates 463
23.2.1 Bound-StateTheorems 463
23.2.2 ComplexEigenvalues: IdentitiesforIm(λn)andRe(λn) 467
23.2.3 FurtherTheorems 468
23.3 Weyl’sTheorem: LimitPointandLimitCircle 471
PARTV SPECIALTOPICSINSCATTERINGTHEORY 475
Chapter24 TheS-MatrixandItsAnalysis 477
24.1 ASquareWellPotential 477
24.1.1 TheBoundStates 480
24.1.2 SquareWellResonance: AHeuristicDerivation
oftheBreit-WignerFormula 480
24.1.3 TheWatsonTransformandReggePoles 481
24.2 MoreDetailsfortheTIRSE 487
24.3 Levinson’sTheorem 489
Chapter25 TheJostSolutions: TechnicalDetails 491
25.1 OnceMoretheTIRSE 491
25.2 TheRegularSolutionAgain 494
25.3 PolesoftheS-Matrix 498
25.3.1 WavepacketApproach 501