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Three-dimensional diffraction and scattering of elastic waves around hemispherical surface topographies

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Three-dimensional diffraction and scattering of elastic waves around hemispherical surface topographies

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Content Three-DimensionalDiffractionandScatteringofElasticWavesaroundHemispherical SurfaceTopographies by GuanyingZhu PhDAdvisor: VincentW.Lee ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (CIVILENGINEERING) August2016 Copyright 2016 GuanyingZhu Acknowledgement I would like to sincerely thank Prof. Vincent Lee for his guidance, help, understanding, patience and friendship. It is indeed my fortune to have Prof. Lee as my PhD advisor. I will not forget the kindness of him and his wife, Jane, to invite many international students to his home party to celebrateThanksgivingeveryyear. MylifeinLAbecomeeasierandhappierbecauseofhim. I would also like to thank Prof. Trifunac. He is one of the most brilliant, knowledgable and charming persons I have ever met. I am really grateful to his ideas and advices for my research, andhiswarmsupporttomyPhDstudies. I would also like to express my appreciation to Prof. Wellford, Prof. Udwadia and Prof. JJ. Lee for sparing their time to be my committee members. Especially to Prof. Wellford for that I havelearntsomuchfromhimforbeinghisTAforfiveyears. I would like to thank the Department of Civil and Environmental Engineering for offering me TAship every semester to support my PhD studies. I also want to thank all my PhD friends, especiallyRayandThang. Idonotfeelalonebecauseofyou. Finally, and most importantly, I would like to thank my family and friends in China and in the USA. I really appreciate the days I spent with my dear friends, and I really thank them for theirunderstanding,theircompromiseandtheirappearanceinmylife. Ialsotreasurethesupport, encouragement and patience from my parents, grandparents, the whole family because these are thebedrockofwhoIamtoday. iii Contents Acknowledgement iii ListofFigures vii ListofTables xviii 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 ABriefReviewoftheWaveTheoryofDiffraction. . . . . . . . . . . . . . . . . . 4 1.3 PreviousWorkinThreeDimensions . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 The“Stress-Free”Analytic-WaveFunctions: ANewApproach . . . . . . . . . . . 12 1.4.1 Theodd-degree-onlyseriesexpansion . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Theboundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.3 Surfacedisplacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5 StructureoftheDissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6 AppendixtoChapter1: DerivationofScatteredandDiffractedWavesforVertical P-WaveIncidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 TheNormalPoint-SourceP-incidence 40 2.1 TheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 TheFree-fieldPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 TheScatteredWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 SurfaceDisplacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.6 Appendix to Chapter 2: Derivation of Scattered Waves for Normal Point-Source P-waveIncidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3 DiffractionaroundaHemisphericalCanyonbyWavesofArbitraryIncidence 71 3.1 ADescriptionoftheCanyonModelandBoundaryConditions . . . . . . . . . . . 71 3.2 PlaneP-waveIncidencewithArbitraryAngle✓ ↵ .................. 76 3.2.1 Free-fieldwavepotentials . . . . . . . . . . . . . . . . . . . . . . . . . . 76 iv 3.2.2 Surfacedisplacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3 PlaneSV-waveIncidencewithArbitraryAngle✓ .................. 92 3.3.1 Free-fieldwavepotentials . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.2 Surfacedisplacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.4 PlaneSH-waveIncidencewithArbitraryAngle✓ .................. 108 3.4.1 Free-fieldwavepotentials . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.2 Surfacedisplacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.6 AppendixtoChapter3: TheApplicationofOdd-Only-TermSeriesExpansiontoa Half-SpaceCanyon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4 DiffractionaroundaHemisphericalAlluvialValleybyWavesofArbitraryIncidence132 4.1 DescriptionoftheModelofValleyandBoundaryConditions . . . . . . . . . . . . 132 4.2 DisplacementsforPlaneP-incidencewithAngle✓ ↵ ................. 140 4.3 DisplacementsforPlaneSV-incidencewithAngle✓ ................ 157 4.4 DisplacementsforPlaneSH-incidencewithAngle✓ ................ 173 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.6 Appendix to Chapter 4: The Application of Odd-Term-Only Series Expansion to AnAlluvialValleyontheHalf-space . . . . . . . . . . . . . . . . . . . . . . . . . 189 5 TheSoil-structure-interaction(SSI)problem 202 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.2 TheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.2.1 Thegroundmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.2.2 Motionofthebuilding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.3 TheBoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.4 ResponseandSurfaceDisplacement . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.4.1 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.4.2 RelativeResponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.4.3 DisplacementsontheVerticalCross-sectionalPlane . . . . . . . . . . . . 222 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 A CoefficientsofSeriesExpansionoftheFree-fieldWaves 232 A.1 RotationofCoordinates: theWignersmall-dFunction . . . . . . . . . . . . . . . . 232 A.2 PlaneP-incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A.3 PlaneSV-incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 A.4 PlaneSH-incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 B DisplacementVectorandStressTensor(MowandPao,1971) 248 B.1 FunctionsofDisplacementfromWavePotential . . . . . . . . . . . . . . . . . . . 249 B.2 FunctionsofStressfromWavePotential . . . . . . . . . . . . . . . . . . . . . . . 250 v C Three-dimensionalGraphsofDisplacementAmplitudesaroundtheCanyon 254 C.1 TheDiffractionaroundaHemisphericalCanyonforthePlaneP-incidence . . . . . 254 C.2 TheDiffractionaroundaHemisphericalCanyonforthePlaneSV-incidence . . . . 259 C.3 TheDiffractionaroundaHemisphericalCanyonforthePlaneSH-incidence . . . . 264 D Three-dimensionalGraphsofDisplacementAmplitudesaroundtheAlluvialValley 269 D.1 TheDiffractionaroundaHemisphericalAlluvialValleyforthePlaneP-incidence . 269 D.1.1 CaseI:µ f /µ=0.25,C ↵f /C ↵ =0.50 .................... 269 D.1.2 CaseII:µ f /µ=0.30,C ↵f /C ↵ =0.60.................... 274 D.1.3 CaseIII:µ f /µ=0.40,C ↵f /C ↵ =0.60 ................... 279 D.1.4 CaseIV:µ f /µ=3.00,C ↵f /C ↵ =1.50 ................... 284 D.2 TheDiffractionaroundaHemisphericalAlluvialValleyforthePlaneSV-incidence 289 D.2.1 CaseI:µ f /µ=0.25,C f /C =0.50 .................... 289 D.2.2 CaseII:µ f /µ=0.30,C f /C =0.60.................... 294 D.2.3 CaseIII:µ f /µ=0.40,C f /C =0.60 ................... 299 D.2.4 CaseIV:µ f /µ=3.00,C f /C =1.50 ................... 304 D.3 TheDiffractionaroundaHemisphericalAlluvialValleyforthePlaneSH-incidence 309 D.3.1 CaseI:µ f /µ=0.25,C f /C =0.50 .................... 309 D.3.2 CaseII:µ f /µ=0.30,C f /C =0.60.................... 314 D.3.3 CaseIII:µ f /µ=0.40,C f /C =0.60 ................... 319 D.3.4 CaseIV:µ f /µ=3.00,C f /C =1.50 ................... 324 Bibliography 329 vi ListofFigures 1.1 Sphericalcoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Three-dimensionalhemisphericalcanyonwithverticalplaneincidence. . . . . . . 8 1.3 Verticaldisplacementamplitudesalongaradiallinefornormalincidence(axisym- metriccase)with⌘ =4,8,15and50 . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =05....... 21 1.5 Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =510....... 22 1.6 Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =1015...... 22 1.7 Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =1520...... 23 1.8 Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =2023...... 23 1.9 Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =2325...... 24 1.10 Radialdisplacement|U r |alongx/afornormalincidence,⌘ =05 ....... 25 1.11 Radialdisplacement|U r |alongx/afornormalincidence,⌘ =510 ....... 26 1.12 Radialdisplacement|U r |alongx/afornormalincidence,⌘ =1015 ...... 26 1.13 Radialdisplacement|U r |alongx/afornormalincidence,⌘ =1520 ...... 27 1.14 Radialdisplacement|U r |alongx/afornormalincidence,⌘ =2025 ...... 27 1.15 Radialdisplacementamplitudeswith⌘ =4,8,15and50 . . . . . . . . . . . . . . 28 1.16 Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =1......... 29 1.17 Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =4......... 29 1.18 Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =8......... 30 1.19 Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =15 ........ 30 1.20 Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =50 ........ 31 1.21 Model0: VerticallyupwardincidentplaneP-waveontosphericalcavity. . . . . . 34 1.22 Model1: VerticallydownwardincidentplaneP-waveontosphericalcavity. . . . . 36 1.23 Half-SpaceModel: VerticallyplaneP-waveontohemisphericalcanyon. . . . . . . 37 2.1 3-Dhemisphericalcanyonwithpoint-sourceincidence.. . . . . . . . . . . . . . . 41 2.2 Adoptionofsphericalcoordinate.. . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Thecoordinatesystemsforthepoint-sourceP-incidence. . . . . . . . . . . . . . . 43 2.4 Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =010andd=1.5a. . . . . . . . . . . . . 53 2.5 Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =1020andd=1.5a. . . . . . . . . . . . 54 vii 2.6 Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =010andd=5.0a. . . . . . . . . . . . . 54 2.7 Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =1020andd=5.0a. . . . . . . . . . . . 55 2.8 Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =010andd=20.0a. . . . . . . . . . . . 55 2.9 Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =1020andd=5.0a. . . . . . . . . . . . 56 2.10 Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5and⌘ =0.02,0.5,1.0and1.5. . . . . 58 2.11 Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5,5,and20for⌘ =0.02........ 59 2.12 Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5,5,and20for⌘ =1.0 ........ 59 2.13 Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5,5,and20for⌘ =20.0........ 60 2.14 Verticaldisplacementamplitude|U z |alongx/ato100forthe normalpoint-sourceincidencewithd/a =1.5,5and20and⌘ =1.0 ........ 61 2.15 Verticaldisplacementamplitudes|U z |alongx/aforvariousdwith⌘ =1.0.. . . . 62 2.16 Verticaldisplacementamplitude|U z |alongx/aforvariousdwith⌘ =4.0..... 63 2.17 Verticaldisplacementamplitude|U z |alongx/aforvariousdwith⌘ =8.0..... 63 2.18 Model0: Upwardpoint-sourceincidentp-waveontosphericalcavity. . . . . . . . 66 2.19 Model1: Downwardpoint-sourceincidentp-waveontosphericalcavity. . . . . . 67 2.20 Half-spacemodel: Upwardpoint-sourceincidentp-waveontohemisphericalcanyon. 69 3.1 Adoptionofsphericalcoordinate.. . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2 A3-Dhemisphericalcanyonwitharbitraryplanewaveincidence. . . . . . . . . . 73 3.3 Field-field and scattered wave potentials in the problem of a 3-D canyon on the half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4 |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidence. . . . . . . . . . . . . 80 3.5 Asketchoftherangeofwhere2-Dplotsdescribe. . . . . . . . . . . . . . . . . . 82 3.6 |U z |and z aty/a=0.0with✓ ↵ =60 ,⌘ =1,3,5ofP-incidence. . . . . . . . . 83 3.7 |U z |and z aty/a=0.0with✓ ↵ =15 ,⌘ =1,3,5ofP-incidence. . . . . . . . . 84 3.8 |U x |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. . . . . . 86 3.9 |U z |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. . . . . . 86 3.10 |U x |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. . . . . . 87 3.11 |U z |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. . . . . . 87 3.12 |U x |aty/a=0.0with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. . . . . . 88 3.13 |U z |aty/a=0.0with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. . . . . . 88 3.14 |U x |atx/a=0.0with✓ ↵ =30 ,⌘ =1,3,5ofP-incidence. . . . . . . . . . . . . 90 3.15 |U z |atx/a=0.0with✓ ↵ =30 ,⌘ =1,3,5ofP-incidence. . . . . . . . . . . . . 91 3.16 |U y |atx/a=0.4with✓ ↵ =15 ,⌘ =1,3,5ofP-incidence. . . . . . . . . . . . . 91 viii 3.17 Thefree-fieldwavesofSV-waveincidence. . . . . . . . . . . . . . . . . . . . . . 93 3.18 |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofcanyonforSV-incidence. . . . . . 97 3.19 |U z |and z aty/a=0.0with✓ ↵ =60 ,⌘ =1,3,5ofSV-incidence. . . . . . . . 99 3.20 |U z |and z aty/a=0.0with✓ ↵ =15 ,⌘ =1,3,5ofSV-incidence. . . . . . . . 100 3.21 |U z |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 102 3.22 |U x |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 102 3.23 |U z |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 103 3.24 |U x |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 103 3.25 |U z |aty/a=0.0with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 104 3.26 |U x |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 104 3.27 |U x |atx/a =0.8with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . 105 3.28 |U y |aty/a=0.5with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 106 3.29 |U y |aty/a=0.5with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 107 3.30 |U y |aty/a=0.5with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. . . . . 107 3.31 |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofcanyonforSH-incidence. . . . . . 112 3.32 |U y |and y aty/a=0.0with✓ =60 ,⌘ =1,3,5ofSH-incidence. . . . . . . . 113 3.33 |U y |and y aty/a=0.0with✓ =15 ,⌘ =1,3,5ofSH-incidence. . . . . . . . 114 3.34 |U y |aty/a=0.0with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 115 3.35 |U y |aty/a=0.0with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 116 3.36 |U y |aty/a=0.0with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 116 3.37 |U x |aty/a=0.5with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 118 3.38 |U x |aty/a=0.5with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 118 3.39 |U x |aty/a=0.5with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 119 3.40 |U z |atx/a=0.0with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 120 3.41 |U z |atx/a=0.0with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 120 3.42 |U z |atx/a=0.0with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. . . . . 121 4.1 Three-dimensionalhemisphericalvalleywitharbitraryplanewaveincidence. . . . 133 4.2 Adoptionofsphericalcoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.3 |U x | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence: µ f /µ = 0.25,C ↵f /C ↵ =0.50 ................................. 142 4.4 |U z |with⌘ =1,3,5of✓ ↵ =90 ofvalleyforP-incidencewithvariousparameters ofµ f /µandC ↵f /C ↵ alongradialdistancer/a=0to 5............... 144 4.5 |U z | with ⌘ =1 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ aty=0.0.. . . . . . . . . . . . . . . . . 145 4.6 |U z | with ⌘ =3 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ aty=0.0.. . . . . . . . . . . . . . . . . 146 4.7 |U z | with ⌘ =5 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ aty=0.0.. . . . . . . . . . . . . . . . . 147 4.8 |U x | with ⌘ =1 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ atx=0.5.. . . . . . . . . . . . . . . . . 150 ix 4.9 |U x | with ⌘ =3 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ atx=0.0.. . . . . . . . . . . . . . . . . 151 4.10 |U x | with ⌘ =5 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ atx=0.0.. . . . . . . . . . . . . . . . . 152 4.11 Exampleofplaneof =45 ............................. 153 4.12 |U y | with ⌘ =1 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ along =45 ............... 154 4.13 |U y | with ⌘ =3 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ along =45 ............... 155 4.14 |U y | with ⌘ =5 of ✓ ↵ =90 ,60 ,30 , and 15 of valley for P-incidence with variousparametersofµ f /µandC ↵f /C ↵ along =45 ............... 156 4.15 |U z | with ⌘ =1 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C aty=0.0.. . . . . . . . . . . . . . . . . 159 4.16 |U z | with ⌘ =3 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C aty=0.0.. . . . . . . . . . . . . . . . . 160 4.17 |U z | with ⌘ =5 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C aty=0.0.. . . . . . . . . . . . . . . . . 161 4.18 |U x | with ⌘ =1 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C aty/a=0.5.. . . . . . . . . . . . . . . . 163 4.19 |U x | with ⌘ =3 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C aty/a=0.2.. . . . . . . . . . . . . . . . 164 4.20 |U x | with ⌘ =5 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofofµ f /µandC f /C aty/a=0.5............... 165 4.21 |U x | with ⌘ =1 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C atx=0.0.. . . . . . . . . . . . . . . . . 166 4.22 |U x | with ⌘ =3 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C atx=0.0.. . . . . . . . . . . . . . . . . 167 4.23 |U x | with ⌘ =5 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C atx=0.0.. . . . . . . . . . . . . . . . . 168 4.24 |U y | with ⌘ =1 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C along =45 ............... 169 4.25 |U y | with ⌘ =3 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C along =45 ............... 170 4.26 |U y | with ⌘ =5 of ✓ =90 ,60 ,30 , and 15 of valley for SV-incidence with variousparametersofµ f /µandC f /C along =45 ............... 171 4.27 |U y | with ⌘ =1 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C aty=0.0.. . . . . . . . . . . . . . . . . 177 4.28 |U y | with ⌘ =3 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C aty=0.0.. . . . . . . . . . . . . . . . . 178 4.29 |U y | with ⌘ =5 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C aty=0.0.. . . . . . . . . . . . . . . . . 179 x 4.30 |U z | with ⌘ =1 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C aty=0.2.. . . . . . . . . . . . . . . . . 180 4.31 |U z | with ⌘ =3 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C aty=0.2.. . . . . . . . . . . . . . . . . 181 4.32 |U z | with ⌘ =5 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C aty=0.2.. . . . . . . . . . . . . . . . . 182 4.33 |U x | with ⌘ =1 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C along =45 ............... 184 4.34 |U x | with ⌘ =3 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C along =45 ............... 185 4.35 |U x | with ⌘ =5 of ✓ =90 ,60 ,30 , and 15 of valley for SH-incidence with variousparametersofµ f /µandC f /C along =45 ............... 186 5.1 3-DSSIproblemwithvertical-planeP-waveincidence. . . . . . . . . . . . . . . . 204 5.2 Thesphericalcoordinatesystem. . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.3 EffectofinteractionwithM b /M s =1,M f /M s =1,✏ = ¯ kH k a =0,2,4.. . . . . . . 218 5.4 EffectofinteractionwithM b /M s =1,M f /M s =2,✏ = ¯ kH k a =0,2,4.. . . . . . . 219 5.5 EffectofinteractionwithM b /M s =4,M f /M s =1,✏ = ¯ kH k a =0,2,4.. . . . . . . 219 5.6 Effect of interaction on relative response with M f /M s =1,✏ =2,M b /M s = 1,2,4and=2 .................................... 221 5.7 Effect of interaction on relative response with M f /M s =1,✏ =4,M b /M s = 1,2,4and=2 .................................... 221 5.8 Shadingareaofcross-sectionalplane. . . . . . . . . . . . . . . . . . . . . . . . . 222 5.9 DisplacementonaradialverticalplanearoundarigidfoundationwithM b /M s = 1,M f /M s =1,✏=2,⌘ =0.5at 0 x/a 5, 0 z/a 5alongfixed..... 224 5.10 Displacements withM b /M s =1, M f /M s =1, ✏=2, ⌘ =0.5 at 0 x/a 5, z/a=0.5,1.0,3.0,5.0alongfixed .. . . . . . . . . . . . . . . . . . . . . . . . . 225 5.11 DisplacementonaradialverticalplanearoundarigidfoundationwithM b /M s = 1,M f /M s =1,✏=2,⌘ =1.0at 0 x/a 5, 0 z/a 5alongfixed..... 226 5.12 Displacements with M b /M s =1, M f /M s =1, ✏=2, ⌘ =1 at 0 x/a 5, z/a=0.5,1.0,3.0,5.0alongfixed .. . . . . . . . . . . . . . . . . . . . . . . . . 227 5.13 DisplacementonaradialverticalplanearoundarigidfoundationwithM b /M s = 1,M f /M s =1,✏=2,⌘ =5.0at 0 x/a 5, 0 z/a 5alongfixed..... 228 5.14 Displacement with M b /M s =1, M f /M s =1, ✏=2, ⌘ =0.5 at 0 x/a 5, z/a=5,1.0,3.0,5.0alongfixed .. . . . . . . . . . . . . . . . . . . . . . . . . . 229 A.1 Sphericalcoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 A.2 Eulerangleofrotationbyy-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 A.3 Thefree-fieldwaveofP-incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A.4 Thefree-fieldwaveofSV-incidence. . . . . . . . . . . . . . . . . . . . . . . . . . 239 A.5 Thefree-fieldwaveofSH-incidence. . . . . . . . . . . . . . . . . . . . . . . . . . 244 xi A.6 ThetransformationfromSV-wavetoSH-wave. . . . . . . . . . . . . . . . . . . . 245 C.1 |U x |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 254 C.2 |U y |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 255 C.3 |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 255 C.4 |U x |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 256 C.5 |U y |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 256 C.6 |U z |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 257 C.7 |U x |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 257 C.8 |U y |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 258 C.9 |U z |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. . . 258 C.10 |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 259 C.11 |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 260 C.12 |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 260 C.13 |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 261 C.14 |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 261 C.15 |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 262 C.16 |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 262 C.17 |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 263 C.18 |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. . 263 C.19 |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 264 C.20 |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 265 C.21 |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 265 C.22 |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 266 C.23 |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 266 C.24 |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 267 C.25 |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 267 C.26 |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 268 C.27 |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. . 268 D.1 |U x | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 269 D.2 |U y | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 270 D.3 |U z | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 270 D.4 |U x | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 271 D.5 |U y | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 271 D.6 |U z | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 272 xii D.7 |U x | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 272 D.8 |U y | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 273 D.9 |U z | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.25,C ↵f /C ↵ =0.50 ............................ 273 D.10 |U x | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 274 D.11 |U y | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 275 D.12 |U z | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 275 D.13 |U x | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 276 D.14 |U y | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 276 D.15 |U z | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 277 D.16 |U x | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 277 D.17 |U y | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 278 D.18 |U z | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.30,C ↵f /C ↵ =0.60 ............................ 278 D.19 |U x | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 279 D.20 |U y | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 280 D.21 |U z | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 280 D.22 |U x | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 281 D.23 |U y | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 281 D.24 |U z | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 282 D.25 |U x | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 282 D.26 |U y | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 283 D.27 |U z | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=0.40,C ↵f /C ↵ =0.60 ............................ 283 xiii D.28 |U x | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 284 D.29 |U y | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 285 D.30 |U z | with ⌘ =1, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 285 D.31 |U x | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 286 D.32 |U y | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 286 D.33 |U z | with ⌘ =3, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 287 D.34 |U x | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 287 D.35 |U y | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 288 D.36 |U z | with ⌘ =5, ✓ ↵ =90 ,60 ,30 , and 15 for P-incidence around the valley: µ f /µ=3.00,C ↵f /C ↵ =1.50 ............................ 288 D.37 |U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 289 D.38 |U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 290 D.39 |U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 290 D.40 |U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 291 D.41 |U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 291 D.42 |U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 292 D.43 |U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 292 D.44 |U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 293 D.45 |U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 293 D.46 |U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 294 D.47 |U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 295 D.48 |U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 295 xiv D.49 |U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 296 D.50 |U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 296 D.51 |U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 297 D.52 |U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 297 D.53 |U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 298 D.54 |U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 298 D.55 |U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 299 D.56 |U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 300 D.57 |U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 300 D.58 |U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 301 D.59 |U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 301 D.60 |U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 302 D.61 |U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 302 D.62 |U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 303 D.63 |U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 303 D.64 |U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 304 D.65 |U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 305 D.66 |U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 305 D.67 |U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 306 D.68 |U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 306 D.69 |U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 307 xv D.70 |U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 307 D.71 |U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 308 D.72 |U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SV-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 308 D.73 |U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 309 D.74 |U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 310 D.75 |U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 310 D.76 |U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 311 D.77 |U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 311 D.78 |U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 312 D.79 |U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 312 D.80 |U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 313 D.81 |U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.25,C f /C =0.50............................. 313 D.82 |U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 314 D.83 |U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 315 D.84 |U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 315 D.85 |U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 316 D.86 |U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 316 D.87 |U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 317 D.88 |U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 317 D.89 |U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 318 D.90 |U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.30,C f /C =0.60............................. 318 xvi D.91 |U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 319 D.92 |U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 320 D.93 |U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 320 D.94 |U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 321 D.95 |U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 321 D.96 |U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 322 D.97 |U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 322 D.98 |U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 323 D.99 |U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=0.40,C f /C =0.60............................. 323 D.100|U x | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 324 D.101|U y | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 325 D.102|U z | with ⌘ =1, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 325 D.103|U x | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 326 D.104|U y | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 326 D.105|U z | with ⌘ =3, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 327 D.106|U x | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 327 D.107|U y | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 328 D.108|U z | with ⌘ =5, ✓ =90 ,60 ,30 , and 15 for SH-incidence around the valley: µ f /µ=3.00,C f /C =1.50............................. 328 xvii ListofTables 3.1 Surfaceamplitudesoffree-fielddisplacementfortheP-incidence. . . . . . . . . . 80 3.2 Summaryof2-DplotsinSection3.2. . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3 Surfaceamplitudesoffree-fielddisplacementfortheSV-incidence.. . . . . . . . . 96 3.4 Summaryof2-DplotsinSection3.3. . . . . . . . . . . . . . . . . . . . . . . . . 98 3.5 Phase y at (x,y,z)=(1,0,1)forvariousanglesandfrequencies. . . . . . . . . . 110 3.6 Summaryof2-DplotsinSection3.4. . . . . . . . . . . . . . . . . . . . . . . . . 112 4.1 Casesofparameterratiosusedinthediscussion. . . . . . . . . . . . . . . . . . . . 138 4.2 Summaryof3-DgraphsinAppendixD.1. . . . . . . . . . . . . . . . . . . . . . . 141 4.3 Summaryof2-DfiguresinSection4.2. . . . . . . . . . . . . . . . . . . . . . . . 143 4.4 Summaryof3-DgraphsinAppendixD.2. . . . . . . . . . . . . . . . . . . . . . . 157 4.5 Summaryof2-DfiguresinSection4.3. . . . . . . . . . . . . . . . . . . . . . . . 158 4.6 Summaryof3-DgraphsinAppendixD.3. . . . . . . . . . . . . . . . . . . . . . . 173 4.7 Summaryof2-DfiguresinSection4.4. . . . . . . . . . . . . . . . . . . . . . . . 175 xviii Chapter1 Introduction 1.1 Background Many structural engineers and researchers are baffled by the observation that earthquakes can causehighlylocalizeddamageareas,aswellasdrasticchangeofgroundmotionamplitudesfrom one point to the next. Relevant research and experiments on this topic have been conducted for many years and are still in progress. Although many numerical-methods achievements exist such as the finite element method (FEM) (Belytschko and Mullen, 1978) (Dasgupta, 1982), the finite difference method (FDM) (Alterman and Karal, 1968), the direct and indirect boundary element method (BEM and IBEM) (Yuan and Liao, 1996) (Yuan and Men, 1992) among others (Bard and Bouchon, 1985) (Sánchez-Sesma and Campillo, 1991) (Miller, 1973), the number of analytic solutions is still limited. The development of fundamental theory, which faces many challenges, will never stop, because fundamental theories (Achenbach, 1984) are the only sound base in the fieldandthestandardtowhichapproximatemethodsandexperimentsmustmeet(Gelietal.,1988). Trifunac (Trifunac, 1973) developed a closed-form solution using wave function expansion to discover patterns of displacement around a canyon on a half-space due to out-of-plane(SH) wave incidence. Trifunac’s paper is a milestone in earthquake engineering because it was the first timetheorthogonalityoftrigonometrywasappliedtohalf-spacetopographyproblems. Following this solution, the problem of a 2-D elliptical (Wong and Trifunac, 1974a) valley on a half-space with an incident SH-wave and the problem of 2-D semi-cylindrical alluvial valley with an inci- dent SH-wave (Trifunac, 1971) were solved. Image methods have also been applied to models 1 of underground topographies with plane SH-waves (Lee and Trifunac, 1979) thus the challenging half-spaceproblemshavebeenconvertedintosolvablefull-spaceboundary-valuedproblems. In all such SH-wave solutions, it has been possible the use the method of images, which cre- atedalineofsymmetryonthehalf-spacesurfaceresultinginzeroanti-planeshearstresses. Sucha method,however,failedforin-planelongitudinalP-andS-wavesduetothemodeconversion,thus any analytic solutions were at first quite difficult to achieve until the introduction of the big-arch approximation. Lee andCao (Leeand Cao,1989) (CaoandLee, 1988)(Cao andLee, 1989)used a large, circular, almost-flat surface to approximate the half-space surface and presented solutions to scattering and diffraction problems of surface circular canyons of various depth for incident plane P- and SV-waves. Todorovska and Lee (Todorovska and Lee, 1990) (Todorovska and Lee, 1991b)(TodorovskaandLee,1991a)usedthesamemethodforanti-planeSH-wavesandforinci- dentRayleigh-wavesoncircularcanyons. Later,LeeandKarl(LeeandKarl,1992)(LeeandKarl, 1993) extended the method to scattering and diffraction of P- and SV-waves by circular under- ground cavities. Lee and Wu (Lee and Wu, 1994a) (Lee and Wu, 1994b) used the same method tostudyarbitrary-shaped2-Dcanyons. Davisetal. (Davisetal.,2001)usedsuchanapproximate methodintheirstudiesofundergroundpipefailureinvolvingthetransverseincidenceofSV-waves. Liang et al. (Liang et al., 2001a) (Liang et al., 2001b) (Liang et al., 2002) (Liang et al., 2003c) (Liang et al., 2003d) (Liang et al., 2003a) (Liang et al., 2003b) (Lee et al., 2004) (Liang et al., 2004a) (Liang et al., 2004b) (Lee et al., 2006) (Liang et al., 2006a) (Liang et al., 2006d) (Liang et al., 2006b) (Liang et al., 2006c) (Liang et al., 2007a) (Liang et al., 2007b) continued the analy- sesbythesamemethodforproblemsinvolvingcircular-arccanyonsandvalleysandunderground pipesinanelasticandporo-elastichalf-space. However, effective solutions of 3-D problems are very limited. Lee (Lee, 1978) developed a solution of wave diffraction around a 3-D canyon by the Laurent series. The 3-D valley case was consequently solved but with considerable numerical complexity. Since then, only a few analytic solutionsof3-Dproblemshavebeendeveloped. 2 In recent years, Lee et al. (Lee et al., 2006) solved the diffraction of SH-waves by a semi- circular cylindrical hill with cosine half-range expansion. Liang et al. (Liang et al., 2010) then applied the same technique to SH-wave diffraction by a underground semi-circular cavity. These papersalsocomparedthenewresultsgeneratedbythehalf-rangeexpansiontechniquewiththeold ones (Lee et al., 2004) obtained from the earlier approximate methods-the methods of moment, or the weighted residue method assisted by Fourier expansion-and their good match strengthens the usefulness of the new methods. Moreover, in addition to the precise calculation with low incident wave frequency, the new method of the half-range expansion can significantly expand the calculable range of wave incidence frequencies. Recently, Lee and Liu (Lee and Liu, 2013) successfullyappliedthehalf-rangeexpansionmethodologytoa2-DmodelofP-ofSV-incidence: the semi-circular canyon in an elastic half-space. The reason for the revisit was that although the big-arc method was used, for many years, to analyze P- and SV-wave incidence, which involved mode conversion, met with some criticism. For example, Lee and Liu (Lee and Liu, 2013) states: ”Ithasbeennotedthatthelargecircularapproximationdoesnotconvergequicklytothatoftheflat half-space when the radius, R, of the circular surface approaches infinity, because as, the Bessel functionsusedinthetransformationapproachzero. Itwasalsosuggestedthatwithoutanaccurate method to satisfy the free-stress boundary conditions at the half-space surface, it might be better to relax the free-stress conditions than to use the large circle approximations”(Todorovska and Al Rjoub, 2006). Similarly, criticism was also received regarding the 3-D solutions since the matrixconsistingoftheLaurentserieshasbeenfoundtobeill-conditionedonoccasion. Forallof thesereasons,thedevelopmentofanewandeffectivetoolforthescatteringanddiffractionof3-D wavesisnecessary. Inspiredbythe2-Dcases,LeeandZhu(LeeandZhu,2013)updatedthe2-Dcosinehalf-range expansionintothe3-DoddLegendrepolynomialhalf-rangeexpansionandobtainedasatisfactory result of the scattering problem of a 3-D spherical canyon with normal plane P-wave incidence. Lateron,LeeandZhuextendedthesametechniquetowavediffractionarounda3-Dhemispherical 3 canyon excited by normal point-sourced P-wave and arbitrary angles of incidence of canyon or alluvial valley for P-, SV- and SH-waves With these developments, the frequencies can be high enough to handle real seismic data so that the new method introduced in this thesis will help to solveagreatnumberofsubsequentmodelswithmorecomplicatedboundaryconditions. 1.2 ABriefReviewoftheWaveTheoryofDiffraction The Cartesian coordinates are assumed to have a downward z-direction, and the spherical coordi- nates(r,✓, )arealsousedasshowninFig.1.1. Define✓ tobetheazimuthalangleinx-yplaneand itismeasuredfromthepositivex-axis. isthepolarangle(alsoknownaszenithangle)measuring fromthepositivez-axis,andristhedistance,whichisalwaysanon-negativenumber,fromapoint totheorigin. Figure1.1. Sphericalcoordinates. Thetransformationfromrectangulartosphericalcoordinatesisthen: x =rcos sin✓, y =rsin sin✓, z =rcos✓. (1.1) The half-space medium is assumed to be elastic, homogeneous, and isotropic. µ and are the Lamé constants and ⇢ is its density. All waves are assumed to be in a steady-state, which 4 means the waves are harmonic with frequency ! so that the corresponding time-dependent term e i!t , i = p 1, will be assumed to be present in all wave terms and omitted. According to the Helmholtz decomposition, the displacement vector U includes two kinds of wave motion. The dilatational part of displacement coming from the scalar potential ' is equal to the gradient of ' denotedas U =r ', (1.2) andtherotationalpartisequalto U =r⇥ , (1.3) where is the potential of shear waves. The dilatational wave speed isC ↵ = p ( +2µ)/⇢ , and the speed of the rotational wave is C = p µ/⇢ (Mow and Pao, 1971). The speed ratio denoted by  = C ↵ /C will be assumed to equal to p 3 in this dissertation. Because C ↵ >C and the particlemotionofdilatationalwaveisalongitstraveldirection,whileparticlemotionsofrotational waves are perpendicular to its travel direction. The dilatational wave is also called a longitudinal wave, compressional wave, primary wave, or P-wave; the rotational wave is called a transverse wave, shear wave, secondary wave, or S-wave. The S-wave usually consists of two independent components: is the potential of shear wave whose normal vector is vertical to the plane (SV- wave); isthepotentialofshearwavewhosenormalvectorishorizontaltotheplane(SH-wave). ThereforethedisplacementvectorUcanbegivenas: U =L+M+N (1.4) where L =r ', M =r (r )⇥ e r , N =lr  @ (r ) @r le r rr 2 , (1.5) 5 andwherel isascalarfactorwillbetakenas1inthisdissertation. In terms of the spherical coordinates (r,✓, ), the components of U (U r ,U ✓ ,U r ) have the fol- lowingrelationshipwiththewavepotentials(MowandPao,1971) Ur = @' @r +l  @ 2 (r ) @r 2 rr 2 , U ✓ = 1 r @' @✓ + 1 rsin✓ @ (r ) @ +l 1 r @ 2 (r ) @✓@r , U = 1 rsin✓ @' @ 1 r @ (r ) @✓ +l 1 rsin✓ @ 2 (r ) @@r . (1.6) BothkindsofwavessatisfytheHelmholtzwaveequation,whichinsphericalcoordinatestakes theform(forthewavefunctionF(r,✓, )exp(i!t)): 1 r 2 @ @r (r 2 @F @r )+ 1 r 2 sin✓ @ @✓ (sin✓ @F @✓ )+ 1 r 2 sin 2 ✓ @ 2 F @ 2 +k 2 F =0, (1.7) where k is the wave number, such that k = k ↵ = !/C ↵ for P-waves and k = k = !/C for S-waves. To use the method of separation of variable to solve (1.7), one can prove that the wave potentialisgiveninthefollowingform z n (kr)P m n (u) cosm sinm , (1.8) wherez n (kr)isthesphericalBesselfunctionorHankelfunctionofthe1 st or2 nd kindandoforder n;P m n (u)istheLegendrepolynomialofdegreenandorderm,andu=cos✓ . Plugging(1.8)into(1.6)onecangetthedisplacement-potentialrelationshipswhicharedefined as the D equations. The detailed expressions for the D equations are listed in Appendix B.1. After obtaining expressions for displacement, one can easily attain the strains and then stresses in terms of wave potential. The stress-potential relationship are denoted as E equations and listed in Appendix B.2. AllD andE equations will not be derived here. A comprehensive explanations 6 canbefoundinTheDiffractionofElasticWavesandDynamicStressConcentrationsbyMowand Pao(MowandPao,1971). 1.3 PreviousWorkinThreeDimensions Many studies (Lee, 1978) (Lee, 1982) (Lee, 1984) have aimed to find the analytical solution of wave scattering by 3-D topographies on the half-space surface. One fundamental solution was developedbyV.W.Lee(Lee,1978). ThegeometryofLee’ssolutionwillbeusedinthispaper,but we will first consider the special case of plane vertically incident longitudinal P-wave in the next sectiontoillustratethenewapproachandmethod. Figure1.2showstheplaneverticallyincidentP-wavesonthehalf-spaceandthehemispherical canyon with radius a. µ and are lamé constants of the elastic half-space and ⇢ is its density. The longitudinal wave speed in the half-space is C ↵ = p ( +2µ)/⇢ , and the shear wave speed is C = p µ/⇢ . Two coordinate systems, the rectangular (x,y,z) and the spherical coordinate (r,✓, )systemswillbeused. Thetwocoordinatesystemsarerelatedby x =rsin✓ cos, y =rsin✓ sin, z =rcos✓ ; or (1.9a) r=(x 2 +y 2 +z 2 ) 1/2 , ✓ =cos 1 (z/r), =cos 1 (x/(x 2 +y 2 ) 1/2 ). (1.9b) Thehalf-spacethustakestheregion: 1 <x<1 , 1 <y <1 , 0 z<1 ; or (1.10a) ra, 0 ✓  ⇡/ 2, ⇡   ⇡. (1.10b) Andthehemisphericalcanyonisintheregionof r a, 0 ✓  ⇡/ 2, ⇡   ⇡. (1.11) 7 Figure1.2. Three-dimensionalhemisphericalcanyonwithverticalplaneincidence. The incident plane P-wave is axis-symmetric with respect to the z-axis. Its potential will be expressedinboththerectangularandsphericalcoordinatesystemsas ' (i) = ' 0 exp(ik ↵ z) = ' 0 1 X n=0 (2n+1)(i) n j n (k ↵ r)P n (u), (1.12) wherej n (k ↵ r) is the spherical Bessel function of the 1 st kind of order n, P n (u) is the (0 th -order) Legendre polynomial of degree n, with u=cos✓ , 0  u  1 for 0  ✓  ⇡/ 2 in the half- space. ' 0 istheamplitudeoftheP-wavepotentialtobescaled,' 0 =1/ik ↵ , sothatitwillhavea displacementamplitudeof“1"atthehalf-spacesurface. Thepresenceofthehalf-spacesurfaceat z=0resultsinreflectedplaneP-wavepotential' (r) propagatingverticallydownwardsandofthe form ' (r) =' 0 exp(+ik ↵ z) =' 0 1 X n=0 (2n+1)(+i) n j n (k ↵ r)P n (u), (1.13) 8 andtherearenoreflectedS-waves. Togethertheincidentandreflectedplanewavesformtheinput free-fieldP-wavepotential ' (ff) = ' (i) +' (r) = ' 0 exp(ik ↵ z)' 0 exp(+ik ↵ z), (1.14) which,insphericalcoordinates,willsimplifyto ' (ff) = ' 0 1 X n=0 a 2n+1 j 2n+1 (k ↵ r)P 2n+1 (u) with a 2n+1 = (8n+6)i 2n+1 ik ↵ = 2(4n+3)(1) n k ↵ . (1.15) The free-field P-potential, ' (ff) , will have a displacement amplitude of “2" at the half-space surface. Itwillalsosatisfythezero-stressboundaryconditionsatthehalf-spacesurface. Forallra zz | z=0 = ⌧ zx | z=0 = ⌧ zy | z=0 =0, (1.16) or ✓✓ | ✓ =⇡/ 2 = ⌧ ✓r | ✓ =⇡/ 2 = X X X X X ⌧ ✓ | ✓ =⇡/ 2 =0. (1.17) For the axisymmetric case here, the waves are not dependent on , hence the stress term ⌧ ✓ | ✓ =⇡/ 2 =0naturally. In the presence of the hemispherical canyon, scattered and diffracted waves are produced and scatteredfromthecanyon,whichtakesthefollowingform(Lee,1978)(Lee,1982): ' (s) = 1 X n=0 A n h (1) n (k ↵ r)P n (u), (s) = 1 X n=0 C n h (1) n (k r)P n (u), (1.18) 9 respectively for the P- and SV-waves. h (1) n (.) are the spherical Hankel functions of the 1 st kind, representing outgoing P- and S- waves. Combining all the waves ' (ff) , ' (s) , and (s) the stresses everywheretaketheform: rr = 1 X n=0 h a 2n+1 E (1) 11 (2n+1,k ↵ r)P 2n+1 (u)+ ⇣ A n E (3) 11 (n,k ↵ r)+C n E (3) 13 (n,k r) ⌘ P n (u) i =0, ✓✓ = 1 X n=0 h a 2n+1 E (1) 21 (2n+1,k ↵ r)P 2n+1 (u)+ ⇣ A n E (3) 21 (n,k ↵ r)+C n E (3) 23 (n,k r) ⌘ P n (u) i =0, ⌧ r✓ = 1 X n=0  a 2n+1 E (1) 41 (2n+1,k ↵ r) dP 2n+1 (u) d✓ + ⇣ A n E (3) 41 (n,k ↵ r)+C n E (3) 43 (n,k r) ⌘ dP n (u) d✓ =0. (1.19) Here the stress functions E (i) jk (i=1,2,3,4;j =1,2,4;k =1,2,3) are used to denote the stress-potentialrelationshipsthatinvolveonlythesphericalBessel/Hankelfunctionscorresponding to various waves. Those are given in (Mow and Pao, 1971). The superscript i is used to denote the type of spherical Bessel functions and/or Hankel functions used. Here (i =1,2,3,4) are respectively for the functions j n ,y n ,h (1) n and h (2) n of degree n . The subscript j is used to denote theparticulartypeofstressfunctions. Herethesubscriptsj=1,2,4arethestresscomponentsfor rr , ✓✓ and⌧ r✓ . Stressesaredueto' and respectfullywhenk=1and3. Detailedexpressions forE (i) jk aregiveninAppendixB.2. At the half-space surface, where = ⇡/ 2,u=cos✓ =0, the following boundary conditions areconsidered: ✓✓ | ✓ =⇡/ 2 = 1 X n=0 h⇣ A n E (3) 21 (n,k ↵ r)+C n E (3) 23 (n,k r) ⌘ P n (0) i =0, ⌧ r✓ | ✓ =⇡/ 2 = 1 X n=0  ⇣ A n E (3) 41 (n,k ↵ r)+C n E (3) 43 (n,k r) ⌘ dP n (0) d✓ =0. (1.20) 10 Thefree-fieldwavetermsarenotincludedherebecausetheyalreadysatisfythehalf-spacezero- stress boundary conditions. Here the term P 1 n (0) is the associated Legendre polynomial (order 1 anddegreen)atu=cos✓ =0fromtheidentity d d✓ P n (u) | ✓ =⇡/ 2 =P 1 n (0). (1.21) Atthesurfaceofthesphericalcanyon,wherer =a,weneed rr | r=a = 1 X n=0 h a 2n+1 E (1) 11 (2n+1,k ↵ a)P 2n+1 (u)+ ⇣ A n E (3) 11 (n,k ↵ a)+C n E (3) 13 (n,k a) ⌘ P n (u) i =0, ⌧ r✓ | r=a = 1 X n=0  a 2n+1 E (1) 41 (2n+1,k ↵ a) dP 2n+1 (u) d✓ + ⇣ A n E (3) 41 (n,k ↵ a)+C n E (3) 43 (n,k a) ⌘ dP n (u) d✓ =0. (1.22) Thereareatotaloffourboundaryconditionsin(1.20)and(1.22)tobesatisfiedbythetwosets of unknown coefficients A n and C n . Additional P- and S-wave functions were thus introduced, in addition to the waves in (1.18) in Lee (Lee, 1982). They are the spherical Bessel functions of the 1 st kind instead of Hankel functions. On the half-space surface, Lee (Lee, 1978) (Lee, 1982) replacedthesphericalBesselandHankelwavefunctionsbytheirLaurentseriesrepresentationsin radialdistancer,andusedtheindependencepropertiesofthepowerseriestosatisfythezero-stress boundaryconditionin(1.20). However,theresultingsystemofequationsisnumericallycomplicatedanddifficulttosolvein thatthecoefficientsofE (i) jk aremadeupoftwosetsof(P-andS-)wavesthatarenotorthogonalto eachother. 11 1.4 The “Stress-Free” Analytic-Wave Functions: A New Approach Inthissection,wewilluseanewformulationtoaccountforthescatteredanddiffractedwavesthat will satisfy the zero-stress boundary conditions at the half-space surface. We will start with the samemodelofthehemisphericalcanyonthatwasdescribedintheprevioussection. Thefollowing isasummaryoftheresultspresentedin(LeeandZhu,2013). 1.4.1 Theodd-degree-onlyseriesexpansion To begin with the scattered and diffracted wave potentials in the form derived (see Appendix 1.6) as ' (s) = 1 X n=0 A 2n+1 h (1) 2n+1 (k ↵ r)P 2n+1 (u), (s) = 1 X n=0 C 2n+1 h (1) 2n+1 (k r)P 2n+1 (u). (1.23) where the wave potentials, as in the case of the free-field plane waves (1.15), are expressed in terms of the odd and only odd degree Legendre polynomials. In contrast to (1.18), those are not expressed by the full set of Legendre functions, {P n (u),n=0,1,2,...}, of both even and odd degrees. It is known that this full set forms an orthogonal set of basis functions in the full space whereu⌘ cos✓ , 0 ✓  ⇡ . However,hereinthehalf-spacewhere 0 ✓  ⇡ (0 u 1),both theodd-degree{P 2n+1 (u)}andeven-degree{P 2n (u)}Legendrepolynomials(andsimilarlyforthe 1 st orderassociatedLegendrepolynomials{P 1 n (u)})eachbythemselvesformanorthogonalsetof basisfunctions,andtheodd-andeven-degreeLegendrepolynomialstogetherarenotorthogonalin thehalf-space. Moreovereveryfunctiondefinedinthehalf-spacecanbeexpandedeitherinterms of a series of theodd degree {P 2n+1 (u)} and/or theeven degree {P 2n (u)} Legendre polynomials. In fact, in the half-space, each odd-degree Legendre polynomial P 2n+1 (u) can be expanded as a series of even-degree {P 2n (u)} Legendre polynomials. This is equivalent to expressing the sine 12 function as a series of the cosine functions in the 2-D half space (Lee et al., 2004) (Lee and Liu, 2013). Sothat P 2m+1 (u)= 1 X n=0 mn P 2n (u), (1.24) where mn = <P 2m+1 (u),P 2n (u)> <P 2n (u),P 2n (u)> ,intermsof (1.25) theinnerproduct <f,g>= Z 1 0 f(u)g(u)du. (1.26) Itisknownthat(Byerly,1893) Z 1 0 P 2m+1 (u)P 2n (u)du = (1) (m+n+1) (2m+1)!(2n)! 2 (m+n) (2n2m+1)(2n+2m+2)(n!) 2 (m!) 2 , (1.27) and Z 1 0 (P 2n (u)) 2 du = 1 4n+1 . (1.28) Using (1.24), the wave potentials in (1.23) (of odd degrees) can be expressed in terms of the Legendrefunctionsofevendegreesas ' (s) = 1 X n=0 A 2n+1 h (1) 2n+1 (k ↵ r)P 2n+1 (u)= 1 X m=0 A 2m+1 h (1) 2m+1 (k ↵ r) 1 X n=0 mn P 2n (u), (s) = 1 X n=0 C 2n+1 h (1) 2n+1 (k r)P 2n+1 (u)= 1 X n=0 C 2m+1 h (1) 2m+1 (k r) 1 X n=0 mn P 2n (u), (1.29) or ' (s) = 1 X n=0 1 X m=0 A 2m+1 h (1) 2m+1 (k ↵ r) mn ! P 2n (u), (s) = 1 X n=0 1 X m=0 C 2m+1 h (1) 2m+1 (k r) mn ! P 2n (u). (1.30) Equations (1.23) and (1.30) thus show that the scattered P- and S-wave potentials in the half- space can be expressed as a series of Legendre polynomials of either the odd or the even degrees, 13 each being a set of wave functions that are complete in the half-space. This is equivalent to, and is an extension of, the 2-D semi-circular canyon problem (Lee and Liu, 2013) using the concept of “half-space expansion” of trigonometric functions, where a function, f(✓ ), defined in a half- range [0,⇡ ], can be expressed both as a sine series and a cosine series, obtained respectively by the odd and even extensions of the function to the full range [⇡,⇡ ]. Here the sine and cosine functionsarereplacedbytheLegendrepolynomialsofoddandevendegree,andthesameconcept of“half-rangeexpansion”isused. 1.4.2 Theboundaryconditions The longitudinal and shear-wave potentials functions as represented by (1.23) and (1.30) can now be shown to be the wave functions that implicitly satisfy the zero-stress boundary conditions on thehalf-spacesurface,namely,(1.17): 1. The zero normal stress boundary condition on the half-space surface zz | z=0 = ✓✓ | ✓ =⇡/ 2 =0. With (1.23) for the P- and S-wave scattered potentials each as an odd-degree series in u⌘ cos✓ ,thenormalstresswillonlybecomputedforthescatteredanddiffractedwavesbecause the free-field incident and reflected plane P-waves already satisfy the zero-stress boundary conditions. From(1.20) ✓✓ | ✓ =⇡/ 2 = 1 X n=1,3,5... h A n E (3) 21 (n,k ↵ r)+C n E (3) 23 (n,k r) i P n (0). (1.31) Equation (1.31) is summed over all the odd integers,n=1,3,5..., whereP n (0) = 0 for all oddn-degreeLegendrepolynomialsandthusforallra, zz | z=0 = ✓✓ | ✓ =⇡/ 2 =0. 2. The zero shear stress boundary condition on the half-space surface ⌧ zx | z=0 = ⌧ zy | z=0 = ⌧ ✓r | ✓ =⇡/ 2 =0. 14 ThistimeboththeP-andS-wavepotentialsarewrittenastheLegendreseriesofevendegrees inu⌘ cos✓ ,using(1.30),thentheshearstresswilltaketheform,from(1.20)as ⌧ ✓r | ✓ =⇡/ 2 = 1 X n=0 " 1 X m=0 ⇣ A 2m+1 E (3) 41 (2m+1,k ↵ r)+C 2m+1 E (3) 43 (2m+1,k r) ⌘ mn # P 1 2n (0). (1.32) Equation (1.32) is summed over all the even 2n=0,2,4,..., where P 1 2n (0) = 0 for all the 1 st order Legendre polynomials of even degrees, and thus for allra, ⌧ zx | z=0 = ⌧ zy | z=0 = ⌧ ✓r | ✓ =⇡/ 2 =0issatisfied. The scattered and diffracted waves ' (s) and (s) of (1.23) are a complete set of the cylindrical-wave functions that each by themselves and together both satisfy the zero-stress boundaryconditionsof(1.20)onthehalf-spacesurface. Thecanyonzero-stressboundaryconditionsarenextconsidered: For 0 ✓  ⇡/ 2,atr = a rr | r=a =0, ⌧ r✓ | r=a =0. (1.33) The P- and SV-wave potentials will now include both the free-field waves and scattered and diffracted waves. Together they will satisfy the zero-stress boundary conditions on the surfaceofthecanyon,sothat rr | r=a = (ff) rr | r=a + (s) rr | r=a =0, ⌧ r✓ | r=a = ⌧ (ff) r✓ | r=a +⌧ (s) r✓ | r=a 0. (1.34) 3. Thezeronormalstressboundaryconditiononthecanyonsurface rr | r=a =0. 15 Fromabove (ff) rr | r=a = 1 X n=0 a 2n+1 E (1) 11 (2n+1,k ↵ a)P 2n+1 (u), (s) rr | r=a = 1 X n=0 ⇣ A 2n+1 E (3) 11 (2n+1,k ↵ a)+C 2n+1 E (3) 13 (2n+1,k a) ⌘ P 2n+1 (u). (1.35) So that rr | r=a = (ff) rr | r=a + (s) rr | r=a =0 gives, using the orthogonality of the Legendre polynomials,foreachn, A 2n+1 E (3) 11 (2n+1,k ↵ a)+C 2n+1 E (3) 13 (2n+1,k a)=a 2n+1 E (1) 11 (2n+1,k ↵ a). (1.36) 4. Thezeroshearstressboundaryconditionsonthecanyonsurface⌧ r✓ | r=a =0. ⌧ (ff) r✓ | r=a = 1 X n=0 a 2n+1 E (1) 41 (2n+1,k ↵ a)P 1 2n+1 (u), ⌧ (s) r✓ | r=a = 1 X n=0 ⇣ A 2n+1 E (3) 41 (2n+1,k ↵ a)+C 2n+1 E (3) 43 (2n+1,k a) ⌘ P 1 2n+1 (u). (1.37) Andthen⌧ r✓ | r=a = ⌧ (ff) r✓ | r=a +⌧ (s) r✓ | r=a =0gives,againusingorthogonality,foreachn, A 2n+1 E (3) 41 (2n+1,k ↵ a)+C 2n+1 E (3) 43 (2n+1,k a)=a 2n+1 E (1) 41 (2n+1,k ↵ a). (1.38) Combining(1.36)and(1.38) 2 4 E (3) 11 (2n+1,k ↵ a) E (3) 13 (2n+1,k a) E (3) 41 (2n+1,k ↵ a) E (3) 43 (2n+1,k a) 3 5 8 < : A 2n+1 C 2n+1 9 = ; =a 2n+1 8 < : E (1) 11 (2n+1,k ↵ a) E (1) 41 (2n+1,k ↵ a) 9 = ; , (1.39) 16 fromwhichthecoefficientsA n andC n aregivenby,forn=1,3,5,..., 8 < : A n C n 9 = ; = a n Det(n) 2 4 E (3) 43 (n,k a) E (3) 13 (n,k a) E (3) 41 (n,k ↵ a) E (3) 11 (n,k ↵ a) 3 5 8 < : E (1) 11 (n,k ↵ a) E (1) 41 (n,k a) 9 = ; , (1.40) withthedeterminantDet(n)=E (3) 11 (n,k ↵ a)E (3) 43 (n,k a)E (3) 13 (n,k a)E (3) 41 (n,k ↵ a). Equation(1.40) is a simple and explicit close-form analytic expression for the coefficients of the P- and S-scattered and diffracted wave functions. This is very uncommon for the 2-D or 3-D elastic-wavepropagationproblemsinahomogeneoushalf-space. For the 2-D half-space cases, Trifunac (Trifunac, 1973) was the first to present an explicit and analytic expression for the coefficients of the scattered and diffracted wave functions for the out-of-planeSH-waves. Thewavefunctionsthere,namely,thecylindricalHankel-functionseries, wereexpressedasthehalf-range,Fourier-cosineseries,whichalsoexplicitlysatisfythehalf-space stress-freeboundaryconditionsfortheout-of-planeSH-waves. The same technique was applied to the corresponding semi-circular, cylindrical alluvial val- ley (Trifunac, 1971) and the semi-circular, cylindrical rigid foundation in a soil-structure inter- action problem (Trifunac, 1972). Later, Trifunac’s work on the 2-D cylindrical coordinate prob- lems was extended to the 2-D elliptical-coordinate problems. Wong and Trifunac (Wong and Tri- funac,1974a)presentedanexplicitandanalyticexpressionforthecoefficientsofthescatteredand diffracted wave functions in the elliptical coordinates for the out-of-plane SH-waves. They later extendedtheirworktoasemi-elliptical,alluvialvalley(WongandTrifunac,1974c),andtoasemi- elliptical,rigidfoundationinasoil-structureinteractionproblem(WongandTrifunac,1974b). All of the above-cited work was for the out-of-plane SH-waves. Most of the other work using the wave-functionseriesoftenresultedinnon-analyticequationsthatneededtobesolvednumerically. Noexplicit,analyticexpressionsforthecoefficientsofthecorrespondingserieswavefunctions existed for the in-plane P- and SV-waves for both the 2-D and 3-D diffraction problems until the writing of this thesis, where shows that the explicit, analytic expressions for the coefficients of 17 3-D spherical waves exist for the case of the normal P-wave incidence. In the next section, the displacementsonandaroundthehalf-spacecanyonwillbecalculated. 1.4.3 Surfacedisplacements (a) VerticalDisplacements The displacement amplitudes at various points along the surface of the half-space and around the hemispherical canyon will help our understanding of amplification of the ground motion. The precise description of the amplitudes of surface-ground motion will allow the determi- nation of the space-dependent transfer functions of the ground motion at and near the hemi- spherical canyon. Complete knowledge of the displacement at each point will also enable us to calculate all the components of strains and stresses. With the scattered and diffracted wave functions in the spherical coordinates, the components of displacement are first calculated in thesphericalcoordinatesandthentransformedbacktotherectangularcoordinates U z =U r cos✓ U ✓ sin✓, (1.41) where U ✓ = 1 X n=1,3,5,..., h a n D (1) 21 (n,k ↵ a)+A n D (3) 21 (n,k ↵ a)+C n D (3) 23 (n,k a) i dP n (u) d✓ , U r = 1 X n=1,3,5,..., h a n D (1) 11 (n,k ↵ a)+A n D (3) 11 (n,k ↵ a)+C n D (3) 13 (n,k a) i P n (u), (1.42) andD (i) jk arethedisplacement-potentialfunctionsgiveninAppendixB.1. For each of the complex components of U, its magnitude is defined as the “displacement amplitude”ofthatcomponentas |U z |=[Re 2 (U z )+Im 2 (U z )] 1/2 . (1.43) 18 For consistency with previous work (Trifunac, 1973), the following dimensionless frequency isdefinedas ⌘ = !a ⇡C = 2fa C = k a ⇡ = 2a , (1.44) where isthewavelengthoftheshearwaves. Figure 1.3 presents the vertical displacement amplitudes on the half-space surface along the radiallineforselecteddimensionlessfrequencies⌘ =4,8,15and50,wherethecaseof⌘ =50 isthehighestfrequencybeingattemptedtoconsider. Thefar-fielddisplacementamplitudeson the half-space surface are equal to two for all frequencies. Equation (1.44) shows that given a half-space with the shear wave speed of C =2km/s and a hemispherical canyon of radius a=1km,thiswouldcorrespondtoelasticwavesatgivencyclicfrequenciesoff =4,8,15and 50Hz. In terms of wavelengths, they would correspond to the shear wavelengths 0.5, 0.25, 0.133and0.04km,or1/4,1/8,1/15and1/50ofthediameterofthehemisphericalcanyon. The dimensionless frequencies used in this chapter are all in a frequency range much higher than those calculated in the related previous work (Lee, 1982), (Lee, 1984) and (Lee and Trifunac, 1982). This is because the new analytic wave functions automatically satisfy the zero-stressboundaryconditionsonthehalf-spacesurface, andtheexplicitexpressionsforthe wavecoefficientsaremuchsimplertocompute. Backthen,theuseofpower-seriesexpansions ofthewavefunctionsresultedincomplicatedmatrixequationsthatweremuchhigherinorder and numerically more difficult to solve. Considering that the highest-dimensionless frequen- cies presented in the 1970s and 1980s were only ⌘ =0.5 and ⌘ =1.0, the results in this thesis representasignificantimprovement. 19 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 Vertical Incidence: η=k β a/π=4.0 with Nterms=15 radius x/a Uz 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 Vertical Incidence: η=k β a/π=8.0 with Nterms=25 radius x/a Uz 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 Vertical Incidence: η=k β a/π=15.0 with Nterms=40 radius x/a Uz 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 Vertical Incidence: η=k β a/π=50.0 with Nterms=80 radius x/a Uz Figure1.3. Verticaldisplacementamplitudesalongaradiallinefornormalincidence (axisymmetriccase)with⌘ =4,8,15and50 InFig.1.3,withaasthehemispherical-canyonradius,thedisplacementamplitudesareplotted along the dimensionless horizontal-radial distance fromx/a=0 tox/a=5. Being axisym- metric, this is also the same plot of amplitudes alongr/a=0 tor/a=5. The rangex/a > 1 corresponds to the half-space surface measured from the center of the canyon. In each of the four graphs for the four dimensionless frequencies, the displacement amplitudes all oscillate aroundthefree-fieldamplitudewhichistwo. Theperiodsofoscillationsscorrespondtothose 20 of the input periods of the waves. Thus, for example, in the 2 nd graph from the bottom, at a dimensionless frequency of ⌘ =2a/ =15, or a wavelength at 0.133akm, there are almost eightpositivepeaks,oralmosteightwavelengths (a/8=0.125a). The amplitudes plotted at distances in the range of x/a=0 to x/a=1 are the amplitudes on the hemispherical canyon. The pointx/a=1 is the rim of the canyon. The displacement amplitudesalsooscillateonthecanyonsurface. Attherimofthecanyonsurface,itisobserved that the displacement amplitude exhibits a “dip and spike” in amplitude. This is the corner- point phenomenon, which is also observed in the semi-cylindrical canyon case for the SH- wave incidence (Trifunac, 1973). It means that the displacement amplitudes on the side of the canyon close to the rim will decay to almost zero, but beyond the rim on the half-space surface, the amplitudes will shoot up to above the free-field amplitude. At the much lower frequencies (⌘  1), in the previous work of Lee (Lee, 1982), such phenomenon was not prominentbecausethewaveswithlowerfrequenciestend“nottosee”thesecornerpoints. As waspointedoutin(LeeandLiu,2013),thiscornerpointeffectinthewave-propagationtheory isoftenknowntoactasasecondarywavesourcepoint. Figure1.4. Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =05. 21 Figure1.5. Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =510. Figure1.6. Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =1015. 22 Figure1.7. Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =1520. Figure1.8. Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =2023. 23 Figure1.9. Verticaldisplacement|U z |alongx/afornormalincidence,⌘ =2325. Figures1.4to1.9showthecorresponding3-Dplotsofthedisplacementamplitudesalongthe same horizontal radial liner/a (= x/a). The plots are for the 6 ranges of ⌘ : ⌘ =05,5 10,1015,1520,2023,and 2325,respectively. Theamplitudesareplotted,fromleft to right, vs. the radial distancex/a from the center of the canyon outward, and from front to back, vs. the dimensionless frequency ⌘ . For 0 x/a 1, the displacement amplitudes are thoseonthehemisphericalsurfaceofthecanyon. Forx/a> 1,theyaretheamplitudesonthe half-spacesurfaceawayfromthecanyon. For the frequency range considered, ⌘ , from 0 to 25 in these six graphs, the amplitudes up to threeareobservedattheflatsurfaceclosetothecanyon,andthesame“dipandspike”behavior isobservedforallfrequenciesatthecornerpoint,whichistherimbetweenthehalf-spaceand the spherical-canyon’s surface. The result of displacement amplitudes close to a constant of two everywhere along the half-space surface is observed when the frequency ⌘ is low and approaches zero—i.e. the long waves do not “see” the inhomogeneity. As the dimensionless frequency increases through these six graphs, the oscillations of the displacement amplitudes increaseasthewavelengthsgetshorter. 24 (b) RadialDisplacements Theradial-displacementamplitudes|U r |of(1.42)inFig. 1.10toFig. 1.14showthe3-Dgraphs along the radial distance x/a from the center of the canyon outward with the dimensionless frequencies ⌘ =05, ⌘ =510,..., ⌘ =2025. Note that |U r | is zero on the half- space surface because the Legendre polynomials of odd degrees are zero at z =0, where u=cos✓ =0 when ✓ = ⇡/ 2. This is also intuitively because only the vertical motions exist for the normal free-field longitudinal waves. For the scattered and diffracted waves, which areconsideredasthesuperpositionofwavesfromModels0and1(Appendix1.6),theradical motions exist for waves from each model, but they cancel out each other on the half-space surface. Therefore,|U r |isonlycaughtonthecanyonsurface,wherex/a 1. Furthermore, it is noted that |U r | is zero everywhere on the canyon surface at the zero fre- quency, that ⌘ =0 (Fig.1.10), corresponding to that an infinitely long period can be interpreted as a static case. As the dimensionless frequency ⌘ increases from Fig. 1.10 to Fig. 1.14, the radialmotionsbecomemoreandmoreoscillatoryalongthecanyonsurface. Figure1.10. Radialdisplacement|U r |alongx/afornormalincidence,⌘ =05 25 Figure1.11. Radialdisplacement|U r |alongx/afornormalincidence,⌘ =510 Figure1.12. Radialdisplacement|U r |alongx/afornormalincidence,⌘ =1015 26 Figure1.13. Radialdisplacement|U r |alongx/afornormalincidence,⌘ =1520 Figure1.14. Radialdisplacement|U r |alongx/afornormalincidence,⌘ =2025 Figure 1.15 shows the corresponding 2-D plots of radial displacements for the dimensionless frequencies of ⌘ =4,8,15, and 50. They are plotted along x/a = r/a from 0 to 1.5 where x/a 1isagainforpointsonthecanyonsurface. Itshowsthesamefeatureofthe3-Dcurves in Fig. 1.10 to Fig. 1.14; namely, that the oscillations increase with the increasing frequency withthemaximumamplitudesof2orless. Also,themagnitudesofoscillationsonthecanyon surface are always between 0.0 and 2.0. At the bottom of the canyon where x/a =0 the 27 oscillation amplitudes are close to 2; however, the decreasing amplitudes are exhibited as the locationgoesuptowardthehalf-spacesurface,andeventuallydecaytozeroattherim,where x/a1 Figure1.15. Radialdisplacementamplitudeswith⌘ =4,8,15and50 28 Figure1.16. Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =1 Figure1.17. Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =4 29 Figure1.18. Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =8 Figure1.19. Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =15 30 Figure1.20. Verticaldisplacement|U z |alongx/aforvariousNtermsfor⌘ =50 (c) DiscussionofConvergence Figures 1.16 through 1.20 illustrate the numerical implementation in calculating the displace- ments at every point using (1.41) to (1.43). Even though the coefficients of all of the wave terms are given in an explicit form, the waves are still given by infinite series. Thus, the infi- nitesumhastobetruncatedtoasummationoffiniteseriesinthenumericalevaluation. Asin allthepreviouswork,ateachgivenvalueoffrequency,asufficientnumberoftermshastobe used to make sure that the sum from the finite series converges to that of the infinite series at each frequency. Figures 1.16 through 1.20 are for dimensionless frequencies ⌘ =1,4,8,15, and 50. These plots for each frequency consist of four subplots all of which are the displace- mentamplitudesalongradialdistanceofx/afrom0to5,butcalculatedwithvariousnumbers offiniteseries. Thetotalnumberoftermsused,N,islabeledasNterminthefigures. Figure1.16for⌘ =1,has Nterm=2,4,6,8whenitalreadyshowsconvergence. Notethatthiswasthehighestfrequency we couldachieve in theolder, related workfrom the 1970s and1980s. Figure 1.17 for⌘ =4, 31 has Nterm=2, 4, 8,15. Figure1.18 for⌘ =8, has Nterm=4, 8,15, 20andwillnotconvergefor Nterm<15. Fig.1.19 for ⌘ =15 needs Nterm around 25 to guarantee convergence. Finally, Figure1.20for⌘ =50needsNtermaround60to80toguaranteeconvergence. 1.4.4 Conclusions BasedontheplotsanddiscussionaboveforFig. 1.4toFig. 1.20,thefollowingcanbeconcluded: 1. By using the Legendre polynomials (of odd degrees) and half-range expansions, the spherical-wave functions that can satisfy the stress-free boundary conditions automatically on the half-space surface have been presented. In the case of the normal incidence, the explicit, analytic expressions were derived for each coefficient of the spherical-wave func- tions. As a result, the numerical calculations become simpler and the results for the much higherfrequenciescanbeobtained. 2. In Lee (Lee, 1982), it was stated that the amplification of the surface displacement ampli- tudes around the hemispherical canyon can be high. Although the same conclusion now holds,itisformuchhigherfrequenciesaswell,asshowninthischapter. 3. Spikesindisplacementamplitudesareobservedattherimofthecanyoninallofthefigures. These spiked displacement amplitudes could not be observed in the previous work because Lee (1982) was not able then to get results at the dimensionless frequencies beyond ⌘ =1. Earlier, calculations were only done at the lower range of frequencies, which tend not to “see” these corner points. Now the frequency as high as ⌘ =50 can be achieved. These cornerpointsareoftenknowntocreatesecondarywavesources. 4. Previously,itwasstatedthat,in(Lee,1982),“Thedimensionlessfrequency⌘ playsanimpor- tant role in determining the displacement patterns. Larger values of ⌘ will result in higher complexity of displacements and in higher amplifications” The same conclusion holds for theresultsinthischapter. 32 5. Inthechapterstofollow,theaboveconceptcanandwillnextbeextendedtoelasticplaneand point-sources with arbitrary-oblique incidence, which will be non-axisymmetric, and both thefree-filedwavesandscatteredP-andS-waveswillbeoftheform,form=0,1,2,...n m z n (kr)P m n (u) cosm sinm , wherez n (kr) is the Bessel or Hankel functions of the 1 st and/or 2 nd kind of degreen, k = k ↵ is for the P-waves and k = k is for the S-waves; P m n (u) is the associated Legendre polynomialsofordermanddegreenwithargumentu=cos✓ . The right form of P- and S-wave functions will again have to be derived and chosen so that theycansatisfythezero-stressboundaryconditionsonthehalf-spacesurface. 1.5 StructureoftheDissertation This dissertation presents the formulation and application of series expansion of wave with the odd-term-only Legendre polynomials in the three-dimensional hemispherical boundary condition problems on the infinite half-space. Chapter 1 introduces the background of this research field in EarthquakeEngineering,andexplainsthisnewmethodologywithasimpleaxisymmetricexample for the normal P-incidence. Chapter 2 solves for the effects of point-source P-wave right under the spherical canyon. The model is advantageous to study on impacts made by earthquakes with shallow epicenter. Chapter 4 is about the arbitrary angles of incidence of P-, SV- and SH-waves by a hemispherical alluvial valley. The resultant displacement amplitudes are presented and dis- cussed. Chapter 3 looks into the general arbitrary angles of incidence of P-, SV- and SH-waves interacting witha hemispherical canyon withoutany sedimented medium. Chapter 5 statesa soil- structure-interaction (SSI) problem with a rigid foundation and vertical P-incidence. The results arecomparedwithpreviouspapersandagoodagreementhasbeenobtained. 33 1.6 Appendix to Chapter 1: Derivation of Scattered and DiffractedWavesforVerticalP-WaveIncidence The scattered and diffracted waves will be derived in what follows as the superposition of waves fromthesetwofullspacemodels: Model0andModel1. Considerthefull-spacemodel,Model0,inFig.1.21,whereaplaneP-waveismovingvertically upwardinthenegativez-direction,propagatingonthex-zplane,incidentontothesphericalcavity (exp(i!t)isassumedtobepresentonallwavetermsinwhatfollows). Figure1.21. Model0: VerticallyupwardincidentplaneP-waveontosphericalcavity. ThepotentialoftheincidentP-waveisgivenby: ' (i) = ' 0 exp(ik ↵ z) = ' 0 1 X n=0 (2n+1)(i) n j n (k ↵ r)P n (u). (1.45) 34 Thepresenceofthesphericalcavityresultsinscatteredanddiffractedwavesoftheform: ' s 0 = ' 0 1 X n=0 A n h (1) n (k ↵ r)P n (u), s 0 = ' 0 1 X n=0 C n h (1) n (k r)P n (u). (1.46) All waves here are independent of the spherical coordinate . ' s 0 and s 0 form a complete solutionofthiswaveproblem. ThecoefficientsA n andC n ofthepotentialscanbesolvedinterms ofthecoefficientsoftheincidentwaveexactlybytheboundaryconditions rr = ⌧ r✓ =0, (1.47) atr = a,whereisthesurfaceofthesphericalcavity. In exactly the same way, consider the 2 nd full space model, Model 1, in Fig. 1.22, where a z plane P-wave is moving vertically downward along the positive z-axis, propagating on the x-z plane,incidentontothesphericalcavity. Withrespecttothe(x 1 ,y 1 ,z 1 )rectangular-coordinatesystem,andthecorrespondingspherical- coordinate system (r 1 ,✓ 1 , 1 ) with thez 1 axis pointing upward, the incident P-wave potential' (i) 1 in Fig.1.22 wouldhave the same form asthe' (i) potential in Fig.1.21, as it would begoing in the negativez 1 direction. Withu 1 =cos✓ 1 ,then ' (i) 1 = ' 1 exp(+ik ↵ z)= ' 1 exp(ik ↵ z 1 ) = ' 1 1 X n=0 (2n+1)(i) n j n (k ↵ r 1 )P n (u 1 ), (1.48) withtheamplitude' 1 . 35 Figure1.22. Model1: VerticallydownwardincidentplaneP-waveontosphericalcavity. The corresponding scattered and diffracted waves ' s 1 and s 1 , with respect to the (r 1 ,✓ 1 , 1 ) coordinates,alsotakethesameformasthe' s 0 and s 0 inFig.1.21that: ' s 1 = ' 1 1 X n=0 A n h (1) n (k ↵ r 1 )P n (u 1 ) s 1 = ' 0 1 X n=0 C n h (1) n (k r 1 )P n (u 1 ) (1.49) As in the first case, the potentials ' s 1 and s 1 form a complete set of solutions for the model in Fig.1.22. The two spherical coordinate systems, (r,✓, ) of Fig.1.21 and (r 1 ,✓ 1 , 1 ) of Fig.1.22 arerelatedby r 1 = r, ✓ 1 = ⇡ ✓, 1 =2⇡ , sothat u 1 =cos✓ 1 =cos✓ =u. (1.50) 36 Figure1.23. Half-SpaceModel: VerticallyplaneP-waveontohemisphericalcanyon. And' s 1 and s 1 inthe (r,✓, )coordinateswouldtaketheform ' s 0 = ' s 0 (r,✓ )= ' 0 1 X n=0 A n h (1) n (k ↵ r)(1) n P n (u), s 0 = s 0 (r,✓ )= ' 0 1 X n=0 C n h (1) n (k r)(1) n P n (u), (1.51) asP n (u 1 )=P n (u)=(1) n P n (u). Noeconsiderahalf-Spacemodelofanhemisphericalcanyoninanelastichalf-spacesubjected to a vertically incident plane P-wave of potential ' i . The presence of the half-space will result in thereflectedplaneP-waveofpotential' r propagatingverticallydownwards,asshowninFig.1.23. TheincidentandreflectedplaneP-wavesare: ' i = ' 0 exp(ik ↵ z), ' r = ' 1 exp(+ik ↵ z), (1.52) 37 which together satisfy the zero-stress boundary conditions on the half-space surface. Take the amplitude of incident potential ' 0 =1. It is then known that the reflected potential has its ampli- tude' 1 =1. The half-space model in Fig. 1.23 can now be considered as a superposition of the model in Fig. 1.21andtheonein1.22suchthat ' i ofHalf-SpaceModel = ' i 0 ofModel0, ' r ofHalf-SpaceModel = ' i 1 ofModel1. (1.53) In other words, the half-space model is the superposition of Model 0 and Model 1. The presence of the hemispherical canyon will result in scattered and diffracted waves ' s and s , which can be takenasthesuperpositionofthescatteredanddiffractedwavesofModel0andModel1: ' s = ' s 0 +' s 1 , s = s 0 + s 1 , (1.54) whichtakestheform,with' 0 =1and' 1 =1. Fromtheaboveequations ' s = ' s 0 +' s 1 = 1 X n=0 A n h (1) n (k ↵ r)P n (u) 1 X n=0 A n h (1) n (k ↵ r)(1) n P n (u) = 1 X n=0 A 0 2n+1 h (1) 2n+1 (k ↵ r)P 2n+1 (u), (1.55) whicharewiththeodd-degree-onlytermsandA 0 2n+1 =2A 2n+1 . Similarly, s = s 0 + s 1 = 1 X n=0 C n h (1) n (k r)P n (u) 1 X n=0 C n h (1) n (k r)(1) n P n (u) = 1 X n=0 C 0 2n+1 h (1) 2n+1 (k r)P 2n+1 (u), (1.56) 38 whichareagainwiththeodd-degree-onlytermsandC 0 2n+1 =2C 2n+1 . Equations (1.55) and (1.56) are now the complete solution for the scattered and diffracted wavestobeusedinthischapter,whichcanbeshowntosatisfythezero-stressboundaryconditions onthehalf-spacesurface. TheyarethesuperpositionofthecompletesolutionsfromModel0and Model1. 39 Chapter2 TheNormalPoint-SourceP-incidence As was discussed previously in Section 1.4, the odd Legendre polynomial of half-range expan- sion can effectively solve for the waves scattered around a 3-D hemispherical canyon in a elastic, homogenous, and isotropic half-space with a normal plane P-wave incidence. In this chapter, the samemethodologywillbeappliedtoasimilarmodelofcanyonbuttheincidenceisapoint-sourced P-wave. In the perspective of seismic technology, a point-sourced incidence is very pertinent for an earthquake with a shallow epicenter. For this, some numerical approaches were developed, such astheboundaryelementmethodby(Altermanetal.,1970),and(Tadeuetal.,2001),andthefinite differenceschemeby(AltermanandKaral,1968). Ananalyticmethodofwavediffractionaround asmallfull-spacespherewasobtainedby(Dassiosetal.,1999),inwhichthemethodoftheimage ofapointsourcedevelopedby(Rudnick,1947)wasborrowed. Nevertheless,noeffectiveanalytic solutions for the P- or SV-incidence in the half-space have been accomplished, because of the difficultiesofthemodeconversiondiscussedinChapter1. Recently in 2013, Kara and Trifunac (Kara and Trifunac, 2013) investigated the finite line source at a distance comparable to the size of the circular alluvial valley for the out-of-plane SH waves. TheycomparedtheresultantdisplacementamplitudeswiththosebytheplaneSH-incidence and provided a positive conclusion that the plane waves are good approximations for studies of earthquake engineering. In the end of this chapter, the same conclusion will be made based on the comparison of results by the plane P-wave incidence and by the point-source with various frequencies. 40 2.1 TheModel Depicted in Fig. 2.1, a canyon is located on the surface of the half-space, which is featured by the density ⇢ , and the Lamé’s parameters µ, and . The canyon is hemispherical with its radius denoted by a. The point source of wave is right below the center of the canyon, and is emitting thelongitudinalwave(P-wave)only. Thedistancefromthepointsourcetothecanyoncenterisd, whichisperpendiculartothehalf-spacesurface. Figure2.1. 3-Dhemisphericalcanyonwithpoint-sourceincidence. Naturally, a spherical coordinates centered at the canyon center is adopted. Figure 2.2 shows therelationshipbetweentheCartesianandthesphericalsystems. Thexy planeisthehalf-space surface, and following the right-hand-rule, thezaxis is downward. The spherical coordinates is composed by the radial distance r, the polar angle ✓ , and the azimuthal angle . ✓ in this case is onlyhalf-ranged—✓ 2 [0,⇡/ 2],and✓ =⇡/ 2onthehalf-spacesurface. 41 Figure2.2. Adoptionofsphericalcoordinate. In order to describe the point-sourced P-wave potential ' (i) , a polar system originated at the point source center is selected. In Fig. 2.3, O refers to the center of this new system of depth d vertically below the center of the canyon, and r is the radial distance measured from O. The incidentpotential' (i) canthenbeexpressedas: ' (i) = ' 0 exp(ik ↵ ri!t) r =i' 0 k ↵ h (1) 0 (k ↵ r)exp(i!t), (2.1) where' 0 isascalefactor,k ↵ isthewavenumberoftheP-wave,andh (1) 0 isthefirstkindofHankel’s functionoforder0forthattheincidentwaveissymmetricallyoutwardpropagated. 42 Figure2.3. Thecoordinatesystemsforthepoint-sourceP-incidence. From Graf’s Addition Theorem provided in (Mow and Pao, 1971) and (Abramowitz and Ste- gun, 1972), ' (i) in (2.1) can be expanded in terms of the (r,✓, ) coordinates originated at the canyoncenterinthefollowingform(withthetimefactor exp(i!t)neglected): i' 0 k ↵ h (1) 0 (k ↵ r)= 8 < : P 1 n=0 i' 0 k ↵ (2n+1)h (1) n (k ↵ d)j n (k ↵ r)P n (u),r<d; P 1 n=0 i' 0 k ↵ (2n+1)j n (k ↵ d)h (1) n (k ↵ r)P n (u),rd, (2.2) where j n and h (1) n are the first kind of spherical Bessel function and the first kind of Hankel’s function of order n, and P n with argument u=cos✓ (0 u 1) is the Legendre polynomial of ordern. Note that the argument ofP n isu, instead ofu, because the z-axis is assumed to be downward. Illuminated by the image methods used in some papers previously, such as (Lee and Trifunac, 1979),thereflectedwavepotentialduetothehalf-spaceboundarycanberepresentedbyanegative 43 (anti-symmetric) source ' (r) with respect to ' (i) relative to the half-space surface. The O in Fig. 2.3indicatesthesourcecenterofthereflectedwaveofdistancedverticallyuponthecenterofthe canyon. Therefore,anotherpolarsystemoriginatedatO isadoptedfortheexpressionof' (r) ,then ' (r) =' 0 exp(ik ↵ ri!t) r =i' 0 k ↵ h (1) 0 (k ↵ r)exp(i!t), (2.3) wherer is the radial distance fromO. The time factor exp(i!t) will be again removed because ofthesteady-stateassumption,andtheGraf’sAdditionTheoremisthenappliedto(2.3)toobtain theexpressionof' (r) bythesphericalcoordinates (r,✓, )intheformof i' 0 k ↵ h (1) 0 (k ↵ r)= 8 < : P 1 n=0 i' 0 k ↵ (2n+1)h (1) n (k ↵ d)j n (k ↵ r)P n (u),r<d; P 1 n=0 i' 0 k ↵ (2n+1)j n (k ↵ d)h (1) n (k ↵ r)P n (u),rd. (2.4) Thestress-freeboundaryconditionsonthehalf-spacesurfacearetobeexamined,toverifythat ' (r) isthereflectedwaveof' (i) inthehalf-space. Theseconditionsatz=0,whichare zz = ⌧ zx = ⌧ zy =0, (2.5) should be satisfied by the summation of ' (i) and ' (r) . In the spherical coordinate system, where -factor is ignored due to the property of axisymmetric for this model, the conditions in (2.5) can beexpressedas ✓✓ | ✓ =⇡/ 2 = ⌧ ✓r | ✓ =⇡/ 2 =0. (2.6) Tosuperimpose' (i) and' (r) ,thetotalwave' (ff) canbeexpressedas ' (ff) = ' (i) +' (r) = 8 < : P 1 n=1,3,5,..., a (r<d) n j n (k ↵ r)P n (u),r<d; P 1 n=1,3,5,..., a (r d) n h (1) n (k ↵ r)P n (u),rd, (2.7) 44 where a (r<d) n =2(2n+1)i' 0 k ↵ h (1) n (k ↵ d) n=1,3,5,..., (2.8a) a (r d) n =2(2n+1)i' 0 k ↵ j n (k ↵ d) n=1,3,5,.... (2.8b) 2.2 TheFree-fieldPotential Nowitisgoingtoprovethat' (ff) naturallysatisfiesbothhalf-spaceboundaryconditionsin(2.6). 1. Thezeronormalstressboundaryconditiononthehalf-spacesurface ✓✓ | ✓ =⇡/ 2 =0. From(MowandPao,1971),whenr<d ✓ | ✓ =⇡/ 2 = 1 X n=0 a (r<d) 2n+1 2µ r 2 2 4 [((2n+1) 2 + (k r) 2 2 )j 2n+1 (k ↵ r)k ↵ rj 2n+2 (k ↵ r)]P 2n+1 (u) + 1 sin 2 ✓ [(2n+1)cos 2 ✓P 2n+1 (u)+(2n+1)cos✓P 2n (u)]j 2n+1 (u) 3 5 , whichisequalto ✓ | ✓ =⇡/ 2 = 1 X n=0 a (r<d) 2n+1 2µ r 2 ✓ (2n+1) 2 + (k r) 2 2 ◆ j 2n+1 (k ↵ r)k ↵ rj 2n+2 (k ↵ r) P 2n+1 (0). (2.9) Similarly,whenrd, ✓ | ✓ =⇡/ 2 = 1 X n=0 a (r d) 2n+1 2µ r 2 ✓ (2n+1) 2 + (k r) 2 2 ◆ h (1) 2n+1 (k ↵ r)k ↵ rh (1) 2n+2 ((k ↵ r) P 2n+1 (0). (2.10) KnownthatP 2n+1 (0)⌘ 0,sotheboundarycondition ✓✓ | ✓ =⇡/ 2 =0isalwaysautomatically satisfied. 2. Thezeroshearstressboundaryconditiononthehalf-spacesurface⌧ ✓r | ✓ =⇡/ 2 =0. 45 Thegeneralrelationshipbetweenthelongitudinalwavepotentialandtheshearstressis ⌧ r✓ = 2µ r ✓ @ 2 ' @r@✓ @' r@✓ ◆ . (2.11) The equations in (2.7) will be re-expressed in terms of the even Legendre polynomials, because all boundary conditions are in half-, not full-, space. Such technique is already discussedChapter1and(LeeandZhu,2013). Bythedefinitionin(1.24)that P 2m+1 (u)= 1 X n=0 mn P 2n (u), (2.12) (2.7)becomes ' (ff) = 8 < : P 1 n=0 P 1 m=0 mn a (r<d) 2m+1 j 2m+1 (k ↵ r)P 2n (u),r<d; P 1 n=0 P 1 m=0 mn a (r d) 2m+1 h (1) 2m+1 (k ↵ r)P 2n (u),rd. (2.13) Accordingtothisnewexpression,thereitis @' (ff) @✓ = 8 < : P 1 n=0 P 1 m=0 mn a (r<d) 2m+1 j 2m+1 (k ↵ r)P 1 2n (u), r<d; P 1 n=0 P 1 m=0 mn a (r d) 2m+1 h (1) 2m+1 (k ↵ r)P 1 2n (u), rd, (2.14) whereP 1 2n (u) is also equal to the the first derivative of the Legendre polynomial with order 2nwithrespectto✓ —i.e. P 1 2n (u)=@P 2n (u)/@✓ . From(2.11)and(2.14),⌧ r✓ | ✓ =⇡/ 2 dueto' (ff) is ⌧ r✓ | ✓ =⇡/ 2 = 8 > < > : 2µ r P 1 n=0 P 1 m=0 mn a (r<d) 2m+1 ⇣ @j 2m+1 (k↵ r) @r 1 r ⌘ P 1 2n (u), r<d; 2µ r P 1 n=0 P 1 m=0 mn a (r d) 2m+1 ✓ @h (1) 2m+1 (k↵ r) @r 1 r ◆ P 1 2n (u), rd. (2.15) 46 SinceP 1 2n (0)⌘ 0at✓ = ⇡/ 2forthehalf-spacesurface,theshearstressboundarycondition ⌧ r✓ | ✓ =⇡/ 2 =0isagainautomaticallysatisfied. Therefore' (r) canbeverifiedtobethereflectedwavefromthehalf-space(withoutanytopog- raphy). In addition, the (ff) defined in (2.7) is the free-field wave for a point-source P-incidence inthehalf-space. 2.3 TheScatteredWaves Thescatteredwavesexistduetothepresenceofthehemisphericalcanyon. Thesamemethodology is applied as in Chapter 1 that the scattered waves potentials will be expressed in terms of odd Legendrepolynomialsonly(see2.6)that ' (s) = 1 X n=0 A 2n+1 h (1) 2n+1 (k ↵ r)P 2n+1 (u), (s) = 1 X n=0 C 2n+1 h (1) 2n+1 (k r)P 2n+1 (u). (2.16) Using (2.12), the wave potentials can be expressed in terms of the Legendre polynomials of evendegreesas ' (s) = 1 X m=0 1 X n=0 A 2m+1 h (1) 2m+1 (k ↵ r) mn ! P 2n (u), (s) = 1 X n=0 1 X n=0 C 2m+1 h (1) 2m+1 (k r) mn ! P 2n (u). (2.17) Thetotalof' (ff) ,' (s) ,and (s) needtosatisfyallboundaryconditionsonboththehalf-space surfaceandthehemisphericalcanyonsurface,whichare ✓✓ | ✓ =⇡/ 2 = ⌧ ✓r | ✓ =⇡/ 2 =0, (2.18) 47 and rr | r=a = ⌧ r✓ | r=a =0. (2.19) Allboundaryconditionswillbediscussedinthefollowing. 1. Thezeronormalstressboundaryconditiononthehalf-spacesurface ✓✓ | ✓ =⇡/ 2 =0. OnlythescatteredP-andS-wavepotentialswillbetakenintoconsiderationwhenevaluating normalstressessince' (ff) isalreadyprovedtosatisfythisboundarycondition. With(2.16), the P- and S- potentials with argument u =cos✓ have the following expressions of the normalstressonthehalf-spacesurface: ✓ | ✓ =⇡/ 2 = 1 X n=1,3,5... h A n E (3) 21 (n,k ↵ r)+C n E (3) 23 (n,k r) i P n (0). (2.20) Equation (2.20) is the summation over all the odd integers, n=1,3,5,..., and P n (0) = 0 for all odd n-degree Legendre polynomials, and thus for all ra, ✓✓ | ✓ =⇡/ 2 =0 is always automatically satisfied. All the E functions above and following in the paper are listed in AppendixB.2. 2. Thezeroshearstressboundaryconditiononthehalf-spacesurface⌧ ✓r | ✓ =⇡/ 2 =0. Similar to Boundary Condition 1, only the scattered P- and S-wave potentials are used to satisfythethisshearstressonhalf-spacesurface. Heretheexpressionsin(2.17)areadopted, andthentheshearstressbytheP-andS-wavesareexpressedas: ⌧ ✓r | ✓ =⇡/ 2 = 1 X n m n=2,4,6... " 1 X m=1,3,5... ⇣ A m E (3) 41 (m,k ↵ r)+C m E (3) 43 (m,k r) ⌘ # P 1 n (0). (2.21) P 1 n (0)isthefirstderivativeofP n (0)sothatP 1 n (0)isoddwhenniseven. Summingupalong n=0,2,4,..., the result is definitely zero since every term is multiplying an odd function 48 withargumentzero. Thereforeconclusioncouldbereachedthatthezeroshearstressisalso satisfiedonthehalf-spacesurface. Now two out of thefour boundary conditions are automatically satisfied (removed), so then the remaining two boundary conditions will next be utilized to solve for the two sets of unknowncoefficients,{A 2n+1 }and{C 2n+1 },oftheP-andS-wavepotentials. 3. Thezeronormalstressboundaryconditiononthecanyonsurface rr | r=a =0. Thetotalnormalstress rr iscomposedofthefree-fieldwaveandthescatteredwavesthat rr | r=a = (ff) rr | r=a + (s) rr | r=a =0, (2.22) thusthefollowingrelationshipisthenobtained: (s) rr | r=a = (ff) rr | r=a . (2.23) where (ff) rr isthenormalstressduetothefree-fieldpotential,and (s) rr isthepartduetothe scatteredwaves. Theirexpressionsinthesphericalcoordinatesystemaregivenas: (ff) r | r=a = 1 X n=0 a 2n+1 E (1) 11 (2n+1,k ↵ a)P 2n+1 (u), (s) r | r=a = 1 X n=0 ⇣ A 2n+1 E (3) 11 (2n+1,k ↵ a)+C 2n+1 E (3) 13 (2n+1,k a) ⌘ P 2n+1 (u). (2.24) Combining(2.23)and(2.24),thereis 1 X n=0 ⇣ A 2n+1 E (3) 11 (2n+1,k ↵ a)+C 2n+1 E (3) 13 (2n+1,k a) ⌘ P 2n+1 (u) = 1 X n=0 ⇣ a 2n+1 E (1) 11 (2n+1,k ↵ a) ⌘ P 2n+1 (u). (2.25) 49 Then to apply the orthogonality of odd Legendre polynomials over the half-space, (2.25) is simplifiedtobe: A 2n+1 E (3) 11 (2n+1,k ↵ a)+C 2n+1 E (3) 13 (2n+1,k a)=a 2n+1 E (1) 11 (2n+1,k ↵ a). (2.26) 4. Thezeroshearstressboundaryconditionsonthecanyonsurface⌧ r✓ | r=a =0. Similar to Boundary Condition 3, the zero shear stress ⌧ r✓ | r=a is from both the free-field waveandthescatteredwaves,sothat ⌧ (s) r✓ | r=a =⌧ (ff) r✓ | r=a . (2.27) Moreover,fromAppendixB.2, ⌧ (ff) r✓ | r=a = 1 X n=0 a 2n+1 E (1) 41 (2n+1,k ↵ a)P 1 2n+1 (u), ⌧ (s) r✓ | r=a = 1 X n=0 ⇣ A 2n+1 E (3) 41 (2n+1,k ↵ a)+C 2n+1 E (3) 43 (2n+1,k a) ⌘ P 1 2n+1 (u). (2.28) Combining(2.27)and(2.28),thereis 1 X n=0 ⇣ A 2n+1 E (3) 41 (2n+1,k ↵ a)+C 2n+1 E (3) 43 (2n+1,k a) ⌘ P 1 2n+1 (u) = 1 X n=0 ⇣ a 2n+1 E (1) 41 (2n+1,k ↵ a) ⌘ P 1 2n+1 (u), (2.29) Afterapplyingtheorthogonalityof{P 1 2n+1 (u)},n=0,1,2,...,,thatisasetofevenLegendre functions,foreachnthecoefficientsA 2n+1 andC 2n+1 havethefollowingrelation: A 2n+1 E (3) 41 (2n+1,k ↵ a)+C 2n+1 E (3) 43 (2n+1,k a)=a 2n+1 E (1) 41 (2n+1,k ↵ a) (2.30) 50 (2.26)and(2.30)canberearrangedtobe 2 4 E (3) 11 (2n+1,k ↵ a) E (3) 13 (2n+1,k a) E (3) 41 (2n+1,k ↵ a) E (3) 43 (2n+1,k a) 3 5 8 < : A 2n+1 C 2n+1 9 = ; =a 2n+1 8 < : E (1) 11 (2n+1,k ↵ a) E (1) 41 (2n+1,k ↵ a) 9 = ; (2.31) Therefore,forn=1,3,5,...,,thecoefficientsA n andC n aregivenby 8 < : A n C n 9 = ; = a n Det(n) 2 4 E (3) 43 (n,k a) E (3) 13 (n,k a) E (3) 41 (n,k ↵ a) E (3) 11 (n,k ↵ a) 3 5 8 < : E (1) 11 (n,k ↵ a) E (1) 41 (n,k a) 9 = ; , (2.32) withthedeterminantDet(n)=E (3) 11 (n,k ↵ a)E (3) 43 (n,k a)E (3) 13 (n,k a)E (3) 41 (n,k ↵ a). Notethata 2n+1 shouldbea (r<d) 2n+1 of(2.8a)becausethesourcelocationmustbelowbottommost partofthecanyon,orinanotherword,r<d. Thenthefree-fieldcoefficientsin(2.32)are a n =(4n+2)i' 0 k ↵ h (1) n (k ↵ d) n=1,3,5,.... (2.33) 2.4 SurfaceDisplacement (a) VerticalDisplacements The amplitude of displacement on a point is a very important assessment of devastation on that point resulted by the seismic waves. The vertical displacement amplitude |U z | has been focused on because the problem presented in this chapter is a axisymmetric problem with the longitudinal wave. U z can be calculated from the spherical components U r and U ✓ by the transformationbetweenthetwocoordinates,therefore, U z =U r cos✓ U ✓ sin✓, (2.34) 51 where U ✓ = 1 X n=1,3,5,..., h a n D (1) 21 (n,k ↵ a)+A n D (3) 21 (n,k ↵ a)+C n D (3) 23 (n,k a) i P n (u) d✓ , U r = 1 X n=1,3,5,..., h a n D (1) 11 (n,k ↵ a)+A n D (3) 11 (n,k ↵ a)+C n D (3) 13 (n,k a) i P n (u). (2.35) andD (i) jk thedisplacement-potentialexpressionsaregiveninAppendixB.1. The displacement amplitude is the magnitude of a complex-valued displacement, so that for U z |U z |=[Re 2 (U z )+Im 2 (U z )] 1/2 . (2.36) Aswasstatedin(2.1)that' 0 isthescalefactorforthewavepotentials,thisscalefactorshould be determined before proceeding the numerical calculation. According to the plane P-wave incidence,' 0 canbedeterminedbythedisplacementamplitudeattheoriginalpointsuchthat suchthat|U z |resultedbythefree-fieldwaveatthecanyoncenterisalways“2”. Given' (i) = ' 0 e ik↵ r r and' (r) =' 0 e ik↵ r r ,theverticaldisplacementduetotheincidentwave is U z = @' (i) @r = ' 0 exp(ik ↵ r)  ik ↵ r 1 r 2 , (2.37) andthenthatduetothereflectedwaveis U z = @' (r) @r = ' 0 exp(ik ↵ r)  ik ↵ r 1 r 2 . (2.38) To sum up (2.37) and (2.38), the total vertical displacement at the original point, where r = r =d,is U z | r=0 =2' 0 exp(ik ↵ d)  ik ↵ d 1 d 2 . (2.39) 52 Therefore,define ' 0 =exp(ik ↵ d)  ik ↵ d 1 d 2 1 , (2.40) to maintainU z at the center of canyon is a dimensionless number “2”, and the coefficients of free-fieldpotentialscanbeexpressedas a (r<d) 2n+1 =(8n+6)ik ↵ exp(ik ↵ d)  ik ↵ d 1 d 2 1 h 2n+1 (k ↵ d),n=0,1,2,..., (2.41a) a (r d) 2n+1 =(8n+6)ik ↵ exp(ik ↵ d)  ik ↵ d 1 d 2 1 j 2n+1 (k ↵ d),n=0,1,2,.... (2.41b) Moreover,a n in(2.42)forthecalculationofA n andC n isthengivenby a n =(4n+2)ik ↵ h (1) n (k ↵ d)exp(ik ↵ d)  ik ↵ d 1 d 2 1 n=1,3,5,.... (2.42) whichcontainsboththesourcelocationd,andthewavenumberk ↵ . Withthecoefficientsabove,thefollowingdimensionlessresultscanbeobtained: Figure2.4. Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =010andd=1.5a. 53 Figure2.5. Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =1020andd=1.5a. Figure2.6. Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =010andd=5.0a. 54 Figure2.7. Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =1020andd=5.0a. Figure2.8. Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =010andd=20.0a. 55 Figure2.9. Verticaldisplacementamplitude|U z |alongx/afor thenormalpoint-sourceincidence: ⌘ =1020andd=5.0a. Figures 2.4 to 2.9 are the vertical displacement amplitude |U z | along the dimensionless radial distance(x/a)withafixedwavesourcelocationandvariousdimensionlessfrequencies⌘ ,fol- lowingthedefinitioninChapter1,that ⌘ = !a ⇡ = 2fa = k a ⇡ = 2a . (2.43) The azimuthal angle will not be considered because this whole model is axisymmetric, the axisx/a is the radial distancer/a along any direction. The range ofx/a from 0 to 1 stands forthedistancefromthecentertotherimonthecanyonsurface. Whenx/a> 1,thehalf-space surfaceisunderdiscussion. Figures 2.4 and 2.5 present the vertical displacement amplitude |U z | from ⌘ =0 to 10, and from ⌘ =10 to 20 with the same source depth of d=1.5a. One can see that |U z | can be as high as 20 when ⌘ is very small, but it decays fast along the radial distance. There is little oscillation observable on the half-space surface in the low ⌘ range when x/a > 1. With the 56 continuously increasing ⌘ , the magnitude of |U z | on the canyon surface obviously decreases, and there are clear wave patterns on the half-space surface meaning that the particles at a relativelyfardistancefromthecanyoncenterstarttovibrate. Figures 2.6 and 2.7 are |U z | from ⌘ =0 to 10, and from ⌘ =10 to 20 with the same source depthofd=5.0a,respectively. Thistimethemaximum|U z |inbothfiguresdonotexceed3.0, andthedecaypatternislessobviouscomparedtothatofFig. 2.4andFig. 2.5when⌘ issmall. It can also be seen that the wave oscillation is substantial, particularly in the high frequency range. At the rim of canyon with ground, where x/a=1, the "dip and spike" phenomenon, whichisdescribedinChapter1and(LeeandZhu,2013),isobserved. Figures 2.8 and 2.9 contain the more intense wave motions at frequencies of ⌘ =010 and ⌘ =1020 respectively, when the depth of source isd=20a below the original point. One can notice that the amplitude of shaking is more frequent, compared with the prior two sets of figures at d=1.5a, and d=5.0a, for a given ⌘ . At the low frequency range where ⌘ is closeto0,theamplitudesalongtheradialdistancearebarelychangedandthisagreeswiththe observation of the normal plane P-incidence in the previous chapter. Moreover, the "dip and spike"phenomenabecomemoreapparentandstable. 57 Figure2.10. Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5and⌘ =0.02,0.5,1.0and1.5. Figure2.10includesfourplotsof|U z |atd/a=1.5for⌘ =0.02,0.5,1.0,and1.5,respectively. It is apparent that |U z | at the origin decreases with the increase of ⌘ . However, for a location outside the canyon, its amplitude for ⌘ =0.02 or ⌘ =0.5 are generally lower than that of ⌘ =1or⌘ =1.5. Toconclude,foranearsourceincidencewithaequivalentlyscaledpotential, the low incident frequencies accumulate energies by the origin, while the higher frequencies transmitenergiesbytheeffectivewavepropagation. Figures 2.11 to 2.13 display the vertical displacement amplitudes |U z | at d/a=1.5, 5, and 20 for⌘ =0.02,1,and20,respectively. 58 Figure2.11. Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5,5,and20for⌘ =0.02 Figure2.12. Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5,5,and20for⌘ =1.0 59 Figure2.13. Verticaldisplacementamplitude|U z |alongx/aforthe normalpoint-sourceincidencewithd/a =1.5,5,and20for⌘ =20.0 It could be demonstrated from Fig.2.11 that for the very low frequency incidence—i.e. ⌘ =0.02, if the epicenter is shallow, the bottommost point of the canyon has the most sub- stantialdeformation. Whileontheotherhand,iftheepicenterisdeep,theplotsappeartohave comparableamplitudesoutsidethecanyonofthoseonthecanyonsurface. Figures 2.12 and 2.13 show that when ⌘ gets higher, the canyon bottom still suffers the most severe damage from the shallow epicenter—atd/a=1.5. The decay rates of all the plots for ⌘ =20 are much lower than their counterparts for ⌘ =1.0 then ⌘ =0.02. In addition, the deep sources do not contribute to a remarkably high amplitude but its impacting area is much expanded. Figure 2.14 provides x/a=0 to 100 to inspect the amplitudes in the perspective of a longer range. Apparently, the amplitudes resulted by a shallow wave source decay faster along the radial distance than that resulted by a deep source. In addition, the amplitudes from all the point-sourcedcaseswilleventuallydroptozero, atsomepointsonthehalf-spacesurfacethat issufficientlydistantfromtheorigin. 60 Figure2.14. Verticaldisplacementamplitude|U z |alongx/ato100forthe normalpoint-sourceincidencewithd/a =1.5,5and20and⌘ =1.0 (b) Comparisonwithplanewaveincidence:d>> a. Theplanewaveincidencescan,intuitively, precisely simulate the point-sourced incidences if the wave sources are deep enough. So the following part is to examine the conditions for a vertical plane incidence to replace a vertical point-sourced incidence. First of all, the free-field coefficients a n of a deep normal point- sourced incidence in (2.42) can be proven equal to the coefficients of the normal plane free- fieldpotentialsof(1.15)inChapter1. From(2.42),whendisverylarge,thecoefficientsa 2n+1 canbesimplifiedas(n=0,1,2,...): a 2n+1 =(8n+6)ik ↵ exp(ik ↵ d)  ik ↵ d 1 h 2n+1 (k ↵ d) d>> a; (2.44) whilethecoefficientsforplanewaveincidenceare: a 2n+1 = (8n+6)i (2n+1) ik ↵ . (2.45) 61 Fromthemathematicalrelationshipthat h 2n+1 (k ↵ d)= 1 k ↵ d e i(k↵ d+(n+1)⇡ ) whend!1 , (2.46) (2.44)and(2.45)canbeprovedequal. In Fig. 2.15 to Fig. 2.17, the results of the normal plane incidence and of the normal point- sourced incidences with various depths are visualized and compared. The patterns of results always agree at the lower range of x/a first, then start to differ when the referenced point movesfarther. Forthecaseof⌘ =1inFig. 2.15, theplotofd=100acanvisuallymatchthe result of the plane incidence up tox/a=2.5; but for ⌘ =4 in Fig. 2.16, the agreeable range shrinkstobearound2alongthehorizontalaxis;andfor⌘ =8inFig. 2.17,thisnumberofthe upper-bound is only less than two. Thus, for a point-sourced incidence at a fixed depth, the highincidentfrequency⌘ narrowstheefficientrangeofasubstitutedplaneincidence. Figure2.15. Verticaldisplacementamplitudes|U z |alongx/aforvariousdwith⌘ =1.0. 62 Figure2.16. Verticaldisplacementamplitude|U z |alongx/aforvariousdwith⌘ =4.0. Figure2.17. Verticaldisplacementamplitude|U z |alongx/aforvariousdwith⌘ =8.0. Forthis,whenthefrequencyishigh,theplaneincidencescannotbeusedforthesimulationof the point-sourced incidence unless it is located deep—such asd=200a for ⌘ =1,d=500a for ⌘ =4, and d=1000a for ⌘ =8 which can guarantee the good matches along the radial distanceupto 5inthesefigurepresented. 63 2.5 Conclusion This chapter applies the series expansion of odd-order Legendre polynomials to the wave diffrac- tion around a hemispherical canyon by a vertical point-sourced P-incidence. Results of various sourcedepthsanddimensionlessfrequenciesarediscussed. Followingarethemajoraspectssum- marizedforthisproblem: • The antisymmetric image method and the odd-order-only Legendre polynomials can be effectively used to relax the zero-stress boundary conditions on the half-space. The cal- culablerangeof⌘ ishigh,althoughthehighest⌘ selectedinthediscussionis 20. • Theoscillationsareintensiveneartherimofcanyon,inparticularforthecaseofhigh⌘ . So thatthe“dipandspike”effectalsoexistsinthepoint-sourcedproblems. • A higher ⌘ must result in a more complex oscillatory pattern on the surface of canyon, and the half-space. The same conclusion was already made for the normal plane P-incidence in thepreviouschapter. • Forincidencewithashallowepicenter,thesubstantialimpactscanbecaughtonthecanyon surface, while the influence on the surface of the half-space is relatively trivial and decays rapidlytozero. • For incidence with a deep epicenter, its impact on the half-space surface in the near range is comparable to that on the canyon surface. The deeper source and higher frequency both helpextendtherangeofimpact. However, theamplitudeswilldroptozeroatafardistance eventuallyaslongastheincidenceispoint-sourced. • The statement in (Kara and Trifunac, 2013) “the results suggest that for the frequencies of interestinearthquakeengineering,theplane-waverepresentationofincidentSHwaveswill lead to reasonably realistic predictions of amplification of strong-motion amplitudes on the 64 groundsurface”stillholdsforthe3-DverticalP-waveincidenceaswellasforalargerrange ofthedimensionlessfrequencies. 2.6 AppendixtoChapter2: DerivationofScatteredWavesfor NormalPoint-SourceP-waveIncidence This part is to prove the validity of the odd-term-only Legendre polynomials of the scattered P- andS-waves. SimilartoAppendix1.6,thescatteredwavesbyahemisphericalcanyoninthehalf- spacecanbeseenasthesuperpositionofwavesfromtwofullspacemodelsofthesphericalcavity: Model0andModel1. Model0isdescribedinFig.2.18,thatthereisasourcelocatedbelowthecavity,onthepositive z-axiswithadistanced,emittingtheP-waveradially. TheP-waveispropagatingonthex-zplane anditinteractsonthesphericalcavitysurfacethengeneratesthescatteredP-andSV-waves. (Term exp(i!t)isomittedinallequationsforconvenience.) ThepotentialoftheincidentP-waveisgivenby ' i 0 = 8 < : P 1 n=0 i' 0 k ↵ (2n+1)h (1) n (k ↵ d)j n (k ↵ r)P n (u),r<d, P 1 n=0 i' 0 k ↵ (2n+1)j n (k ↵ d)h (1) n (k ↵ r)P n (u),rd, (2.47) where r is the radius and u=cos✓ that ✓ is the polar angle of the spherical coordinates that is sharingthesameoriginwiththeCartesiancoordinatesinFig.2.18. The scattered waves due to presence of the spherical cavity can be expressed in the following form: ' s 0 = ' 0 1 X n=0 A n h (1) n (k ↵ r)P n (u), X s 0 = ' 0 1 X n=0 C n h (1) n (k r)P n (u). (2.48) 65 Figure2.18. Model0: Upwardpoint-sourceincidentp-waveontosphericalcavity. Bothexpressionsareindependentofthesphericalcoordinate sincethemodelisaxisymmet- ric. CoefficientsA n andC n canbesolvedexactlybytheboundaryconditions rr = ⌧ r✓ =0, (2.49) atr = a where is the surface of the spherical cavity. Thus ' s 0 and s 0 form a complete solution of thiswaveproblemthatwasalreadysolvedinthe1970s(MowandPao,1971). Similarly, the 2 nd full space model will be considered. Different from Model 0, Model 1 in Fig.2.19depictsasourcerightabovethesphericalcavitywiththesameconditions. Thenarotated Cartesiancoordinatesystemisadoptedherethat x 1 =x, y 1 =y, z 1 =z. (2.50) 66 Figure2.19. Model1: Downwardpoint-sourceincidentp-waveontosphericalcavity. (r 1 ,✓ 1 , 1 )isthesphericalcoordinatesystemcorrespondingtotheCartesiancoordinatesystem (x 1 ,y 1 ,z 1 ). Withu 1 =cos✓ 1 ,theincidentP-wavepotential' i 1 inFig.2.19canbeexpressedas ' i 1 = 8 < : P 1 n=0 i' 1 k ↵ (2n+1)h (1) n (k ↵ d)j n (k ↵ r 1 )P n (u 1 ),r 1 <d, P 1 n=0 i' 1 k ↵ (2n+1)j n (k ↵ d)h (1) n (k ↵ r 1 )P n (u 1 ),r 1 d, (2.51) with' 1 asitsamplitude. Theresultantscatteredwavepotentials' s 1 and s 1 inthecoordinatesystem (r 1 ,✓ 1 , 1 )aregivenby ' s 1 = ' 1 1 X n=0 A n h (1) n (k ↵ r 1 )P n (u 1 ) s 1 = ' 0 1 X n=0 C n h (1) n (k r 1 )P n (u 1 ) (2.52) 67 whichareinthesameformof' s 0 and s 0 . Thecoefficientsofthescatteredpotentialareagaincom- pletely solved because Model 1 can be taken as an adapted case of Model 0. From the following relationshipsbetweencoordinates (r,✓, )and (r 1 ,✓ 1 , 1 )from(2.50)that r 1 = r, ✓ 1 = ⇡ ✓, 1 =2⇡ , thereis u 1 =cos✓ 1 =cos✓ =u. (2.53) And' s 1 and s 1 arethenre-expressedas ' s 0 = ' s 0 (r,✓ )= ' 0 1 X n=0 A n h (1) n (k ↵ r)(1) n P n (u), s 0 = s 0 (r,✓ )= ' 0 1 X n=0 C n h (1) n (k r)(1) n P n (u), (2.54) forthatP n (u 1 )=P n (u)=(1) n P n (u). Nowthehalf-spacemodelisconsidered. Figure2.20showsahemisphericalcanyonlocatedat the original point of the Cartesian coordinate (x,y,z) that of the same description as before, and the entire x-y plane is an elastic half-space. The point-sourced P-wave of potential ' i originating vertically below the canyon will result in the point-sourced P-wave of potential ' r , so that, from (2.47)and(2.51), ' i = 8 < : P 1 n=0 i' 0 k ↵ (2n+1)h (1) n (k ↵ d)j n (k ↵ r)P n (u),r<d, P 1 n=0 i' 0 k ↵ (2n+1)j n (k ↵ d)h (1) n (k ↵ r)P n (u),rd, and ' r = 8 < : P 1 n=0 i' 1 k ↵ (2n+1)h (1) n (k ↵ d)j n (k ↵ r 1 )P n (u 1 ),r 1 <d, P 1 n=0 i' 1 k ↵ (2n+1)j n (k ↵ d)h (1) n (k ↵ r 1 )P n (u 1 ),r 1 d. 68 Figure2.20. Half-spacemodel: Upwardpoint-sourceincidentp-waveontohemispherical canyon. Because the superposition of ' i and ' r need to satisfy the zero-stress half-space boundary conditions,then' 1 =' 0 . Take' 0 =1sothat' 1 =1. Nowthewholesituationisexactlythe samewithAppendix1.6. Therefore,thesamewordsandsentenceswillbeusedhere. The half-space model in Fig. 2.20 can now be considered as a superposition of the models in Fig. 2.18and2.19suchthat ' i ofHalf-SpaceModel = ' i 0 ofModel0, ' r ofHalf-SpaceModel = ' i 1 ofModel1. (2.55) 69 In other words, the half-space model is the superposition of Model 0 and Model 1. The presence ofthehemisphericalcanyonwillresultinthescatteredwaves' s and s ,whichcanbetakenasthe superpositionofthescatteredwavesofModel0andModel1: ' s = ' s 0 +' s 1 , s = s 0 + s 1 , (2.56) whichtakestheform,with' 0 =1and' 1 =1. Fromtheaboveequations ' s = ' s 0 +' s 1 = 1 X n=0 A n h (1) n (k ↵ r)P n (u) 1 X n=0 A n h (1) n (k ↵ r)(1) n P n (u) = 1 X n=0 A 0 2n+1 h (1) 2n+1 (k ↵ r)P 2n+1 (u), (2.57) whichiswiththeodd-degree-onlytermsandA 0 2n+1 =2A 2n+1 . Similarly, s = s 0 + s 1 = 1 X n=0 C n h (1) n (k r)P n (u) 1 X n=0 C n h (1) n (k r)(1) n P n (u) = 1 X n=0 C 0 2n+1 h (1) 2n+1 (k r)P 2n+1 (u), (2.58) whichisalsowiththeodd-degree-onlytermsandC 0 2n+1 =2C 2n+1 . Equations (2.57) and (2.58) are now the complete solution for the scattered waves to be used in the paper, which can be shown to satisfy the zero-stress boundary conditions at the half-space surface. TheyarethesuperpositionofthecompletesolutionsfromModel0andModel1. 70 Chapter3 DiffractionaroundaHemisphericalCanyon byWavesofArbitraryIncidence The impact of seismic waves around a 3-D canyon located on the half-space will be evaluated. In thischapter,theproblemofplanewaveswitharbitraryangleofincidencewillbefocusedon. Sec- tion 3.1 provides a general description of the canyon model and the related boundary conditions. ResultantamplitudesofdisplacementsbyP-,SV-,andSH-incidenceswillbediscussedinSection 3.2, 3.3, and 3.4 respectively. Section 3.6 presents the technical details of the series expansion of wave potential applications on every boundary condition to solve for the coefficients of scattered waves. Finally, the complete sets of 3-D graphs of resultant displacement amplitudes are given in AppendiceD.1,D.2,andD.3. 3.1 A Description of the Canyon Model and Boundary Condi- tions Figure 3.1 shows the model for wave diffraction around a hemispherical canyon with radius a on an infinite homogeneous, isotropic, and elastic half-space. The half-space medium has density ⇢ , Lamé parameters , and µ. The space above half-space is a vacuum so that both the half-space and the canyon surface are stress-free. Spherical coordinates (r,✓, ) are adopted wherer0 is the radial distance from the origin; ✓ , that 0 ✓  ⇡ is the polar angle being measured from the z-axis,and 2 [⇡,⇡ ]istheazimuthalanglefromthepositivex-axis. Inthishalf-spacemodel,z isalwayspositivesothatangle✓ onlyrangesfrom0to⇡/ 2. Thefree-fieldwavepotential,denoted 71 as' (ff) , (ff) , and (ff) , andwhosecoefficientsofseriesexpansioninsphericalcoordinateswill begiveninthenextsections,aredependentonthetypesofincidentwavesandanglesofincidence ✓ ↵ forP-waveand✓ forS-wavesasshowninFig. 3.2. Moreover,theyalwaystogethersatisfythe stress-freeboundaryconditionsonthehalf-spacesurfacewhere✓ =⇡/ 2 ✓✓ = ⌧ ✓r = ⌧ ✓ =0. (3.1) Figure3.1. Adoptionofsphericalcoordinate. Due to the existence of hemispherical canyon, the potentials of scattered wave are introduced. All types of scattered wave potentials are assumed to include the first kind of Hankel function h (1) n (kr)asoutgoingwaves,wherekiswavenumbereitherk ↵ ork ,dependingonthewavetypes. The summation of free-field potential and scattered potential is going to satisfy not only the threeboundaryconditionsonthehalf-spacesurfacein(3.1)butalsothethreestress-freeboundary conditionsonthehemisphericalcanyonsurfacedescribedasfollows: onthehemisphericalcanyonsurface,wherer = a, rr = ⌧ r✓ = ⌧ r =0. (3.2) 72 Figure3.2. A3-Dhemisphericalcanyonwitharbitraryplanewaveincidence. Withthetimefactore i!t droppedforconvenience,thescatteredwavesfromthehemispherical canyoninthehalf-spacewillbe: ' (s) = 1 X m=0 ' (s) m cosm sinm , with' (s) m = 1 X n m m+n=odd A (s) mn h (1) n (k ↵ r)P m n (u), (s) = 1 X m=0 (s) m sinm cosm , with (s) m = 1 X n m m+n=odd k B (s) mn h (1) n (k r)P m n (u), (s) = 1 X m=0 (s) m cosm sinm , with (s) m = 1 X n m m+n=odd C (s) mn h (1) n (k r)P m n (u), (3.3) whereh (1) n (.) is the spherical Hankel’s function of ordern that represents for outgoing waves and u=cos✓ . NotethatthefirstrowsareusedincasesofP-waveorSV-waveincidencebecausethey arein-planewavesandthesecondrowsareusedforout-of-planeSH-waveincidence. 73 Figure3.3. Field-fieldandscatteredwavepotentialsintheproblemofa3-Dcanyononthe half-space. Similar to findings by Lee and Zhu (Lee and Zhu, 2013), where onlyP 1 n (u) withn+1 is odd areselectedintheexpressionsofwavesscatteringaroundthehemisphericalcanyoninthecaseof verticalincidence,thepotentialsofscatteredwavesforarbitraryangleofincidenceareassumedto consistofassociatedLegendrepolynomialsP m n (u)withm+nisoddinwhichmistheorderand n is the degree ofP m n (u). The reason of adopting the odd-only Legendre polynomials is that in a half-space where ✓ is from 0 to ⇡/ 2 oru=cos✓ is from 0 to 1, both {P m n : n +m isodd} and {P m n : n+m iseven}canseparatelyformacompleteorthogonalsetoffunctions. Moreover, on thehalf-space,any{P m n : n+m isodd}canbeexpandedintermsof{P m n : n+m iseven}as P m n (u)= 1 X l m l+m=even m ln P m l (u), (3.4) where coefficients of expansion m ln will be derived and given in Section 3.6, the appendix to Chapter3,attheendofthischapter. 74 Therefore,thescatteredpotentialsin(3.3)taketheform ' (s) m = 1 X n m m+n=odd 1 X l m l+m=even A (s) mn h (1) n (k ↵ r) m ln P m l (u); (s) m = 1 X n m m+n=odd 1 X l m l+m=even k B (s) mn h (1) n (k r) m ln P m l (u); (s) m = 1 X n m m+n=odd 1 X l m l+m=even C (s) mn h (1) n (k r) m ln P m l (u). (3.6) One can show in Section 3.6 that the stress-free boundary conditions on the half-space given in (3.1) can be naturally satisfied, while the remaining three boundary conditions of (3.2) will be utilized to solve for the coefficients of scattered wave potentials A (s) mn , B (s) mn , and C (s) mn . The dimensionlessfrequency⌘ willcontinuetobeusedinthefollowingdiscussion. From(1.44), ⌘ = !a ⇡C = 2fa C = k a ⇡ = 2a , (3.7) where isthelengthoftheshearwaves. AlltechnicaldetailsaredescribedinSection3.6,which istheappendixattheendofthischapter. 75 3.2 PlaneP-waveIncidencewithArbitraryAngle✓ ↵ 3.2.1 Free-fieldwavepotentials Begin with a plane P-wave incidence with potential ' (i) and incident angle ✓ ↵ with respect to the horizontalx-axis. Herethetimefactore i!t isomittedbecauseofthesteadystateassumption ' (i) =exp(ik ↵ (xcos✓ ↵ zsin✓ ↵ )), (3.8) whichcanbeexpressedinthesphericalcoordinateas ' (i) = 1 X m=0 1 X n m a (i) mn j n (k ↵ r)P m n (u)cosm, (3.9) wherej n (.) is the spherical Bessel function of ordern andP m n (u) with argumentu=cos✓ (0 u 1)istheassociatedLegendrepolynomialofdegreenandorderm. Theexpressionfora (i) mn is givenin(A.5)ofAppendixA.2attheendofthethesis. The presence of the stress-free half-space surface atz=0 results in reflected a plane P-wave with potential ' (r) and a reflected plane S-wave which can be polarized into two potentials (r) and (r) inmutuallyorthogonaldirections. ThereflectedP-andS-wavesareexpressedinspherical coordinatesas ' (r) = 1 X m=0 1 X n m a (r) mn j n (k ↵ r)P m n (u)cosm, (3.10a) (r) = 1 X m=0 1 X n m k b (r) mn j n (k r)P m n (u)sinm, (3.10b) (r) = 1 X m=0 1 X n m c (r) mn j n (k r)P m n (u)cosm, (3.10c) 76 wherek ↵ =!/C ↵ andk =!/C arethewavenumbersofP-wavesandS-waves;C ↵ andC are the wave speeds of P- and S-waves. The coefficients of reflected potentiala (r) mn , b (r) mn , andc (r) mn are derivedin(A.10),(A.15)and(A.16)ofAppendixA.2attheendofthethesis. Thefree-fieldwavepotentialsaregivenby: ' (ff) = ' (i) +' (r) = 1 X m=0 1 X n m a (ff) mn j n (k ↵ r)P m n (u)cosm, (ff) = (r) = 1 X m=0 1 X n m k b (ff) mn j n (k r)P m n (u)sinm, (ff) = (r) = 1 X m=0 1 X n m c (ff) mn j n (k r)P m n (u)cosm, (3.11) whereforanym0,nm, a (ff) mn =a (i) mn +a (r) mn , b (ff) mn =b (r) mn , c (ff) mn =c (r) mn , Consequently,theuniqueandcompletesolutionofscatteredwavecoefficientsA (s) mn ,B (s) mn ,andC (s) mn in(3.58)and(3.59)ofSection3.6canbeobtained. 77 3.2.2 Surfacedisplacements The resultant displacements are a linear combination of free-field waves displacements and scat- tered waves displacements. The displacements of free-field (incident and reflected) waves are giveninthefollowingvectorialform: ˜ U (i) =ik ↵ (cos✓ ↵ ˜ e x sin✓ ↵ ˜ e z )expik ↵ (xcos✓ ↵ +zsin✓ ↵ ); (P): ˜ U (r) 1 =iK 1 k ↵ (cos✓ ↵ ˜ e x +sin✓ ↵ ˜ e z )expik ↵ (xcos✓ ↵ zsin✓ ↵ ); (S): ˜ U (r) 2 =iK 2 k (sin✓ ˜ e x cos✓ ˜ e z )expik (xcos✓ zsin✓ ); (3.12) whereK 1 andK 2 arethereflectioncoefficientsgivenin(A.8)inAppendixA.2. Thedisplacement vector of reflected waves in (3.12) is divided into two components—the one from P-wave and the one from S-wave, and ˜ e x and ˜ e z are unit vectors of x and z in the Cartesian coordinate system showninFig. 3.2. Displacements from scattered waves are naturally given in the (r,✓, ) coordinates by the Bessel-Legendre series expansion because of the form of potentials in (3.3). According to AppendixB.1,thecorrespondingcomponentsare U (s) r = 1 r 1 X m=0 1 X n m m+n=odd h A (s) mn D (3) 11 +C (s) mn D (3) 13 i P m n (u)cosm, U (s) ✓ = 1 r 1 X m=0 1 X n m m+n=odd " ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ dP m n (u) d✓ + mrB (s) mn D (3) 22 sin✓ P m n (u) # cosm, U (s) = 1 r 1 X m=0 1 X n m m+n=odd  m sin✓ ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ P m n (u)rB (s) mn D (3) 22 dP m n (u) d✓ sinm, (3.13) whereu=cos✓ ,andthedetailedexpressionsofD (i) jk functionsreflectingtherelationshipsbetween the displacements and potentials are derived and given in Appendix B.1. The superscript (3) of 78 D refers to the inclusion of the first kind of Hankel function h (1) n (.) in the scattered wave poten- tials. The spherical components (U (s) r ,U (s) ✓ ,U (s) ) can be transformed to (U (s) x ,U (s) y ,U (s) z ) in the Cartesiansystemwiththefollowing: 2 6 6 6 4 U (s) x U (s) y U (s) z 3 7 7 7 5 = 2 6 6 6 4 sin✓ cos cos✓ cos sin sin✓ sin cos✓ sin cos cos✓ sin✓ 0 3 7 7 7 5 2 6 6 6 4 U (s) r U (s) ✓ U (s) 3 7 7 7 5 . (3.14) Asummationof(3.12)and(3.14)givesthetotaldisplacements(U x ,U y ,U z )byaplaneP-incidence at given frequency. Similar to the case of the vertical P-incidence in Section 1.4, the amplitudes of displacements are calculated for the assessment of oscillations because the displacements are complexnumbers. Therefore,theamplitudes |U x |= p Re(U x ) 2 +Im(U x ) 2 , |U y |= q Re(U y ) 2 +Im(U y ) 2 , |U z |= p Re(U z ) 2 +Im(U z ) 2 , (3.15) andthephasesare x =tan 1 (Im(U x )/Re(U x )), y =tan 1 (Im(U y )/Re(U y )), z =tan 1 (Im(U z )/Re(U z )), (3.16) whereRe(.)andIm(.)arerealandimaginarypartsofacomplexnumber. Four angles of incidence ✓ ↵ =90 ,60 ,30 and 15 were selected to illustrate the result- ing displacements. To standardize the results, all displacement amplitudes will be multiplied by 1/ik ↵ .Thenormalizedamplitudesoffree-fielddisplacement,foreach✓ ↵ onthehalf-spacesurface, arelistedinthefollowingtable: 79 ✓ ↵ |U x ||U y ||U z | 90 002 60 1.1211 0 1.6901 30 1.7321 0 1 15 1.6126 0 0.6584 Table3.1. Surfaceamplitudesoffree-fielddisplacementfortheP-incidence. Figure3.4. |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidence. In Appendix C.1, the 3-D graphs of amplitudes of displacements with ⌘ =1,3,5, and ✓ ↵ = 90 ,60 ,30 ,15 are shown. From (1.44), ⌘ , the dimensionless frequency, can also be interpreted astheratioofthecanyondiameterandlengthoftheincidentP-wave. As it can be seen in Fig. 3.4, which is an example graph in Appendix C.1, x/a and y/a are normalizedhorizontalaxesscaledbya,theradiusofthecanyon,andall|U x |,|U y |and|U z |arenor- malizedthathavealreadybeendividedbyik ↵ . Rangesofallgraphsare2ato2aforbothxandy 80 axes. Notethatalthoughtheorientationoftheyaxisisreversedbecausethepositivezdirection defined in the model is downward and the Cartesian system follows the right-hand rule, all of the amplitudesareplottedonthexy planeforthereaders’convenience. Thedistributionsofampli- tudesofallkindsofdisplacementsaresymmetricrelativetothexz planebecausetheincidence ofplaneP-waveisalsosymmetricrelativetotheverticalxzplane. Thisalsoexplainswhy|U y |is zero alongy=0. The vertical displacement amplitudes |U z | are always axisymmetric and ampli- tudes of the x-component of horizontal displacement |U x | and amplitudes of the y-component of horizontaldisplacement|U y |alwayshavesimilarpatternsforaverticalincidence. The graphs of ✓ ↵ =60 ,30 , and 15 in Fig. 3.4 show that non-vertical incidences no longer have the property of cylindrical symmetry. In Table 3.2, some 2-D plots are shown for a clearer demonstration of displacement amplitudes and phases associated with various angles ✓ ↵ or fre- quencies ⌘ . Figure 3.5 depicts a cross-section perpendicular to thexy plane with the bold line indicatingthecanyonsurfaceandhalf-spacethatthe2-Dplotsdescribe. Fig. no. Component ⌘ ✓ ↵ x/a y/a 3.6,3.7 |U z |& z 1,3,5 60 ,15 [5,5] 0.2 3.8,3.9 |U x |,|U z | 1 90 ,60 ,30 ,15 [5,5] 0.0 3.10,3.11 |U x |,|U z | 3 90 ,60 ,30 ,15 [5,5] 0.0 3.12,3.13 |U x |,|U z | 5 90 ,60 ,30 ,15 [5,5] 0.0 3.14,3.15 |U x |,|U z | 1,3,5 30 0.0 (0,5] 3.16 |U y | 1,3,5 15 0.4 (0,5] Table3.2. Summaryof2-DplotsinSection3.2. Before continuing the discussion, let’s define the negative range ofx/a astheillumatedside, because the incident wave impacts on the canyon directly from this side. On the other hand, we usetheshadowside to refer to the positive range ofx/a, because the positive range is “blocked” bythecanyonfromtheincidence. 81 Figure3.5. Asketchoftherangeofwhere2-Dplotsdescribe. In the absence of any topographies, the free-field plane P- and SV-waves at the half-space surfacewherez=0or✓ =⇡/ 2wouldallhavetermsoftheform: (i) | z=0 =exp(ik ↵ xcos✓ ↵ ); (i) | z=0 =K 1 exp(ik xcos✓ ); (i) | z=0 =K 2 exp(ik xcos✓ ). (3.17) And sincek ↵ cos✓ ↵ = k cos✓ (L’Hospital Rule), meaning that the free-field waves all have the same horizontal wave number or same horizontal phase velocity. This means all components of thefree-fieldwaveshavethesamephasegivenby x = y = z =(k ↵ cos✓ ↵ )x=(k cos✓ )x. (3.18) This equation show that at the half-space surface, the free-field phase will be a linear function of x,withslopegivenbyk ↵ cos✓ ↵ =(k cos✓ ). Thisslopemeasureswithincreasingwavenumber offrequency. Inotherwords,thehigherthefrequency,thesteeperitis. Italsoshowsthattheslope isdependentontheincidentangle,✓ ↵ ,oftheP-wave. Foragivenfrequency,thecos✓ ↵ termshows thattheslopeishighestat✓ ↵ =0,thecaseofhorizontal(glazing)incidence. Astheangleof 82 Figure3.6. |U z |and z aty/a=0.0with✓ ↵ =60 ,⌘ =1,3,5ofP-incidence. 83 Figure3.7. |U z |and z aty/a=0.0with✓ ↵ =15 ,⌘ =1,3,5ofP-incidence. 84 incidenceincreases, cos✓ ↵ decreases,andsotheslopeofthephasefunctiondecreases. Atvertical incidence,when✓ ↵ =⇡/ 2, cos✓ ↵ =0andsotheslopeofthephasefunctioniszero,meaningthat thewavesarriveatthehalf-spacesurfaceallatthesametimeeverywhere. Thepresenceofthehemisphericalcanyonresultsinscatteredwavesandtogetherwiththefree- field waves, the resultant waves will have the phase deviating from the linear shape. Close to the canyon, andalongthecanyonsurface, thedeviationwillbehighest, asseenfromFig. 3.6. Asthe amplitudes of the waves oscillate along the surface fro low to high, the corresponding phases will also deviate. At points on the surface where the amplitude is at its minimum, the corresponding phase will exhibits a change of almost ⇡ , corresponding to a reversal of the directin of motion in bothsidesofthepoint. In Fig. 3.6 and Fig. 3.7, the amplitudes and phases ofU z alongy/a=0.0 for ⌘ =1,3, and 5 with ✓ ↵ =60 and 15 are presented. The traveling direction of the incident wave is always from lefttoright. Amplitudes|U z |aregivenatthetoppartofthefigures,andtheircorrespondingphases are plotted on the bottom part of figures with the same color and line style. Note that the positive z-direction is defined downward, thus an ascending phase plot implies the travel direction of the wave from below to the surface of the half-space. When ⌘ is high, the slope of the phase plot is also high. It can be seen that ⌘ =5 always provides the highest slope in both phase figures while the curves of ⌘ =1 appear to be relatively flat. Moreover, phases have obvious interferences by thescatteredwavesnearbothrims(x/a = ± p 10.2 2 = ±0.98). Forthecaseof✓ ↵ =60 inFig. 3.6, although only slight fluctuations can be found at the rims from the ⌘ =1 plot, obvious dips (change of phases) are observed on the ⌘ =3 and ⌘ =5 plots. In Fig. 3.7 in which the incident angle✓ ↵ ismorehorizontal,thechangesinslopesaremoresubstantialirrespectiveofthevalueof thefrequency⌘ . Therefore,theimpactonphasesareinfluencedbynotonly⌘ butalsotheangleof incidence✓ ↵ asstatedintheabove. 85 Figure3.8. |U x |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. Figure3.9. |U z |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. 86 Figure3.10. |U x |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. Figure3.11. |U z |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. 87 Figure3.12. |U x |aty/a=0.0with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. Figure3.13. |U z |aty/a=0.0with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 ofP-incidence. 88 Figures 3.8 to 3.13 are the amplitudes |U x | and |U z | along x/a =5 to 5 at y=0 for ✓ ↵ = 90 ,60 ,30 , and 15 with ⌘ =1,3, and 5, respectively. The displacement amplitudes for the relative long-period incident wave, ⌘ =1 in Fig. 3.8 and Fig. 3.9, are higher on the illuminated side than on the shadow side of the canyon surface. With the increase of ⌘ (Fig. 3.12 and 3.13), onecanclearlyobservethatthedisplacementsareintensivelyamplifiedalsoontheshadowsideof thehemisphericalcanyon. From Fig. 3.9 where ⌘ =1, the maxima |U z | for ✓ ↵ =60 , ✓ ↵ =30 , and ✓ ↵ =15 are respectivelyequalto3.77,5.15,and4.76. InFig. 3.13of⌘ =5,themaxima |U z |arerespectively 4.13, 4.45, and 4.14. From Table 3.1, the corresponding free-field |U z | of ✓ ↵ =60 , ✓ ↵ =30 , and ✓ ↵ =15 are equal to 1.7, 1, and 0.66, the peak of ✓ ↵ =15 is amplified by 4.76/0.66 = 7.2 times, which is higher than the 5.15 times of ✓ ↵ =30 even though ✓ ↵ =30 has the absolute highestvalue. Therefore,themorehorizontaltheincidentwave,thelargeramplificationofvertical displacement is observed. Oscillations of |U z | can be seen as distorted on the shadow side where x/a> 1 behind the canyon along the traveling path of the incident wave. This observation, which is pronounced in particular with ✓ ↵ =30 and ✓ ↵ =15 when the angle of incidence is almost horizontal,canbeinterpretedasthephenomenonofthewavediffraction. Inaddition,the“dipand spike” phenomenon (Trifunac, 1973) is again found at the rims especially when ⌘ is high. Please refertoSection1.4fordetailsonthiscorner-pointeffect. The lengths of waves are dependent both on frequency ⌘ and incident angle ✓ ↵ . As was pre- viously discussed in Chapter 1, when ✓ ↵ =90 , for a given dimensionless frequency ⌘ , the wave- length of oscillation of displacement would roughly be 1/⌘ of the diameter of the hemispherical canyon. For the oblique incidences, wave cycles can be observed compacted on the illuminated side of the canyon but the cycles are expanded on the shadow side. For instance, in Fig. 3.11 in which⌘ =3,theplotof|U z |for✓ ↵ issymmetricrelativetox=0,andalongx/a<1orx/a> 1 everytwounitsofx/aapproximatelycontainsthreewaveperiods. Thewavelengthsontheleftfor ✓ ↵ =60 ,30 and 15 gradually decrease, while on the shadow side for ✓ ↵ =15 the plot shows 89 waveswiththelargestwavelength. Nevertheless,allthevibrationsrelativetothefree-fieldampli- tudesinTable3.1willeventuallyrestatthefree-fieldamplitudesatadistancewhereitistoofarto beinfluencedbythescatteredwavepotentials. Figures 3.14, 3.15 and 3.16 exhibit the displacement amplitudes for different dimensionless frequencies ⌘ =1,3, 5 all at the same given angle of incidence. The planes x/a =0.0 and x/a=0.4areselected,andonlythepositiverangeofy/a=05isplottedbecauseallresultsare symmetric relativetoy=0. Evidently, the oscillation of |U z | of✓ ↵ =30 inFig. 3.15isstronger than |U x | in Fig. 3.14 for all corresponding values of ⌘ . Although this is intuitive because |U z | is the vertical displacement on the stress-free half-space surface, on the other hand the movement of particles along thexdirection is rigorously hindered by the massive half-space medium. |U y | decays to zero quickly outside the canyon territory because the incident waves only propagate in thexz plane so that there exists very little potential to “push” the particle to the side along the y-directionoutsidethecanyon. Figure3.14. |U x |atx/a=0.0with✓ ↵ =30 ,⌘ =1,3,5ofP-incidence. 90 Figure3.15. |U z |atx/a=0.0with✓ ↵ =30 ,⌘ =1,3,5ofP-incidence. Figure3.16. |U y |atx/a=0.4with✓ ↵ =15 ,⌘ =1,3,5ofP-incidence. 91 3.3 PlaneSV-waveIncidencewithArbitraryAngle✓ 3.3.1 Free-fieldwavepotentials In this section, the in-plane shear (SV-) wave will be investigated. The expression of incident SV-potentialintherectangularcoordinatesgiveninFig. 3.1is,therectangularpotential, (i) = 1 ik e ik ( xcos✓ +zsin✓ ) i!t ˜ e y , (3.19) where1/ik isthescalefactorforthenormalizationofdisplacementamplitudes. Theseriesexpan- sionof(3.19)insphericalcoordinatesisbrokendownintotwosphericalpotentials(usingthesame notation (i) ): (i) = 1 X m=0 1 X n m k b (i) mn j n (k r)P m n (u)sinm, (i) = 1 X m=0 1 X n m c (i) mn j n (k r)P m n (u)cosm, (3.20) where b (i) mn and c (i) mn are given in (A.22), and (A.23) in Appendix A.3 at the the of the thesis, and the time factor e i!t has been omitted. Note that although (i) in (3.19) and (3.20) has different definitions, although the identical symbol is used. The displacement on the half-space surface for incidentwaves ˜ U (i) isequaltothefollowingvector ˜ U (i) =ik (sin✓ ˜ e x cos✓ ˜ e z )expik (xcos✓ +zsin✓ ), (3.21) which is independent of y. As in (3.12), ˜ e x and ˜ e z are unit vectors of xand zdirections. The stress-freeboundaryconditionsonhalf-spacesurfaceare ✓✓ | z=0 = ⌧ ✓r | z=0 = ⌧ ✓ | z=0 =0. 92 This results in the reflected wave potentials in the spherical system ' (r) , (r) and (r) in the form of ' (r) = 1 X m=0 1 X n m a (r) mn j n (k ↵ r)P m n (u)cosm, (r) = 1 X m=0 1 X n m k b (r) mn j n (k r)P m n (u)sinm, (r) = 1 X m=0 1 X n m c (r) mn j n (k r)P m n (u)cosm, (3.22) wherek ↵ andk arethewavenumbersoftheP-andS-wavesaspreviouslydefined. Figure3.17. Thefree-fieldwavesofSV-waveincidence. BecausetheSV-incidenceisin-plane,itsreflectedpotentialsareintheformofbothlongitudinal and shear waves. Figure 3.17 depicts the angle of incident shear wave, the angle of reflected longitudinalwave,andtheangleofreflectedshearwaves. BySnell’slaw,therelationshipbetween ✓ ↵ and✓ are: cos✓ ↵ C ↵ = cos✓ C (3.23) or ✓ ↵ =arccos( C ↵ C cos✓ ). (3.24) 93 However, whenC ↵ /C > cos✓ , the reflected angle ✓ ↵ becomes imaginary and will not be given bySnell’slaw. Consequently,twocasesshouldbetakenintoconsiderationseparatelyinthedeter- minationofa (r) mn ,b (r) mn ,andc (r) mn in(3.22). CaseI isthecaseinwhichtheincidentangle✓ (w.r.t. horizontal)isgreaterthanorequaltothecrit- ical angle ✓ cr =arccos( ) defined in Appendix A.3 and Subsection 3.3.2, the displacement vectorsofthereflectedP-waveandS-wavearerespectivelyequalto (P): ˜ U (r) 1 =iK 1 k ↵ (cos✓ ↵ ˜ e x +sin✓ ↵ ˜ e z )expik ↵ (xcos✓ ↵ zsin✓ ↵ ), (S): ˜ U (r) 2 =iK 2 k (sin✓ ˜ e x cos✓ ˜ e z )expik (xcos✓ zsin✓ ), (3.25) and where K 1 and K 2 are coefficients of reflection in (A.24). a (r) mn is of the form given in (A.10)ofAppendixA.2,andb (r) mn andc (r) mn aregivenin(A.15)and(A.16)ofAppendixA.3. CaseII is the case in which the incident angle (w.r.t. horizontal) is less than the critical angle—i.e. ✓ <✓ cr —the reflected P-wave has become a surface wave. The reflected waves then take theform: (P): ˜ U (r) 1 =iK 1 k ↵ (s ˜ e x +c ˜ e z )expik ↵ (xs zc ), (S): ˜ U (r) 2 =iK 2 k (sin✓ ˜ e x cos✓ ˜ e z )expik (xcos✓ zsin✓ ), (3.26) where K 1 and K 2 are coefficients of reflection given in (A.25), a (r) mn is derived in (A.27), the expressions of b (r) mn and c (r) mn are given in (A.15) and (A.16), and steps of the detailed derivationofthesecoefficientsaregiveninA.3. 94 Thefree-fieldpotentialsarethengivenby: ' (ff) = ' (i) = 1 X m=0 1 X n m a (ff) mn j n (k ↵ r)P m n (u)cosm, (ff) = (i) + (r) = 1 X m=0 1 X n m k b (ff) mn j n (k r)P m n (u)sinm, (ff) = (i) + (r) = 1 X m=0 1 X n m c (ff) mn j n (k r)P m n (u)cosm, (3.27) where a (ff) mn =a (r) mn , b (ff) mn =b (i) mn +b (r) mn , c (ff) mn =c (i) mn +c (r) mn , with m0, and nm. Again, as in Section 3.1, A (s) mn , B (s) mn and C (s) mn can be solved by (3.58) and(3.59)giveninthelastsectionofthischapter. 3.3.2 Surfacedisplacements  = C ↵ /C is the ratio of the speeds of P- and S-waves that has been assumed to be p 3 as in Section 1.2, so that ✓ cr =arccos(1/ )=0.9553⇡ 0.3⇡ (54 ). Therefore, the cases of ✓ =90 and 60 are related to Case I of the reflection waves where the reflected P-waves are pure waves; and the cases of ✓ =30 and 15 are related to Case II where the reflected P-waves are surface waves. The normalized amplitudes of free-field displacements for the four angles of incidences citedabovearelistedonTable3.3. 95 ✓ |U x ||U y ||U z | 90 200 60 1.7321 0 1 30 0.5 0 1.1180 15 0.8656 0 0.8893 Table3.3. Surfaceamplitudesoffree-fielddisplacementfortheSV-incidence. SimilartotheP-incidence,thesphericalcomponentsofdisplacementduetothescatteredwaves areofthefollowingform,whereu=cos✓ ,that: U (s) r = 1 r 1 X m=0 1 X n m m+n=odd h A (s) mn D (3) 11 +C (s) mn D (3) 13 i P m n (u)cosm, U (s) ✓ = 1 r 1 X m=0 1 X n m m+n=odd " ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ dP m n (u) d✓ + mrB (s) mn D (3) 22 sin✓ P m n (u) # cosm, U (s) = 1 r 1 X m=0 1 X n m m+n=odd  m sin✓ ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ P m n (u)rB (s) mn D (3) 22 dP m n (u) d✓ sinm. (3.28) A (s) mn ,B (s) mn ,C (s) mn arecontinuedasthecoefficientsofthescatteredwavesdefinedin(3.3),whichcan be solved by boundary conditions. The final displacements (U x ,U y ,U z ) are the total of displace- ments by the incident wave in (3.20), and by the reflected waves in (3.25) or (3.26)and, with the sum of the displacements (U (s) x ,U (s) y ,U (s) z ) due to the scattered waves, which are transformed by (3.14)from (U (s) r ,U (s) ✓ ,U (s) )in(3.28). The 3-D graphs of normalized |U x |, |U y | and |U z | are given in Appendix C.2. The ranges of vertical axes x/a and y/a in the graphs are from2 to 2 as before. Selected dimensionless frequenciesof⌘ =1,3,5arepresented. Figure3.18isanexampleofthe3-Dgraphshowing |U z | for ⌘ =1, with ✓ =90 ,60 ,30 , and 15 . The result of normal incidence is always the first 96 subfigure of every 3-D graph. One can see that when the incident angle is 90 (vertical), although this vertical incidence no longer brings about axisymmetric results, all of its resultant amplitudes are still symmetric relative to both the xz plane and the yz plane. The cause of this xz symmetry is because of the setup of the model, and the cause of the yz symmetry is because particles activated by the free-field waves move equally in both the +x andx directions when ✓ =90 ,therebytheequalamplitudesappear. Moreover,bothU z andU y havezeroamplitudesat x=0foranormalincidence. ThiswillbeshowninSection3.4. Figure3.18. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofcanyonforSV-incidence. Table 3.4 summarizes the 2-D plots shown in this section for a better demonstration of the detailsofdisplacementamplitudesresultantbytheSV-incidence. 97 Fig. no. Component ⌘ ✓ x/a y/a 3.19,3.20 |U z |& z 1,3,5 60 ,15 [5,5] 0.0 3.21,3.22 |U z |,|U x | 1 90 ,60 ,30 ,15 [5,5] 0.0 3.23,3.24 |U z |,|U x | 3 90 ,60 ,30 ,15 [5,5] 0.0 3.25,3.26 |U z |,|U x | 5 90 ,60 ,30 ,15 [5,5] 0.0 3.27 |U x | 5 90 ,60 ,30 ,15 -0.8 (0,5] 3.28,3.29,3.30 |U y | 1,3,5 90 ,60 ,30 ,15 [5,5] 0.5 Table3.4. Summaryof2-DplotsinSection3.3. Figures 3.19 and 3.20 are the amplitudes and phases of the vertical displacementU z with ⌘ = 1,3,and 5aty/a=0forthetwoobliqueincidences✓ =60 and✓ =15 ,respectively. Similar to the P-incidence, the two figures have the displacement amplitudes on the top, and the plots of their corresponding phases with the same color and line style at the bottom. The positive slopes still indicate the oblique incidence of the waves from the bottom left to the top right of the half- space. Theplotsofthephasesfor✓ =15 inFig. 3.20havehigherslopesthantheircounterparts for✓ =60 inFig. 3.19. Higher⌘ alsoresultsinsteeperslopes. Foralloftheanglesofincidence and the dimensionless frequencies, plots of phases are always almost monotone and increasing, exceptwheninterferedwithbythescatteredwavesinthevicinityofthecanyon. 98 Figure3.19. |U z |and z aty/a=0.0with✓ ↵ =60 ,⌘ =1,3,5ofSV-incidence. 99 Figure3.20. |U z |and z aty/a=0.0with✓ ↵ =15 ,⌘ =1,3,5ofSV-incidence. 100 As shown by the 3-D graphs in Appendix C.2, all of the peaks of |U x | and |U z | for oblique incidencesarelocatedalongy=0. Therefore,Fig. 3.21toFig. 3.26arethe2-Dplotsof|U z |and |U x |aty/a=0intherangeofx/a =5to 5for✓ =90 , 60 , 30 ,and 15 and⌘ =1,3,and 5. Forallofthenormalincidencesat✓ =90 ,|U z |and|U x |arealwayssymmetricrelativetox=0 in these figures as what was previously observed. Moreover, within the canyon boundary, where x/a=[1,1], |U z | can be observed to bear a more intensive oscillation with higher amplitudes. On the other hand, the plots of |U z | for normal incidences (✓ =90 ) do not show any patterns of wave oscillations on the half-space surface; instead, all of these plots decay to zero gradually for all⌘ asthewavesarefurtherawayfromthecanyoninFig. 3.21,Fig. 3.23andFig. 3.25. When the incidence is oblique, |U z | always has strong oscillations on the half-space surface with the amplitudes beyond or below its corresponding free-field amplitude given in Table 3.3, and will eventually converge to its free-field level. The oscillations of |U z | on the shadow side are obviously interfered with by the scattered waves, which is explained by the phenomenon of wave diffraction. Surprisingly, from Fig. 3.21, Fig. 3.23 and Fig. 3.25, there are the intermediate anglesofincidences,neithertheverticalnorthemosthorizontalangles,thatcontributetothemost pronouncedreactiononthecanyonsurface—i.e. ✓ =60 and✓ =30 . Nevertheless,foralmost horizontal incidence, ✓ =15 , it shows the most explicit phenomenon of wave diffraction when ⌘ ishigh-i.e. ⌘ =3or 5. 101 Figure3.21. |U z |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. Figure3.22. |U x |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. 102 Figure3.23. |U z |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. Figure3.24. |U x |aty/a=0.0with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. 103 Figure3.25. |U z |aty/a=0.0with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. Figure3.26. |U x |aty/a=0.0with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofSV-incidence. 104 In Fig. 3.22, Fig. 3.24 and Fig. 3.26, |U x | converges to its free-field level instantaneously outside the canyon rim for all cases of ✓ and ⌘ . Figures 3.24 and 3.26 demonstrate that on the canyonsurfacewhenthefrequencyishigh,thealmostverticalangleofincidence(✓ =60 )hasits strongest oscillations intensively distributed on the illuminated side, while the almost horizontal angles (✓ =30 and 15 ) have their most amplified results on the shadow side of the canyon surface. Figure3.27. |U x |atx/a =0.8with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. Obvious oscillations of |U x | on the half-space surface are most significant on theyz plane, thoughnotsosignificantfromtheprecedingfiguresonthexzplane. Figure3.27presents|U x |at x/a =0.8alongy/afrom 0to 5forthefouranglesofincidenceat⌘ =5. Becauseofsymmetry relative to the xz plane, y/a only contains a positive range from 0 to 5 in this figure. Similar to |U z |, |U x | are also oscillating by their free-field amplitudes. In addition, from the figure, it is observed that the wavelengths of the resultant waves are slightly higher for the vertical incidence (✓ =90 )whencomparedwiththatfromthealmosthorizontalincidence(✓ =15 ). Therefore, 105 for the SV-incidence, the resultant wavelengths depend on not only the frequency of incidence ⌘ , butalsoontheangleofincidence✓ . Figure3.28. |U y |aty/a=0.5with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. The third displacement component, the out-of-plane component, |U y |, is given in Fig. 3.28 to Fig. 3.30 plotted aty/a=0.5 fromx/a from5 to 5. The magnitude of |U y | is significantly lower than that of |U z | and |U x | under the same conditions of incidence, because as in the P-incidence, thereisnofree-fielddisplacementintheydirectionforanSV-incidence. Theimpactby✓ =90 is always the smallest at each ⌘ in all figures, and as ⌘ increases, the peaks of |U y | are moving toward the center, which is the bottommost point of the canyon. Such interesting phenomenon can be more obviously observed from the 3-D graphs in Appendix C.2. For oblique incidences of ✓ =60 ,30 ,and 15 ,massivespikesarealwaysseenneartherim,thussupportingthetheoryof asecondarywavesourceatthecorner. 106 Figure3.29. |U y |aty/a=0.5with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. Figure3.30. |U y |aty/a=0.5with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSV-incidence. 107 3.4 PlaneSH-waveIncidencewithArbitraryAngle✓ 3.4.1 Free-fieldwavepotentials In this section, the SH-incidence, or the out-of-plane shear wave incidence, will be discussed. Denote the angle of incidence to be ✓ . Similar to the SV-incidence in Section 3.3, the SH- incidence is expressed by two polarized shear wave potentials in the spherical coordinates in the followingform: (i) = 1 X m=0 1 X n m k b (i) mn j n (k r)P m n (u)cosm, (i) = 1 X m=0 1 X n m c (i) mn j n (k r)P m n (u)sinm, (3.29) whereb (i) mn andc (i) mn aregivenin(A.34)and(A.35)ofAppendixA.4. Notethatthepositionsof sin and cosareexchangedfor and comparedwiththeexpressionsinprevioussections,andthisis becausetheSH-waveinananti-planewave,whiletheothertwo—P-andSV-waves—arein-plane waves. Thisexchangewillbecarriedonthroughoutthissection,sothatpotentials' and contain cosm ,and composesof sinm . Inthepresenceofthehalf-spacesurfacez=0,reflectedplaneSH-wavesaregenerated,which inthesphericalcoordinates,theirpotentials (r) and (r) areexpressedas: (r) = 1 X m=0 1 X n m k b (r) mn j n (k r)P m n (u)cosm, (r) = 1 X m=0 1 X n m c (r) mn j n (k r)P m n (u)sinm, (3.30) where the coefficients b (r) mn and c (r) mn satisfying the stress-free boundary conditions are given in (A.36)and(A.37)ofAppendixA.4. 108 Thefree-fieldwavepotentialsarethesuperpositionoftheincidentandreflectedpotentialsthat ' (ff) =0, (ff) = (i) + (r) = 1 X m=0 1 X n m k b (ff) mn j n (k r)P m n (u)cosm, (ff) = (i) + (r) = 1 X m=0 1 X n m c (ff) mn j n (k r)P m n (u)sinm, (3.31) with a (ff) mn =0, b (ff) mn =b (i) mn +b (r) mn , c (ff) mn =c (i) mn +c (r) mn , for anym0, andnm. Note that the sinm and cosm terms are interchanged between the SH-incidence and the SV-incidence. The scattered waves will take the form, that ' (s) , (s) , and (s) willbeexpressedbythesecondrowsasin(3.3),with: ' (s) = 1 X m=0 ' (s) m sinm, (s) = 1 X m=0 (s) cosm, (s) = 1 X m=0 (s) sinm. (3.32) Nevertheless,theapplicationofthewavepotentialstothecanyonsurfaceboundaryconditions will generate an identical set of equations—(3.58) and (3.59) —to solve for the scattered wave coefficientsA (s) mn ,B (s) mn , andC (s) mn as for cases of P- and SV-waves. The detailed steps of obtaining (3.58)and(3.59)areprovidedinSection3.6. 109 3.4.2 Surfacedisplacements Both the incident and reflected waves are shear waves, and their displacements are the following vectorsintheCartesiansystem: ˜ U (i) =ik ˜ e y expik (xcos✓ +zsin✓ ), ˜ U (r) =ik ˜ e y expik (xcos✓ zsin✓ ). (3.33) Thefree-fielddisplacementistheout-of-planedisplacementU y ,asmotionsofparticlesinthiscase arealwaysperpendiculartothexzplane. Normalizationistakenbymultiplyingthedisplacement with1/ik . Sothatthenormalizedfree-fielddisplacementamplitude|U y |alwaysequalsto“2”for all values of ✓ and ⌘ , but the phase of U y , denoted by y , is affected by the change of ✓ and ⌘ . Table 3.5 uses the point (x,y,z)=(1,0,1) as an example to illustrate how the relative phase y varieswith✓ and⌘ . a a a a a a a ⌘ ✓ 90 60 30 15 1 1.50⇡ 0 1.31⇡ 1.47⇡ 3 1.50⇡ ⇡ 0.05⇡ 0.40⇡ 5 1.50⇡ ⇡ 0.83⇡ 0.33⇡ Table3.5. Phase y at (x,y,z)=(1,0,1)forvariousanglesandfrequencies. The spherical components of displacement by the scattered waves are given in (3.34). The detailed D functions, which are the displacement-potential relationships, have been provided in AppendixB.1. 110 U (s) r = 1 r 1 X m=0 1 X n m m+n=odd h A (s) mn D (3) 11 +C (s) mn D (3) 13 i P m n (u)sinm, U (s) ✓ = 1 r 1 X m=0 1 X n m m+n=odd " ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ dP m n (u) d✓ mrB (s) mn D (3) 22 sin✓ P m n (u) # sinm, U (s) = 1 r 1 X m=0 1 X n m m+n=odd  m sin✓ ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ P m n (u)rB (s) mn D (3) 22 dP m n (u) d✓ cosm. (3.34) Similar to the previous sections, (U (s) r ,U (s) ✓ ,U (s) ) will be converted into (U (s) x ,U (s) y ,U (s) z ) in Cartesiancoordinates,andthencombinedwiththedisplacementsoffree-fieldwavestoobtainthe total displacements by the SH-incidence around the hemispherical canyon. Appendix C.3 shows the3-Dgraphsofdisplacementamplitudes|U x |,|U y |,and|U z |for✓ =90 ,60 ,30 ,and 15 and ⌘ =1,3,and5. ThecomparisonbetweengraphsbytheverticalSV-incidenceinAppendixC.2and verticalSH-incidenceinAppendixC.3impliesthattheirresultsareessentiallythesame,matching the|U x |ofSV-incidencewiththe|U y |ofSH-incidence,andviseversa. NotethattheSH-incidence is antisymmetric relative to the xz plane due to the phase of displacement. In addition, this explainswhy |U y |oftheverticalSH-incidenceisnotzeroalongy=0inFig. C.20(whichisFig. 3.31below),C.23andC.26ofAppendixC.3. Moreover,because U (s) x =U (s) r sin✓ cos +U (s) ✓ cos✓ cos U (s) sin, U (s) z =U (s) r cos✓ U (s) ✓ sin✓, and from the expressions of the values of U (s) r and U (s) ✓ in (3.34), |U x | and |U z | along y =0 must be equal to zero. This is also the reason that |U y | and |U z | are zero at x=0 in the vertical SV-incidenceintheprevioussection. 111 Figure3.31. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofcanyonforSH-incidence. Note that all of the graphs in Appendix C.3 remain symmetric in spite of the antisymmetry of the SH-incidence, because the displacement amplitudes are the magnitudes of the complex- numbered displacements, and then all kinds of results should be positive. Table 3.6 summarizes the2-DfiguresusedinthediscussionofresultsbytheSH-incidence. Fig. no. Component ⌘ ✓ x/a y/a 3.32,3.33 |U y |& y 1,3,5 60 ,15 [5,5] 0.0 3.34,3.35,3.36 |U y | 1,3,5 90 ,60 ,30 ,15 [5,5] 0.0 3.37,3.38,3.39 |U x | 1,3,5 90 ,60 ,30 ,15 [5,5] 0.5 3.40,3.41,3.42 |U z | 1,3,5 90 ,60 ,30 ,15 0.0 (0,5] Table3.6. Summaryof2-DplotsinSection3.4. 112 Figure3.32. |U y |and y aty/a=0.0with✓ =60 ,⌘ =1,3,5ofSH-incidence. 113 Figure3.33. |U y |and y aty/a=0.0with✓ =15 ,⌘ =1,3,5ofSH-incidence. 114 Figures3.32and3.33includetheamplitudesandphasesofU y with⌘ =1,3,and5for✓ =60 and 15 , respectively. The displacement amplitudes |U y | are presented on the top, and their corre- spondingphasesareplottedwiththesamecolorandlinestyleonthebottomofeachfigure. Same as the previous sections, the positive slopes of phase plots indicate that the waves are propagating from the illuminated side to the shadow side (left to right). All of the slopes of the phases with ⌘ =1,3, and 5 for the almost horizontal incidence (✓ =15 ) remain higher than their coun- terparts for the almost vertical incidence of ✓ =60 . Moreover, a higher ⌘ results in a steeper phase slope for each angle of incidence. In the vicinity of the rim of the canyon, the phases are notably distorted by the scattered waves. However, the impact on phases by the scattered waves decays rapidly outside the canyon territory, though the impact on displacement amplitudes are pronouncedonthehalf-spacesurfaceonboththeilluminatedandshadowsides. Figure3.34. |U y |aty/a=0.0with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. 115 Figure3.35. |U y |aty/a=0.0with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. Figure3.36. |U y |aty/a=0.0with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. 116 Figures 3.34 to 3.36 are |U y | at y =0 along x/a from5 to 5 for ✓ =90 ,60 ,30 , and 15 and ⌘ =1,3. As was previously discussed, the resultant |U y | for an SH-incidence is more pronouncedatthisxz plane(y=0). The3-DgraphsinAppendixC.3showthatthehighest|U y | is located alongy=0 only when ✓ =90 , and all of the oblique incidences cause the “humps” distributedthroughoutthecanyonsurfaceparticularlyforthehighdimensionlessfrequencies—⌘ = 3,and5. Forthenormalincidences,theamplificationofthewavesincreasedfromthecaseof⌘ =1 to⌘ =3,andthehighestamplitudesfor⌘ =3and 5arebothatthecanyoncenter. Fortheoblique incidences,theresultantamplitudesontherimsinthesefigures(x/a = ±1)areprominentlyhigher than that on other points on the canyon and half-space surface. When the angle of incidences is almost horizontal—i.e. ✓ =15 —|U y | on the canyon surface has a decreasing trend from the illumatedsidetotheshadowsideforallthe⌘ presented. Ontheotherhand,theresultsof✓ =60 are relatively evenly amplified through the canyon zone. Therefore, similar to the conclusion for theP-incidence,foralmosthorizontalincidenceofwaves,the|U y |ontheshadowsideareshielded bytheilluminatedsideofthecanyon. The effect of wave diffraction also exists in the SH-incidence. It is clear that |U y | on the shadow side are diffracted and de-amplified when ✓ is almost horizontal (✓ =15 or 30 ) in every plot from Fig. 3.34 to Fig. 3.36. In addition, as was concluded in the previous sections, the almosthorizontalangle✓ =15 hasshorterwavelengthontheilluminatedside,whilethevertical incidence✓ =90 rsultsinwaveswithlongerwavelength. FromFig. C.19,Fig. C.22,andFig. C.25inAppendixC.3,onecanseethatahigher⌘ clearly brings higher oscillations and higher amplifications of |U x |. When ✓ =90 , |U x | is symmetric relativetoboththexz planeandyz plane. Because |U x |alongy=0areallzero,thecross- sectionofy/a=0.5isadoptedforthe2-Dplotsof|U x |. Figures3.37to3.39are|U x |aty/a=0.5 andx/afrom5to 5for✓ =90 ,60 ,30 ,and 15 ,and⌘ =1,3,and 5,respectively. 117 Figure3.37. |U x |aty/a=0.5with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. Figure3.38. |U x |aty/a=0.5with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. 118 Figure3.39. |U x |aty/a=0.5with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. For oblique incidences, the resultant amplitudes |U x | on the illuminated side can be seen, in mostcases,lowerthanthatontheshadowside,andthepeakof|U x |increaseswiththedecreaseof ✓ . However,onthecanyonsurfacethemagnitudeofamplitudesaveragedontheilluminatedside (from x/a =1 to x/a=0) and that on the shadow side (from x/a=0 to x/a=1) are quite equalwhentheangleofincidenceisgettingalmosthorizontal,suchastheplotsof✓ =15 inFig. 3.38 and Fig. 3.39. From the 3-D graphs in Appendix C.3, the highest amplitudes of |U x |, which aresymmetricrelativetothexz plane,arefoundmovingalongthexz planewhen⌘ increases, andaretowardtheshadowsidewhen✓ decreases. Therefore,toconclude,the|U x |ingeneralhas anascendingtrendfromtheilluminatedsidetotheshadowside,whichiscontrarytotheproperty of |U y |; in addition, when the dimensionless frequency is higher the highest amplitudes show up closer to thexz plane, when the angle of incidence is almost horizontal the highest amplitudes are found along the positive range of x/a, and both of these occasions cause the distribution of “humps”throughouttheentirecanyonsurfacemorehomogenous. 119 Figure3.40. |U z |atx/a=0.0with⌘ =1,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. Figure3.41. |U z |atx/a=0.0with⌘ =3,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. 120 Figure3.42. |U z |atx/a=0.0with⌘ =5,✓ =90 ,60 ,30 ,and 15 ofSH-incidence. Figures C.21, C.24 and C.27 in Appendix C.3 are the 3-D vertical displacement |U z | graphs with ✓ =90 ,60 ,30 , and 15 for ⌘ =1,3, and 5, respectively. Similar to the |U y | and |U x | graphsoftheSH-incidence,theoscillationsaroundthecanyonboundaryformexplicitwavecycles andhavepronouncedamplitudes. Thehighestpeaksonthecanyonsurfacearemovingtowardthe shadowsideastheangleofincidenceismorehorizontal. Figures3.40to3.42arethecorresponding 2-D plots of |U z | at x=0 fromy/a=0 to 5. Here, the vertical (✓ =90 ) and almost vertical angles of incidences (✓ =60 ) contribute to the highest amplitudes along the canyon surface in allofthesefigures. Onthehalf-spacesurface,theresultantamplitudesfromhightolowarestrictly ordered by the angles of incident from vertical (✓ =90 ) to almost horizontal (✓ =15 ). Also, thedecayof|U z |onthehalf-spacesurfaceismuchslowerthanthatofthecorresponding|U x |. And thisisbecausethereisnomediumabovethehalf-spacetopreventtheparticlesmovingvertically, whilethesoilmediumhindersthemotionofparticlesinthehorizontalxdirection. 121 3.5 Conclusion In this chapter, the wave scattering around a hemispherical canyon on the stress-free half-space has been investigated. The odd-only-term series expansion of wave potential is adopted, thus the zero-stress boundary conditions on the infinite half-space are relaxed. The remaining boundary conditions on the canyon surface are utilized to solve for the coefficients of the scattered wave potentials. The resultant displacement phases and amplitudes have been demonstrated and dis- cussedinSection3.2,3.3,and3.4fortheP-,SV-andSH-incidences,respectively. Thesimilarities anddifferencesamongthesecasesaresummarizedinthefollowing: • All resultant wave amplitudes are symmetric relative to thexz plane since the free-field wavesarewavesinorparalleltothexzplane,butonlythenormalincidenceofP-wavehas axisymmetric results. The normal incidences of SV- and SH-waves are essentially identical caseswiththexandydirectionsswapped. • Thepronouncedeffectofwavediffractioncanbeobservedforobliqueincidences,inpartic- ular in the cases with an almost horizontal angle of incidence—i.e. ✓ ↵ or ✓ =15 —and a largedimensionlessfrequency—⌘ =5. The|U z |and|U x |amplitudesofthein-plane(P-and SV-)waveincidences,aswellasthe|U y |amplitudeoftheanti-plane(SH-)incidence,areall scatteredanddiffractedonthehalf-spacesurfacebehindthehemisphericalcanyon. • The wavelengths of the displacement amplitudes depend not only on the dimensionless fre- quencybutalsoontheangleofincidences. Foralltypesofwaveincidence,ahigher⌘ always contributes to the higher frequencies of the oscillatory displacement amplitudes, hence the shorterwavelengths. Inaddition,amorehorizontalangleofincidencecausesamore“com- pacted”wavepattern,sothatthewavelengthiscomparativelyshorter. • Theresultant |U y |amplitudeofanin-planeincidenceandtheresultant |U x |amplitudeofan anti-plane incidence decay rapidly to zero outside the canyon boundary on the half-space 122 surface. The reason is that there is little driving force to move the particles along the men- tioned direction associated to each case. For a specific case, the oscillation of the vertical |U z |isalwaysstrongerthanthatofthehorizontal |U x |onthehalf-space,andthisisbecause the vacuum above the half-space does not hinder the vertical motion of a particle, while the half-spacemediumfunctionsasadampertodecaythehorizontalamplitudeseffectively. • Thestatementin(Lee,1982)“largervaluesof⌘ willresultinhighercomplexityofdisplace- mentsandinhigheramplifications”stillholdsfortheresultsofarbitraryanglesofincidence. Thealmosthorizontalangleofincidencealso, inmostcases, causeshigheramplitudesthan thatofthealmostverticalangleofincidence. • There are strong amplifications around the hemispherical canyon surface in all the 3-D graphs for |U x |, |U y |, and |U z | when the dimensionless frequency is high. There, the “spike and dip” effect exists at the rim of the canyon in all types of incidence and supports the hypothesis that a secondary wave source is generated at the discontinuity of geometry or material. • InthecaseofP-orSV-incidence,thehighestamplitudesof|U x |and|U z |arealwayslocated in the plane y=0. However, for the SH-incidence, the peaks of displacement amplitudes arespreadonthecanyonsurface,with|U x |and|U z |aty=0beingzero. • The coefficients of scattered wave potentialsA (s) mn , B (s) mn , andC (s) mn for specificm andn can all be solved by the finite size (2⇥ 2 and 1⇥ 1) matrices in (3.58) and (3.59), respectively. This neat expression of solution contributes to a simpler numerical calculation as well as theresultsofahigherrangeof⌘ thanthepreviousmethodsin(Lee,1978)and(Lee,1982). Notethatthereisstillalimitofcomputationforextremelyhighfrequencies⌘ becauseofthe roundofferror. Therefore,theanalysespresentedinthischapterareupto⌘ =5. 123 3.6 Appendix to Chapter 3: The Application of Odd-Only- TermSeriesExpansiontoaHalf-SpaceCanyon As was previously discussed in Section 3.1, the scattered wave potentials can be assumed to be a complete form of series expansions with odd-only-term Hankel-Legendre polynomials as shown below: ' (s) = 1 X m=0 ' (s) m cosm sinm , with' (s) m = 1 X n m m+n=odd A (s) mn h (1) n (k ↵ r)P m n (u), (3.35a) (s) = 1 X m=0 (s) m sinm cosm , with (s) m = 1 X n m m+n=odd k B (s) mn h (1) n (k r)P m n (u), (3.35b) (s) = 1 X m=0 (s) m cosm sinm , with (s) m = 1 X n m m+n=odd C (s) mn h (1) n (k r)P m n (u). (3.35c) Moreover, on the half-space, any odd Legendre polynomial P m n (n +m is odd) can be expanded asasummationofaseriesofevenLegendrepolynomialsP m l (l +miseven)as P m n (u)= 1 X l m l+m=even m ln P m l (u),m+n = odd, (3.36) where m ln = <P m n (u),P m l (u)> <P m l (u),P m l (u)> = R 1 0 P m n (u)P m l (u)du R 1 0 P m l (u)P m l (u)du . TheintegralofassociatedLegendrepolynomialswithargumentufrom0to1iscalculatedwiththe equation below, with n to be an odd number and l to be an even number as shown in the equation belowfromPg.169(Erdélyietal.,1953): 124 Z 1 0 P m n (u)P m l (u)du = 1 (nl)(n+l+1) 8 < : (l +m)P m n (0)P m l 1 (0), ifmisodd (n+m)P m n 1 (0)P m l (0), ifmiseven (3.37) AndfromEq. 8.14.13,Pg. 338(AbramowitzandStegun,1972): Z 1 0 P m l (u)P m l (u)du = (l +m)! 2(l+0.5)(lm)! , (3.38) sothat, ' (s) m = 1 X n m m+n=odd 1 X l m l+m=even A (s) mn h (1) n (k ↵ r) m ln P m l (u); (3.39a) (s) m = 1 X n m m+n=odd 1 X l m l+m=even k B (s) mn h (1) n (k r) m ln P m l (u); (3.39b) (s) m = 1 X n m m+n=odd 1 X l m l+m=even C (s) mn h (1) n (k r) m ln P m l (u). (3.39c) Rememberthatthetwosetsofboundaryconditionofthehalf-spacecanyon,whichare • onthehalf-spacesurfacewherez=0or✓ =⇡/ 2: ✓✓ = ⌧ ✓r = ⌧ ✓ =0. • onthecanyonsurfacewherer = a,theradiusofcanyon: rr = ⌧ r✓ = ⌧ r =0. Theboundaryconditionswillbediscussedinthefollowing. 125 1. Thezeronormalstressboundaryconditiononthehalf-spacesurface ✓✓ | ✓ =⇡/ 2 =0. Oneonlyneedstoshow ✓✓ | ✓ =⇡/ 2 = (s) ✓✓ | ✓ =⇡/ 2 =0. From(MowandPao,1971), ✓✓ | ✓ =⇡/ 2 dueto' (s) m in (3.35a): 1 X n m m+n=odd A (s) mn ✓ 2µ r 2 ◆ ⇣ E (3) 21 P m n (u)+ ˆ E (3) 21 ˆ P m n (u) ⌘ , ✓✓ | ✓ =⇡/ 2 dueto (s) m in (3.39b): 1 X n m m+n=odd 1 X l m l+m=even ±k B (s) mn ✓ 2µ r ◆ mh (1) n (k r) 1u 2 ! m ln ⇥ (n1)uP m l (u)(n+m)P m l 1 (u) ⇤ , ✓✓ | ✓ =⇡/ 2 dueto (s) m in (3.35c): 1 X n m m+n=odd C (s) mn ✓ 2µ r 2 ◆ ⇣ E (3) 23 P m n (u)+ ˆ E (3) 23 ˆ P m n (u) ⌘ , (3.40) where ˆ P m n (u)= 1 1u 2 ⇥ (m 2 u 2 n)P m n (u)+(n+m)uP m n 1 (u) ⇤ . (3.41) Note that for the “±” in the expression of ✓✓ | ✓ =⇡/ 2 dueto (s) m , “+” is used in the in-plane P- or SV-incidence, and “-” is for the out-of-plane SH-incidence. This two-row convention, whichisthesameforscatteredwavepotentials,willbecarriedoninthissection. Intheexpressionsoftheabovestresses,functionsE (i) jk =E (i) jk (n,k ↵ r)arethestressfunctions involvingthesphericalBessel/Hankelfunctionscorrespondingtovariouswaves. Thek=1 in(3.40)isusedtodenotethepotentialtypetobeaP-wave. Whenk=2,3theE functions refertotheshearwavepotentialoftheout-of-planewaveandthein-planewave,respectively. The derivations are given in (Mow and Pao, 1971). The superscript i is used to denote the typeofsphericalBesselfunctionsand/orHankelfunctionsused. Here,(i=1,2,3,4)arefor thefunctionsj,y,h (1) ,andh (2) ,respectively. Thesubscriptj isusedtodenotetheparticular type of a stress function. Here, the subscripts j =1,2,3,4,5,6 are the stress components in the sequence rr , ✓✓ , , ⌧ r✓ , ⌧ r , and ⌧ ✓ . Stresses are associated with ' , , and respectfullywhenk=1,2and3. DetailedexpressionsofE (i) jk aregiveninAppendixB.2. 126 Notethatat✓ =⇡/ 2,u=cos✓ =0soform+n =odd ˆ P m n (0) =P m n (0) = 0, (3.42) andform+l =even, (n1)uP m l (u)(n+m)P m l 1 (u)=(n+m)P m l 1 (0) = 0. (3.43) Thus it can be shown that all equations in (3.40) are naturally equal to zero. Therefore, the zeronormalstressboundaryconditionisautomaticallysatisfied. 2. Thezeroradialshearstressboundaryconditiononthehalf-spacesurface⌧ ✓r | ✓ =⇡/ 2 =0. Because the free-field waves have already satisfied the surface boundary conditions, only scatteredwavepotentialsneedtobediscussed: ⌧ ✓r | ✓ =⇡/ 2 dueto' (s) m in (3.39a): 1 X n m m+n=odd 1 X l m l+m=even m ln A (s) mn ✓ 2µ r 2 ◆ E (3) 41 dP m l (u) d✓ , ⌧ ✓r | ✓ =⇡/ 2 dueto (s) m in (3.35b): 1 X n m m+n=odd k B (s) mn ✓ 2µ r ◆✓ ±m sin✓ ◆ E (3) 42 P m n (u), ⌧ ✓r | ✓ =⇡/ 2 dueto (s) m in (3.39c): 1 X n m m+n=odd 1 X l m l+m=even m ln C (s) mn ✓ 2µ r 2 ◆ E (3) 43 dP m l (u) d✓ . (3.44) Forl +miseven,at✓ =⇡/ 2 dP m l (0) d✓ =P m l 1 (0) = 0. (3.45) Using(3.45),allequationsin(3.44)canbeshowntosatisfythezeroboundarycondition. 127 3. The zero azimuthal shear stress boundary condition on the half-space surface ⌧ ✓ | ✓ =⇡/ 2 =0. ⌧ ✓ | ✓ =⇡/ 2 dueto' (s) in (3.39a): 1 X n m m+n=odd 1 X l m l+m=even m ln A (s) mn ✓ 2µ r 2 ◆✓ ⌥ m sin✓ ◆ E (3) 61 ✓ dP m l (u) d✓ cot✓P m l (u) ◆ , ⌧ ✓ | ✓ =⇡/ 2 dueto (s) in (3.35b): 1 X n m m+n=odd k B (s) mn ✓ 2µ r ◆ E (3) 52 sin 2 ✓ ! ✓✓ n 2 n 2 sin 2 ✓ +nm 2 ◆ P m n (n+m)cos✓P m n 1 ◆ , ⌧ ✓ | ✓ =⇡/ 2 dueto (s) in (3.39c): 1 X n m m+n=odd 1 X l m l+m=even m ln C (s) mn ✓ 2µ r 2 ◆✓ ⌥ m sin✓ ◆ E (3) 63 ✓ dP m l (u) d✓ cot✓P m l (u) ◆ . (3.46) Again, all equations in (3.46) can be shown to satisfy the zero stress boundary condition from(3.45)and(3.43). 4. Thezeronormalstressboundaryconditiononthecanyonsurface rr | r=a =0. rr | r=a consists of (ff) rr | r=a , which is normal stress due to free-field waves, and (s) rr | r=a , whichisnormalstressduetoscatteredwaves: rr | r=a = (ff) rr | r=a + (s) rr | r=a = ✓ 2µ a 2 ◆ 1 X m=0 ⇥ (ff) rr (m)+ (s) rr (m) ⇤ cosm sinm =0, (3.47) 128 where (ff) rr (m) and (s) rr (m) are terms of rr due to free-field waves and scattered waves respectivelyatr = awithorderm=0,1,2,.... TheirexpressionsasaBessel-Fourierseries expansionaregiveninthefollowing(MowandPao,1971): (ff) rr (m)= 1 X n m h a (ff) mn E (1) 11 (n)+c (ff) mn E (1) 13 (n) i P m n (u); (3.48a) (s) rr (m)= 1 X n m m+n=odd h A (s) mn E (3) 11 (n)+C (s) mn E (3) 13 (n) i P m n (u), (3.48b) sothat 1 X n m m+n=odd h A (s) mn E (3) 11 (n)+C (s) mn E (3) 13 (n) i P m n (u)= 1 X l m h a (ff) ml E (1) 11 (l)+c (ff) ml E (1) 13 (l) i P m l (u). (3.49) Forn = m+1,m+3, ... such thatn +m is odd, multiply both sides of (3.49) byP m n (u) andintegrateu=cos✓ from0to1. TheorthogonalityoftheP m n (u)functionsgives A (s) mn E (3) 11 (n)+C (s) mn E (3) 13 (n)= R 1 0 (ff) rr (m)P m n (u)du R 1 0 P m n (u) 2 du = 1 X l m h a (ff) ml E (1) 11 (l)+c (ff) ml E (1) 13 (l) i m ln . (3.50) Nowtheright-handsideof(3.50)isasummationofknowncoefficients. 5. Thezeropolarshearstressboundaryconditiononthecanyonsurface⌧ r✓ | r=a =0. Let ⌧ r✓ | r=a = ✓ 2µ a 2 ◆ 1 X m=0 h ⌧ (ff) r✓ (m)+⌧ (s) r✓ (m) i cosm sinm =0, (3.51) 129 where,similartoboundarycondition4,form=0,1,2,... ⌧ (ff) r✓ (m)= 1 X n m ✓ ⌧ (1)+ mn dP m n (u) d✓ ±⌧ (1) mn maP m n (u) sin✓ ◆ , (3.52a) ⌧ (s) r✓ (m)= 1 X n m m+n=odd ✓ ⌧ (3)+ mn dP m n (u) d✓ ±⌧ (3) mn maP m n (u) sin✓ ◆ , (3.52b) withfornm ⌧ (1)+ mn =a (ff) mn E (1) 41 (n)+c (ff) mn E (1) 43 (n), ⌧ (1) mn =b (ff) mn E (1) 42 (n); (3.53) and ⌧ (3)+ mn =A (s) mn E (3) 41 (n)+C (s) mn E (3) 43 (n), ⌧ (3) mn =B (s) mn E (3) 42 (n). (3.54) Thisboundaryconditionwillbecombinedwithboundaryconditioninthefollowingdiscus- sion. 6. Thezeroazimuthalshearstressboundaryconditiononthecanyonsurface⌧ r | r=a =0. Let ⌧ r | r=a = ✓ 2µ a 2 ◆ 1 X m=0 h ⌧ (ff) r (m)+⌧ (s) r (m) i sinm cosm =0, (3.55) whereform=0,1,2,... ⌧ (ff) r (m)= 1 X n m ✓ ⌥ ⌧ (1)+ mn mP m n (u) sin✓ +⌧ (1) mn adP m n (u) d✓ ◆ , (3.56a) ⌧ (s) r (m)= 1 X n m m+n=odd ✓ ⌥ ⌧ (3)+ mn mP m n (u) sin✓ +⌧ (3) mn adP m n (u) d✓ ◆ , (3.56b) with n = m,m+1,m+2,...⌧ (1)+ mn ,⌧ (1) mn ,⌧ (3)+ mn and ⌧ (3) mn defined as in (3.53) and (3.54). FromPg. 812-813(EringenandSuhubi,1975), dP m n (u) d✓ and P m n (u) sin✓ arealsoindependentinthe 130 half spaceu=cos✓ =[0,1). Therefore, to combine (3.52a), (3.52b), (3.56a) and (3.56b), thefollowingequationsareobtained: 1 X n m m+n=odd ⌧ (3)+ mn P m n (u) sin✓ = 1 X n m m+n=odd ⇣ A (s) mn E (3) 41 (n)+C (s) mn E (3) 43 (n) ⌘ P m n (u) sin✓ = 1 X n m ⌧ (1)+ mn P m n (u) sin✓ , (3.57a) 1 X n m m+n=odd ⌧ (3) mn P m n (u) sin✓ = 1 X n m m+n=odd B (s) mn E (3) 42 (n) P m n (u) sin✓ = 1 X n m ⌧ (1) mn P m n (u) sin✓ . (3.57b) Because sin✓ on both sides of (3.57a) and (3.57b) are canceled out, identical forms of (3.49) are obtained. Note that all of the coefficients on the right-hand sides are known and the pair ⇣ A (s) mn ,C (s) mn ⌘ foranym=0,1,2,...,andnmcanbesolvedfrom(3.50)and(3.57a),whichcan eventuallybearrangedasthematrixformbelow: 2 4 E (3) 11 (n) E (3) 13 (n) E (3) 41 (n) E (3) 43 (n) 3 5 2 4 A (s) mn C (s) mn 3 5 = 1 X l m m ln 2 4 E (1) 11 (l) E (1) 13 (l) E (1) 41 (l) E (1) 43 (l) 3 5 2 4 a (ff) ml c (ff) ml 3 5 . (3.58) Ontheotherhand,{B (s) mn }canbesolvedsolelyby(3.57b)withthefollowingexpression B (s) mn = P 1 l m b (ff) ml E (1) 42 (l) m ln E (3) 42 (n) . (3.59) 131 Chapter4 DiffractionaroundaHemisphericalAlluvial ValleybyWavesofArbitraryIncidence In this chapter, the wave diffraction around a hemispherical valley on the infinite half-space have been discussed. Similar to the structure of Chapter 3, Section 4.1 describes the valley model and adoptstheodd-term-onlyseriesexpansionstosolveforthescatteredanddiffractedwavepotentials, andthedetailedstepsarepresentedinSection4.6attheendofthischapter. Sections4.2,4.3,and 4.4 contain the P-, SV-, and SH-incidences with arbitrary angles, respectively. Properties and features of the resultant displacement amplitudes are investigated. The 3-D graphs of |U x |, |U y |, and|U z |withvariousratiosofcoefficientsareshowninAppendixD.1toAppendixD.3attheend ofthethesis. 4.1 Description of the Model of Valley and Boundary Condi- tions In Fig. 4.1, a hemispherical alluvial valley is located on the surface of the half-space. The media of both the half-space and the valley are assumed to be elastic, homogeneous and isotropic. The half-space medium is featured by the density ⇢ , and Lamé parameters , and µ; while the valley medium has its density ⇢ f , and Lamé parameters f , andµ f . Vacuum is assumed above the half- space,thusallkindsofstressesarezeroonthehalf-spaceorvalleysurface. InthecontextofCivil Engineering,thehalf-spacemediumissometimescalled“soil”,andthevalleymediumcanalsobe afoundation(elastic)andmaybereferredtoassuch. 132 Figure4.1. Three-dimensionalhemisphericalvalleywitharbitraryplanewaveincidence. Like the canyon problem, both the Cartesian coordinates and the spherical coordinates are utilized as shown in Fig. 4.2. The origin of both coordinates are the center of the hemispherical valley. Thezdirection is vertically downward, so that the planez=0 is the half-space surface. Forthein-planewaves—i.e. theP-andSV-waves—theparticlesmoveparalleltothexz plane, andfortheSH-wave,theparticlesmoveperpendiculartothexz plane. Thesphericalcoordinate systemconsistsofr whichistheradialdistance,✓ (0 ✓  ⇡/ 2)whichisthepolarangle, and (⇡   ⇡ ) which is the azimuthal angle. On the half-space or valley surface whenz=0, the polarangle✓ =⇡/ 2. All the three types of waves will be considered in this problem. The angle of incidence with respect to the xaxis associated to the longitudinal (P-)wave is denoted by ✓ ↵ , and the angle of incidence for the shear (SV- or SH-)wave is denoted by ✓ . Due to the stress-free boundary conditionsonthehalf-spacesurfacethat zz | z=0 = ⌧ zx | z=0 = ⌧ zy | z=0 =0, 133 the free-field wave potentials can be determined to satisfy the boundary conditions above. The detailed expressions depending on the wave type of the incidence are presented in Appendix A, whicharethesameofthatforthecanyonprobleminChapter3. Figure4.2. Adoptionofsphericalcoordinates. Inthepresenceofthealluvialvalley, asetofscatteredwavesandasetofrefractedwaveswill be generated, respectively, in the half-space. The expressions of the scattered wave potentials are givenas(thetimefactore i!t hasbeendroppedforconvenience): ' (s) = 1 X m=0 ' (s) m cosm sinm , with' (s) m = 1 X n m m+n=odd A (s) mn h (1) n (k ↵ r)P m n (u), (s) = 1 X m=0 (s) m sinm cosm , with (s) m = 1 X n m m+n=odd k B (s) mn h (1) n (k r)P m n (u), (s) = 1 X m=0 (s) m cosm sinm , with (s) m = 1 X n m m+n=odd C (s) mn h (1) n (k r)P m n (u), (4.1) whereh (1) n (.)isthefirstkindofsphericalHankel’sfunctionofordernthatcanbeusedtorepresent theoutgoingwaves,k ↵ ,andk arethewavenumbersofthelongitudinalwave,andtheshearwaves 134 forthemediumofthehalf-space(soil),respectively. Thetopsetofthetrigonometricfunctionsare fortheP-andSV-wavepotentials,whilethebottomsetoftrigonometricfunctionsarefortheSH- wave potentials. While for the refracted waves, the first kind of spherical Bessel function j n is adoptedbecauseallkindsofthepotentialsarefiniteatthecanyoncenter. Thentheexpressionsof therefractedwavespotentialsaregiveninthefollowing: ' (R) = 1 X m=0 ' (R) m cosm sinm , with' (R) m = 1 X n m m+n=odd A (R) mn j n (k ↵f r)P m n (u), (R) = 1 X m=0 (R) m sinm cosm , with (R) m = 1 X n m m+n=odd k f B (R) mn j n (k f r)P m n (u), (R) = 1 X m=0 (R) m cosm sinm , with (R) m = 1 X n m m+n=odd C (R) mn j n (k f r)P m n (u), (4.2) wherek ↵f ,andk f arethewavenumbersofthelongitudinalwave,andshearwavesforthemedium ofthevalley(foundation). Similar to the methodology applied to the canyon cases, any odd-term Legendre polynomial P m n (n +m is odd) can be expanded as a summation of the even-term Legendre polynomialsP m l (l +miseven)onthehalf-space,thatis: P m n (u)= 1 X l m l+m=even m ln P m l (u), (4.3) wheretheexpansioncoefficients m ln arealreadydefinedin(3.36). 135 Therefore, the scattered wave potentials can also be expressed in terms of the even Legendre polynomialssuchthat ' (s) m = 1 X n m m+n=odd 1 X l m l+m=even A (s) mn h (1) n (k ↵ r) m ln P m l (u), (s) m = 1 X n m m+n=odd 1 X l m l+m=even k B (s) mn h (1) n (k r) m ln P m l (u), (s) m = 1 X n m m+n=odd 1 X l m l+m=even C (s) mn h (1) n (k r) m ln P m l (u). (4.4) As in the canyon problem, they can thus automatically satisfy all the stress-free boundary condi- tionsonthehalf-spacesurface. Sodotherefractedwavepotentialsintheformthat ' (R) m = 1 X n m m+n=odd 1 X l m l+m=even A (R) mn j n (k ↵f r) m ln P m l (u), (R) m = 1 X n m m+n=odd 1 X l m l+m=even k f B (R) mn j n (k f r) m ln P m l (u), (R) m = 1 X n m m+n=odd 1 X l m l+m=even C (R) mn j n (k f r) m ln P m l (u), (4.5) andasthescatteredwaves,therefractedwavescanthusautomaticallysatisfythehalf-spacebound- ary conditions on the valley surface. Therefore, there are in total six wave potentials hence six series of coefficients—the A (s) mn ,B (s) mn ,C (s) mn ,A (R) mn ,B (R) mn , and C (R) mn —to be solved by the boundary conditionslistedbelow: 136 I. Thezero-stressboundaryconditionsonthehalf-spacesurface,where✓ =⇡/ 2andra: (ff) ✓✓ + (s) ✓✓ =0, ⌧ (ff) ✓r +⌧ (s) ✓r =0, ⌧ (ff) ✓ +⌧ (s) ✓ =0. (4.6) II. Thezero-stressboundaryconditionsonthevalleysurface,where✓ =⇡/ 2andr< a: (R) ✓✓ =0, ⌧ (R) ✓r =0, ⌧ (R) ✓ =0. (4.7) Asstatedabove,IandIIaresatisfiedrespectivelybythescatteredandrefractedwaves. III. Thecontinuityboundaryconditionsontheinteractingsurfaceofthehalf-spaceandthevalley, wherer = aand 0 ✓  ⇡/ 2: (ff) rr + (s) rr = (R) rr , ⌧ (ff) r✓ +⌧ (s) r✓ = ⌧ (R) r✓ , ⌧ (ff) r +⌧ (s) r = ⌧ (R) r , U (ff) r +U (s) r =U (R) r , U (ff) ✓ +U (s) ✓ =U (R) ✓ , U (ff) +U (s) =U (R) . (4.8) Notethatforthesuperscriptsofthestressordisplacementcomponents, (ff) referstothecom- ponent caused by the free-field potentials, (s) refers to that caused by the scattered potentials, and (R) is for the refracted wave potentials. Section 4.6 is the appendix of this chapter that 137 includes all the technical details of the application of the odd-term-only potentials to the above boundary conditions. Finally (4.56) in Section 4.6 contains the closed-form analytic solution for (A (s) mn ,C (s) mn ,A (R) mn ,C (R) mn ), and (4.57) presents the solution for (B (s) mn ,B (R) mn ), for m=0,1,2,..., nm. As to (Lee, 1984), four groups of the ratios of µ f µ and C ↵f C↵ are selected, whereC ↵ is the speed of the longitudinal wave in the half-space medium and C ↵f is the speed of longitudinal wave in the valley medium. Then the corresponding wave numbers arek ↵ = !/C ↵ for the half-space and k ↵f = !/C ↵f for the valley, where ! is a constant because of the steady state assumption. In addition, from (Gercek, 2007), it is also valid to assume the poisson’s ratios to be 0.25 in both media,sothat ,thespeedratioofthelongitudinalwaveandtheshearwaveinthesamemedium, remainstobetheconstant p 3aswasstatedinChapter1. Consequently,theratioofthelongitudinal wavenumbersinvariousmediamustbeequalthethatofshearwavenumbersuchthat k f k = k ↵f k↵ , where C is the shear wave speed in the half-space medium, C f is the shear wave speed in the valleymedium,k istheshearwavenumberinthehalf-spacethatk =!/C ,andk f istheshear wavenumberinthevalleythatk f =!/C f . Table4.1liststheratiosadoptedforthefourcasesintheresults. AlthoughonlytheC ↵f /C ↵ for the P-incidence is shown,C f /C shares the same value for the cases of shear incidence because of the assumptions above. The ratios for case I to III are, according to (Lee, 1984), the soft alluvial media in granite rock, while case IV, the only case whose µ f µ and C ↵f C↵ are greater than one, represents a hard medium in granite rock. So that these four cases are ordered by the valley mediumfromsofttostiff. Caseno. µ f /µ C ↵f /C ↵ k ↵f /k ↵ I 0.25 0.5 2 II 0.3 0.6 1.67 III 0.4 0.6 1.67 IV 3 1.5 0.67 Table4.1. Casesofparameterratiosusedinthediscussion. 138 Sameasthepreviouswork,thefollowingdimensionlessvariablesareusedintheanalysis: - Thedimensionlessdistancesx/aandy/a. - Thedimensionlessfrequency⌘ = k a ⇡ . - Theratioofpoisson’sratiosforthehalf-spaceandforthevalley µ f µ . - Theratioofwavenumbersforthehalf-spaceandforthevalley k ↵f k↵ or k f k . 139 4.2 DisplacementsforPlaneP-incidencewithAngle✓ ↵ The formulation of the normalized free-field wave potential in the valley problem is exactly the same as that of the canyon problem stated in Section 3.2.1. And the same angles of incidence are used in this chapter ✓ ↵ =90 ,60 ,30 , and 15 , with their corresponding free-field displacement amplitudes presented in Table 3.1. The calculation of surface displacements will be divided into twopartsbecauseofthedifferentmediaandthedifferentsourcesofwaves. 1. In the half-space wherer> a, the resultant displacements are the superposition of the displacements caused by the free-field potentials and that by the scattered potentials. The components of displacements due to the scattered waves in the spherical coordinates are givenby,asin(3.13): U (s) r = 1 r 1 X m=0 1 X n m m+n=odd h A (s) mn D (3) 11 +C (s) mn D (3) 13 i P m n (u)cosm, U (s) ✓ = 1 r 1 X m=0 1 X n m m+n=odd " ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ dP m n (u) d✓ + mrB (s) mn D (3) 22 sin✓ P m n (u) # cosm, U (s) = 1 r 1 X m=0 1 X n m m+n=odd  m sin✓ ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ P m n (u)rB (s) mn D (3) 22 dP m n (u) d✓ sinm, (4.9) whereu=cos✓ andD (i) jk arefunctionsofthepotential-displacementrelationshipswhichare shortforD (i) jk (n,k ↵ r)ofthelongitudinalwaveorD (i) jk (n,k r)oftheshearwavesinthehalf- space. The completed and detailed expressions ofD (i) jk function are given in Appendix B.1 attheendofthethesis. Thesphericalcomponents(U (s) r ,U (s) ✓ ,U (s) )canbethentransformed into the Cartesian components (U (s) x ,U (s) y ,U (s) z ) by the relationship given in (3.14) to sum withthefree-fielddisplacementsoftheplaneP-wavegivenin(3.12). 140 2. In the valley medium wherer a, the displacements are only determined by the refracted wave potentls ' (R) , (R) , and (R) . The displacements in spherical coordinates can be expressedas: U (R) r = 1 r 1 X m=0 1 X n m m+n=odd h A (R) mn D (1) 11 +C (R) mn D (1) 13 i P m n (u)cosm, U (R) ✓ = 1 r 1 X m=0 1 X n m m+n=odd " ⇣ A (R) mn D (1) 21 +C (R) mn D (1) 23 ⌘ dP m n (u) d✓ + mrB (R) mn D (1) 22 sin✓ P m n (u) # cosm, U (R) = 1 r 1 X m=0 1 X n m m+n=odd  m sin✓ ⇣ A (R) mn D (1) 21 +C (R) mn D (1) 23 ⌘ P m n (u)rB (R) mn D (1) 22 dP m n (u) d✓ sinm, (4.10) where u =cos✓ the D (i) jk functions here are short for D (i) jk (n,k ↵f r) or D (i) jk (n,k f r) for the medium of valley. (U (R) r ,U (R) ✓ ,U (R) ) can be transformed into the Cartesian system as (U (R) x ,U (R) y ,U (R) z )usingthesametransformationmatrixin(3.14). Caseno. Fig. no. µ f /µ C ↵f /C ↵ Type I D.1toD.9 0.25 0.50 softestvalley II D.10toD.18 0.30 0.60 softervalley III D.19toD.27 0.40 0.60 softvalley IV D.28toD.36 3.00 1.50 stiffvalley(foundation) Table4.2. Summaryof3-DgraphsinAppendixD.1. Notethatonthesurfaceofthehalf-spaceorthevalley,thereis✓ =⇡/ 2,sothatu=cos✓ =0 in(4.9)and(4.10)forallcomponentsofdisplacementsonthesurface. Inthefollowingdiscussion, theresultantamplitudesofthesurfacedisplacementsofallthefourcasesinTable4.1andTable4.2 here are shown and compared. The 3-D graphs of |U x |, |U y |, and |U z | are presented in Appendix 141 D.1 at the end of the thesis, and Table 4.2 contains the figure numbers and their related ratios of parameters. The 3-D figures in Appendix D.1 are plotted in the ranges ofx/a andy/a both from2 to 2, with various angles of incidence that ✓ ↵ =90 ,60 ,30 , and 15 . The case of ✓ ↵ =90 , which is alsothenormalincidence,alwayscausesthetwocomponentsofhorizontaldisplacements|U x |and |U y |equaltozeroasinthetopleftfigureinFig. 4.3. ThisisbecausetheverticalplaneP-incidence causesaxisymmetryandasimilarproofhasbeenshowninthecanyonprobleminChapter1. Figure4.3. |U x |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence: µ f /µ=0.25,C ↵f /C ↵ =0.50 Tobetterillustratetheresults,thecorresponding2-Dplotsareincludedinthediscussion. Table 4.3liststhecomponent,thedimensionlessfrequency⌘ ,andtheangleofincidence✓ ↵ usedineach figure. All of the 2-D figures contain four subfigures for the four corresponding cases of media giveninTable4.1. 142 Fig. no. Component ⌘ ✓ x/a y/a 4.4 |U z | 1,3,5 90 (0,5] 0.0 4.5to4.7 |U z | 1,3,5 90 ,60 ,30 ,15 [5,5] 0.0 4.8 |U x | 1 90 ,60 ,30 ,15 0.5 (0,5] 4.9,4.10 |U x | 3,5 90 ,60 ,30 ,15 0.5 (0,5] Fig. no. Component ⌘ ✓ r/a 4.12to4.14 |U y | 1,3,5 90 ,60 ,30 ,15 [5,5] 45 Table4.3. Summaryof2-DfiguresinSection4.2. Figure 4.4 shows the vertical displacement amplitudes |U z | for the normal P-wave incidence, ✓ ↵ =90 aty=0 alongx/a from 0 to 5, because the results are symmetric relative to theyz plane. The subfigure on the top has the vertical |U z | for ⌘ =1, ⌘ =3, and ⌘ =5, respectively, with the media properties of Case I, which represents the softest valley medium selected in this problem. The next two subfigures in the middle of Fig. 4.4 are |U z | on the medium firm valley givenbyCaseII(softer)andCaseIII(soft),for⌘ =1,3,and5. Thelastsubfigureshowstheplots of |U z | for ⌘ =1,3, and 5, on the stiffest valley/foundation medium in Case IV. When the valley is softer than the half-space (Case I, II, and III), the resultant |U z | on the valley surface, where x/a=0 to 1, is significantly greater than that on the half-space surface, wherex/a > 1. Only in thelastsubfigureforCaseIV,theresultantamplitudesonthecanyonsurfacebetweenx/a=0and 1aresmallerandlessoscillatorythanthatonthehalf-spacesurface. Asthefrequency⌘ increases, theplotsof |U z |areofhighercomplexitybothonthevalleysurfaceandthehalf-spacesurface, as showninallthesubfiguresforall✓ ↵ andallkindsofmedia. 143 Figure4.4. |U z |with⌘ =1,3,5of✓ ↵ =90 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ alongradialdistancer/a=0to 5. 144 Figure4.5. |U z |with⌘ =1of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ aty=0.0. 145 Figure4.6. |U z |with⌘ =3of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ aty=0.0. 146 Figure4.7. |U z |with⌘ =5of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ aty=0.0. 147 Figures 4.5 to 4.7 demonstrate the 2-D plots of |U z | aty=0 for ✓ ↵ =90 ,60 ,30 , and 15 , and ⌘ =1,3, and 5, respectively. The range x/a = 5 to 5 is selected because the results for the oblique angles of incidence are no more symmetric relative to theyz plane. Each of these figures still contains four subfigures corresponding to the four cases of media properties same as in Fig. 4.4. Evidently, the soft valley media result in higher displacement amplitudes than that of the stiff valley/foundation medium. In Fig. 4.5 for ⌘ =1, the resultant |U z |, on both the valley surface and the half-space surface, decreases with the change of the incident angle from vertical—at✓ ↵ =90 —to almost horizontal—at✓ ↵ =15 . Moreover, the incidences of✓ ↵ =30 and ✓ ↵ =15 contribute to obvious spikes on the rims of the valley (x/a = ±1) for ⌘ =1, and the amplitudes on the (left) illuminated side can be seen higher than the amplitudes on the (right) shadow side, when valley is softer that the half-space as shown in the three subfigures on the top. Notethatthesamephenomenonhasbeenobservedintheresultsofthecanyonproblempresented inChapter3. Sameasthereasonforthecanyonproblem,hereisbecausewhenthewavesincident from the left, the valley rim is acting as a shield to prevent potential energy traveling to the right (Lee,1984). Nevertheless, the |U z | amplitudes for ✓ ↵ =30 and ✓ ↵ =15 on the shadow side of the valley are higher than that on the illuminated side with increasing ⌘ , such as the ⌘ =3 in Fig. 4.6 and ⌘ =5 in Fig. 4.7. Although the almost vertical incidence ✓ ↵ =60 have its highest |U z | values relative evenly distributed on the soft valley surface (x/a =1 to 1) for all the selected ⌘ , the displacement amplitudes on the half-space surface are observed, like other oblique incidences ✓ ↵ =30 and15 ,severelyinterferedbythescatteredwavesontheshadowside(x/a> 1). Again, similartothehemisphericalcanyon,thisobservationcanbeexplainedasthephenomenonofwave diffraction, which is prominent for the almost horizontal angles of incidence, in particular when theincident frequency⌘ ishigh. Thereactionson thebottomsubfigure, whichrepresentsthe stiff valley/foundationmedium,arealwayslowonthevalleysurfaceforvariousfrequencies⌘ ,however the oscillations of amplitudes on the half-space can be also seen on the shadow side. Therefore, 148 the phenomenon of wave diffraction exists in cases of not only the soft valley media but ask the hardmedium. Figures4.8to4.10containthe2-Dhorizontaldisplacementamplitudes |U x |ofthevarious✓ ↵ , and the various ratios of µ and C ↵ , at a given x/a, a plane in from of the canyon perpendicular to the plane of incident waves. Note that the horizontal displacements for the normal P-wave incidencearealwayszeroonthexy plane,sotheplotsof|U x |for✓ ↵ =90 arenotnoticeablein the figures. It is always not easy to use a 2-D cross-section to efficiently represent the whole 3-D graph. The3-Dgraphsof|U x |areallinAppendixD.1. The2-Dplotsof|U x |for⌘ =1areplotted atx/a=0.5, whilethe2-Dplotsfor⌘ =3, and5areplottedatx/a=0. Onlythepositiverange ofy/a=0to 5isselected,becausealltheresultantamplitudesaresymmetricrelativetothexz plane. Similartotheverticalcomponent|U z |,the|U x |alsohaspronouncedamplitudesforthesoft valley medium, while the amplitudes of |U x | on the surface of the stiff valley/foundation given by CaseIVisrelativelytrivial,asshowninthelastsubfigureofeachofthe|U x |figure. Inmostcases, the angle of incidence ✓ ↵ =30 results in the highest amplitudes on the surface of the valley and thehalf-spaceineachsubfigure. Theangle✓ ↵ =60 hasthelowestfree-fieldamplitudeamongthe three oblique incidences—✓ ↵ =60 ,30 , and 15 , however its amplification on the valley surface ishigherthanthatof✓ ↵ =15 ,andsometimeshigherthanthatof✓ ↵ =30 (inthefirstsubfigurein 4.10). The increase of frequency⌘ causes the complexity of the resultant |U x |, but the amplitudes are not obviously amplified, except for Case I that for the frequency ⌘ =5, the amplitudes |U x | canbeashighas 8asshowninFig. 4.10. 149 Figure4.8. |U x |with⌘ =1of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ atx=0.5. 150 Figure4.9. |U x |with⌘ =3of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ atx=0.0. 151 Figure4.10. |U x |with⌘ =5of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ atx=0.0. 152 Figure4.11. Exampleofplaneof =45 . Because of the pattern of distribution of |U y | in the 3-D graphs in Appendix D.1, the 2-D |U y | figures will be plotted along the =45 /225 diagonal plane, which has been illustrated in Fig. 4.11, in order to cover more fluctuations on the valley surface. Figures 4.12 to 4.14 are the 2-D plotsof|U y |oftheanglesofincidence✓ ↵ =90 ,60 ,30 ,and15 for⌘ =1,3,and5,respectively. The range r/a =5 to 0 refers to the position of =225 , and the range r/a=0 to 5 is for =45 . Note that the |U y | for the normal P-wave incidence is also zero on the xy plane, so thatthereareonlythreenon-zerocurvesineachsubfigure. One could notice that Case II, instead of Case I, gives the largest response of ⌘ =3. This reflectsthe non-monotonicpropertyoftheproblemsofwavediffractionandrefraction. Whenthe wavelengthissmallenough(⌘ =5),theamplitudesonthenegativerangeandpositiverangehave quite equal patterns which means that the resultant |U y | of high-frequency incidence is affected less by whether the incident wave comes from the x or thex direction. Moreover, the out- of-plane component |U y | is not exhibiting the phenomenon of wave diffraction even though the dimensionlessfrequencyofP-waveishigh,incontrasttotheclearobservationin|U z |and|U x |. 153 Figure4.12. |U y |with⌘ =1of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ along =45 . 154 Figure4.13. |U y |with⌘ =3of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ along =45 . 155 Figure4.14. |U y |with⌘ =5of✓ ↵ =90 ,60 ,30 ,and 15 ofvalleyforP-incidence withvariousparametersofµ f /µandC ↵f /C ↵ along =45 . 156 4.3 DisplacementsforPlaneSV-incidencewithAngle✓ Thefree-fieldwavepotentialsofthearbitraryangleofSV-incidencecanbeformulatedinthesame way as that of the canyon problem in Section 3.3.1. The reflected waves can be categorized into two cases depending on ✓ ✓ cr or ✓ <✓ cr . Because the SV-incidence is an incidence with in- planedisplacement, thecalculationsofdisplacementsduetothescatteredwavesinthehalf-space andthevalleyaresimilartothatoftheplaneP-incidence. Andthefree-fielddisplacementsonthe half-spaceareagaingivenby(3.25)or(3.26),dependingonwhether✓ ✓ cr or✓ <✓ cr . The angle of incidence is denoted by ✓ , and ✓ =90 ,60 ,30 , and 15 will be adopted in thediscussionstofollow. AppendixD.2presentsthe3-Dgraphsofthenormalized|U x |, |U y |,and |U z |for✓ =90 ,60 ,30 ,and 15 ,and⌘ =1,3,and 5intherangesofx/aandy/afrom-2to2. Table4.4summarizestheratiosofparametersusedinthecalculationandthecorrespondingfigure numbersoftheresults. Caseno. Fig. no. µ f /µ C f /C Type I D.37toD.45 0.25 0.50 softestvalley II D.46toD.54 0.30 0.60 softervalley III D.55toD.63 0.40 0.60 softvalley IV D.64toD.72 3.00 1.50 stiffvalley Table4.4. Summaryof3-DgraphsinAppendixD.2. Fromthe3-DgraphsinAppendixD.2,theresultanthorizontalamplitudes|U x |and|U y |forthe normalincidence—✓ =90 —arenon-zerobothonthevalleysurfaceandthehalf-spacesurface. Moreover, both the |U x | and |U y | are symmetric relative to not only the xz plane but also the yz planewhen✓ =90 , andthiswasshownintheSH-incidenceofthehemisphericalcanyon probleminSection3.4. 157 The 2-D plots are presented to assist the discussion of the resultant displacement amplitudes for the SV-incidence. Information of the component of displacements, the frequency of incidence ⌘ , the angle of incidence ✓ , and the ranges of location of the 2-D figures has been summarized andlistedinTable4.5. Fig. no. Component ⌘ ✓ x/a y/a 4.15to4.17 |U z | 1,3,5 90 ,60 ,30 ,15 [5,5] 0.0 4.18,4.20 |U x | 1,5 90 ,60 ,30 ,15 [5,5] 0.5 4.19 |U x | 3 90 ,60 ,30 ,15 [5,5] 0.2 4.21to4.23 |U x | 1,3,5 90 ,60 ,30 ,15 0.5 (0,5] Fig. no. Component ⌘ ✓ r/a 4.24to4.26 |U y | 1,3,5 90 ,60 ,30 ,15 [5,5] 45 Table4.5. Summaryof2-DfiguresinSection4.3. Figures4.15to4.17aretheverticaldisplacementamplitudes|U z |aty=0alongx/afrom5 to 5 for✓ =90 ,60 ,30 , and 15 , and⌘ =1,3, 5, respectively. Similarto theP-incidence, here thesoftmediumofvalleycontinuestoresultinthehighestamplitudesamongallthefourcasesof parameterratios,andtheincreaseof⌘ alwaysresultsinmoreamplifiedandcomplexdisplacement amplitudesonthevalleysurface. When⌘ =1,forthesoftmediaofvalley,themostprominent|U z | amplitudes are always found on the surface of the valley, as shown in the top three subfigures of Fig. 4.15;whileforthestiffmediumofvalleyinthebottomsubfigure,thehighest|U z |canbeseen attherimsatx/a = ±1. As⌘ increases,theobliqueincidences—✓ =60 ,30 ,and 15 —startto havetheirsignificantamplitudesmovingawayfromthecentertotherimsofthevalley. InFig. 4.16when⌘ =3,thealmostverticalincidence(✓ =60 )hasitshighestamplitudeson theilluminatedsideonthesurfaceofthevalleyinthefirstsubfigurewhichisfortheforthesoftest mediumofvalley,whileinthetwosubfiguresforCaseIIandCaseIII,theamplificationson 158 Figure4.15. |U z |with⌘ =1of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C aty=0.0. 159 Figure4.16. |U z |with⌘ =3of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C aty=0.0. 160 Figure4.17. |U z |with⌘ =5of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C aty=0.0. 161 both sides of the valley are equivalent on the plots of ✓ =60 . For almost horizontal incidences, ✓ =15 at ⌘ =3, all the soft valley media result in the most amplified displacements close to the rim of the shadow side (x/a=1). In Fig. 4.17 for |U z | at ⌘ =5, the highest amplitudes of eachplotareintensivelydistributedaroundtherims,wheretheamplitudesontheilluminatedside are lower than that on the shadow side of the valley. Also the waves of the |U z | amplitudes on thehalf-spacesurfaceontheshadowsidecanbeseendistortedbytheinterferenceofthescattered waves, in particular for the almost horizontal incidence ✓ =15 . So that the wave diffraction phenomenonclearexistsintheSV-incidenceofthehemisphericalalluvialvalley. Figures 4.18 to 4.20 are the 2-D plots of the horizontal displacement amplitude |U x | for ✓ = 90 ,60 ,30 , and 15 , and⌘ =1,3, 5 alongx/a from5 to 5. From the 3-D graphs in Appendix D.2,theplaney/a=0.5hasbeenselectedforplotsof⌘ =1inFig. 4.18andplotsof⌘ =5inFig. 4.20, and the results for ⌘ =3 in 4.19 are plotted aty/a=0.2. Unlike the |U z | amplitudes that obviouslyoscillationonthehalf-spacesurface,the|U x |amplitudesconvergetotheircorresponding free-fieldamplituderapidlyonthehalf-spacesurfacealongx/aforanyincidentangles✓ ,andany frequencies⌘ . Nevertheless,inanotherperspective,the|U x |atx=0alongy/afrom 0to 5forthe variousangles✓ andfrequencies⌘ arepresentedabitmoreonFig. 4.21toFig. 4.23(followthat ofFig. 4.8toFig. 4.10forP-waveincidence). Slightfluctuationofthehorizontal|U x |amplitudes canbeseeninthisgroupoffigures. Theresultantamplitudes |U x |onthesurfaceofthestiffvalley/foundationmediumarestillde- amplified, as shown in the bottom subfigures in all these figures. The softer medium of valley againresultsinmoresubstantialdisplacementamplitudes,and,however,forthehorizontal|U x |,it istheincidentangle✓ =30 thatbringsthemostpronouncedresultscomparedwithother✓ for thesame⌘ andcaseofparameterratios. The3-Dgraphsof|U x |inAppendixD.2givecompelling evidence for this, although the free-field |U x | amplitude for ✓ =30 is the lowest among all the selectedanglesofincidence. Also,althoughitisnotasclearasfortheresultsof|U z |,thewaves 162 Figure4.18. |U x |with⌘ =1of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C aty/a=0.5. 163 Figure4.19. |U x |with⌘ =3of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C aty/a=0.2. 164 Figure4.20. |U x |with⌘ =5of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofofµ f /µandC f /C aty/a=0.5. 165 Figure4.21. |U x |with⌘ =1of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C atx=0.0. 166 Figure4.22. |U x |with⌘ =3of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C atx=0.0. 167 Figure4.23. |U x |with⌘ =5of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C atx=0.0. 168 Figure4.24. |U y |with⌘ =1of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C along =45 . 169 Figure4.25. |U y |with⌘ =3of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C along =45 . 170 Figure4.26. |U y |with⌘ =5of✓ =90 ,60 ,30 ,and 15 ofvalleyforSV-incidence withvariousparametersofµ f /µandC f /C along =45 . 171 of|U x |amplitudesareshadowingonthehalf-spacesurfaceontheshadowsideofthecanyon(x/a from 1to 5)showninFig. 4.18toFig. 4.20. Figures 4.24 to 4.26 are the out-of-plane horizontal component |U y | at the diagonal plane, = 45 /225 ,alongr/afrom5to5for✓ =90 ,60 ,30 ,and15 ,and⌘ =1,3,and5,respectively. Same as the P-incidence, the ranger/a from 0 to -5 indicates the direction of the azimuthal angle =225 along the radial distancer/a from 0 to 5, and the ranger/a from 0 to 5 in these figures isinthedirectionof =45 . As the horizontal |U y | is the out-of-plane displacement component that affected less by the in-plane SV-wave incidence, the amplitudes on the half-space surface decay rapidly to zero away from the vicinity of the spherical valley. The softest valley medium in the first subfigure of each figure continues to contribute to the most pronounced resultant amplitudes compared with that of the other media, and the increase of frequency ⌘ again causes the more complex pattern on the valley surface, and the stiff valley/foundation medium gives the de-amplified results as before. Liketheotherhorizontalcomponent|U x |,theincidentangle✓ =30 mostofthetimesresultsin thehighestamplitudefor|U y |thatcanbeseenonthe3-DgraphsinAppendixD.2. Onecanseethatforthesoftvalleymedia,whentheanglesofincidencearevertical(✓ =90 ), oralmostvertical(✓ =60 ),thehighestpeaksof|U y |aredistributednearthecanterofthevalley, but when the angles are almost horizontal (✓ =30 and 15 ), the most significant results appear ontherimsofthevalley,asshowninFig. 4.24toFig. 4.26. 172 4.4 DisplacementsforPlaneSH-incidencewithAngle✓ Thepotentialsofout-of-planefree-fieldSH-wavescanbeformulatedinthesamewayasinSection 3.4, and the same selections of angles of incidence ✓ =90 ,60 ,30 and 15 , and the selections oftheincidentfrequencies⌘ =1,3,and 5willbepresentedinthediscussionoftheresults. Table 4.6 categorizes the four cases of parameter ratios and their corresponding numbers of figures for the3-DgraphspresentedinAppendixD.3attheendofthethesis. Caseno. Fig. no. µ f /µ C f /C Type I D.73toD.81 0.25 0.50 softestvalley II D.82toD.90 0.30 0.60 softervalley III D.91toD.99 0.40 0.60 softvalley IV D.100toD.108 3.00 1.50 stiffvalley Table4.6. Summaryof3-DgraphsinAppendixD.3. Since the SH-waves are the out-of-plane waves, the calculation of surface displacements will be different from that of the in-plane P- and SV-waves. As before, the calculation should be partitioned into two areas—within the valley where r  a, and on the surface of the half-space wherer> a. The trigonometric terms sinm and cosm are again interchanged, so that the scatteredwavepotentialsinthehalf-spacearegivenas,sameas(3.32): ' (s) = 1 X m=0 ' (s) m sinm, (s) = 1 X m=0 (s) m cosm, (s) = 1 X m=0 (s) m sinm, (4.11) 173 andtherefractedwavesinthevalleyaregivenby ' (R) = 1 X m=0 ' (R) m sinm, (R) = 1 X m=0 (R) m cosm, (R) = 1 X m=0 (R) m sinm, (4.12) where ' (s) m , (s) m , and (s) m , and ' (R) m , (R) m , and (R) m and have been already defined in (4.1) and (4.2). The expressions of displacement components in the spherical coordinates are then of the followingform: 1. In the half-space wherer> a, the spherical components of displacements due to the scat- teredwavepotentialsare U (s) r = 1 r 1 X m=0 1 X n m m+n=odd h A (s) mn D (3) 11 +C (s) mn D (3) 13 i P m n (u)sinm, U (s) ✓ = 1 r 1 X m=0 1 X n m m+n=odd " ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ dP m n (u) d✓ mrB (s) mn D (3) 22 sin✓ P m n (u) # sinm, U (s) = 1 r 1 X m=0 1 X n m m+n=odd  m sin✓ ⇣ A (s) mn D (3) 21 +C (s) mn D (3) 23 ⌘ P m n (u)rB (s) mn D (3) 22 dP m n (u) d✓ cosm, (4.13) whereD (i) jk areshortforthedisplacement-potentialrelationshipsD (i) jk (n,k ↵ r)orD (i) jk (n,k r) in the half-space medium.Then the Cartesian components (U (s) x ,U (s) y ,U (s) z ) due to the scat- tered waves can be obtained by the transformation in (3.14). To sum up (U (s) x ,U (s) y ,U (s) z ) and the free-field displacements of plane the SH-wave given in (3.33) the total resultant displacementsinthehalf-spacearethenachieved. 174 2. In the valley where r  a, the displacements are only determined by the refracted waves ' (R) , (R) ,and (R) ,withtheexpressionsinthesphericalcoordinatesas: U (R) r = 1 r 1 X m=0 1 X n m m+n=odd h A (R) mn D (1) 11 +C (R) mn D (1) 13 i P m n (u)sinm, U (R) ✓ = 1 r 1 X m=0 1 X n m m+n=odd " ⇣ A (R) mn D (1) 21 +C (R) mn D (1) 23 ⌘ dP m n (u) d✓ mrB (R) mn D (1) 22 sin✓ P m n (u) # sinm, U (R) = 1 r 1 X m=0 1 X n m m+n=odd  m sin✓ ⇣ A (R) mn D (1) 21 +C (R) mn D (1) 23 ⌘ P m n (u)rB (R) mn D (1) 22 dP m n (u) d✓ cosm, (4.14) where theD (i) jk functions here are short forD (i) jk (n,k ↵f r) orD (i) jk (n,k f r) in the medium of valley. AllthedetailedexpressionsofD (i) jk aregiveninAppendixB.1attheendofthisthesis. TheCartesiancomponents(U (R) x ,U (R) y ,U (R) z )canbeobtainedfrom(U (R) r ,U (R) ✓ ,U (R) )bythe transformationof(3.14). The 2-D plots are utilized for the discussion of resultant displacements. Table 4.7 lists the information of the 2-D plots in each figure. Note that in this section, because the wave incidence isout-of-plane,theout-of-plane|U y |componentamplitudeswillbethefirsttoinvestigate. Fig. no. Component ⌘ ✓ x/a y/a 4.27to4.29 |U y | 1,3,5 90 ,60 ,30 ,15 [5,5] 0.0 4.30to4.32 |U z | 1,3,5 90 ,60 ,30 ,15 [5,5] 0.2 Fig. no. Component ⌘ ✓ r/a 4.33to4.35 |U x | 1,3,5 90 ,60 ,30 ,15 [5,5] 45 Table4.7. Summaryof2-DfiguresinSection4.4. As was stated in Chapter 3 for the hemispherical canyon, the vertical (✓ =90 ) SV-wave incidenceandthevertical(✓ =90 )SH-waveincidenceareanalyticallyandnumericallyidentical 175 except that their xand ydirections are interchanged. From the 3-D graphs in Appendix D.3, onecanseethatthehorizontalcomponents |U x |and|U y |are,therefore,symmetrictonotonlythe xz planebutalsotheyz planewhentheincidenceisvertical. Moreover,theout-of-plane|U y | amplitudeshaveallthehighestoscillationsdistributedalongthexz planeforallincidentangles ✓ ,frequencies⌘ ,andthepropertiesofmediagiveninthefourcases. Incontrasttotheincidences ofin-planewaves(P-orSV-waves)thatthemostsignificantcomponentsofdisplacementsare|U z | and|U x |,herethe|U y |inthisSH-incidencepresentsthemostprominentresults. Figures 4.27 to 4.29 are the 2-D plots of the out-of-plane component |U y | aty=0 alongx/a from5 to 5, for incident angles t✓ =90 .60 ,30 , and 15 , and ⌘ =1,3, and 5, respectively. Each figure still contains four subfigures for the four corresponding cases of the valley and the half-space media given in Table 4.6. The softest valley medium (the top subfigure in each figure) mostly results in the largest resultant amplitudes for any incident angles ✓ and frequencies ⌘ , while the stiff valley medium (the bottom subfigure in each figure) always has the de-amplified oscillations on the valley surface. Also, the increase of ⌘ brings more complex patterns of the resultantamplitudesonthevalleysurface. Fortheobliqueincidence—✓ =60 ,30 ,and15 ,theresultant|U y |amplitudesontheshadow side of the valley are always higher than that on the illuminated side, for all incident frequencies ⌘ . Nevertheless for the soft valley media, at low frequency (⌘ =1) in Fig. 4.27, the highest amplitudes appear on the shadow side, almost half way between the valley center and the right rim (x/a =1); on the other hand, when ⌘ is as high, at ⌘ =3 and 5 in Fig. 4.28 and Fig. 4.29, the highest peaks are closer to the right toward the vicinity of the right rim, and the almost horizontalanglesofincidence(✓ =30 and 15 )contributetomorepronouncedresultsthanthat of the vertical (✓ =90 ) or almost vertical (✓ =60 ) incidence. The waves of |U y | for oblique incidencescanbeseendistortedonthehalf-spacesurfacebothontheilluminatedandtheshadow sides,similartotheobservationofthe|U z |plotsintheP-andSV-incidences. 176 Figure4.27. |U y |with⌘ =1of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C aty=0.0. 177 Figure4.28. |U y |with⌘ =3of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C aty=0.0. 178 Figure4.29. |U y |with⌘ =5of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C aty=0.0. 179 Figure4.30. |U z |with⌘ =1of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C aty=0.2. 180 Figure4.31. |U z |with⌘ =3of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C aty=0.2. 181 Figure4.32. |U z |with⌘ =5of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C aty=0.2. 182 Figures4.30to4.32presentthe2-Dplotsoftheverticalcomponent|U z |attheplaney/a=0.2 along x/a from5 to 5 for ✓ =90 ,60 ,30 , and 15 , and ⌘ =1,3, and 5. Apparently, the softest valley medium results in the most significant amplitudes for all ✓ and ⌘ on the surface of the valley. The oscillations decay fast to zero on the half-space surface, although the amplitudes on the shadow side can be observed much higher than that on the illuminated side in particular when⌘ ishigh(3and 5)andtheangleofincidenceisalmosthorizontal—✓ =15 . When⌘ =1, theresultant|U z |onthevalleysurfacefor✓ =90 ,verticalincidence,ishighestwhencompared to that for oblique incidences ✓ =60 ,30 , and 15 , and the highest amplitude of each curve is near the yz plane at x/a=0 for all the soft valley media. As ⌘ increases, the results for the almosthorizontalincidence,✓ =30 and15 ,starttoexceedthatoftheverticaloralmostvertical incidences,✓ =90 and 60 ,asshowninFig. 4.31and4.32when⌘ =3and 5. Moreover,fromthe3-DgraphsinAppendixD.3,onecanseethatwhentheincidenceisvertical (✓ =90 ),themostpronounced |U z |alwaysoccuralongtheyz plane,andthetwopeaksnear thevalleycenterbecomemoreandmoredominantwiththeincreaseofthefrequency⌘ . However when the incidence is almost horizontal, the “humps” throughout the valley surface can be seen homogeneous distributed and the highest peaks appear near the the rims at x/a± 1 on the xz plane. Figures4.33to4.35aretheother(in-plane)horizontalcomponent|U x |alongr/afrom5to5 at =45 /255 , the diagonal plane, for the various ✓ and ⌘ . In Fig. 4.33 when the frequency is lowat⌘ =1,the|U x |amplitudesonthevalleysurfaceontheshadowside,forobliqueincidences, are higher than the amplitudes on the illuminated side. But as ⌘ grows, as shown in Fig. 4.34 and Fig. 4.35, the amplitudes on both sides on the valley surface become equally weighted, and the resultsforthevertical(✓ =90 )oralmostvertical(✓ =60 )incidenceare,however,higherthan thatofthealmosthorizontalincidences(✓ =30 and 15 ). 183 Figure4.33. |U x |with⌘ =1of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C along =45 . 184 Figure4.34. |U x |with⌘ =3of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C along =45 . 185 Figure4.35. |U x |with⌘ =5of✓ =90 ,60 ,30 ,and 15 ofvalleyforSH-incidence withvariousparametersofµ f /µandC f /C along =45 . 186 4.5 Conclusion The diffraction and refraction of waves around a hemispherical alluvial valley on the elastic half- spacehasbeenstatedinthischapter. Theodd-only-termseriesexpansion,whichwasutilizedinthe hemisphericalcanyoninChapter3,isappliedtosatisfythestress-freeboundaryconditionsonthe half-spacesurfaceautomatically. Theremainingboundaryconditionsontheinter-surfacebetween the half-space and the valley provide sufficient information to solve for the unique coefficients of thescatteredwavesandrefractedwavescompletely. TheresultantamplitudesfortheP-, SV-, and SH-incidencesarepresentedanddiscussedinSection4.2,4.3,and4.4,respectively. Followingare thesimilaritiesanddifferencesofthesecasesofincidences: • Allresultantwaveamplitudes,similartothatofthecanyonproblem,aresymmetricrelative to the xz plane. The normal incidence of P-wave (✓ ↵ =90 ) results in axisymmetric results,whiletheresultsofthenormalSV-andSH-incidencesaresymmetricrelativetoboth thexz andtheyz planes. • Whenthemediumofthevalleyissoft,theresultant|U x |,|U y |and|U z |amplitudesarehighly amplified. But when the medium of the valley is hard, the resultant displacements are de- amplifiedonthevalleysurface. • For the high frequencies—⌘ =5 and almost horizontal angles of incidence ✓ ↵ or ✓ at 15 , the shadowing of waves is significant on the half-space surface on the shadow side. Similar tothecanyonproblem,thisphenomenonispronouncedinthegraphsof|U x |and|U z |forthe in-planeP-orSV-incidences,andthegraphsof|U y |fortheanti-planeSH-incidence. • The higher frequency and almost horizontal incident angle also result in more complex and amplified patterns of the displacement amplitudes on both the half-space surface and the valleysurface,sothatthestatementin(Lee,1982)stillholdshere. 187 • The displacements amplitudes |U x |, |U y |, and |U z | at the rim of the valley on the half-space surface can be seen strongly amplified and de-amplified in the 3-D graphs. Thus the “spike anddip”effectalsoexistsinthehemisphericalvalleyproblem. • Similar to the canyon problem, the highest amplitudes of |U x | and |U z | in the 3-D graphs alwaysappearinthexzplanefortheP-orSV-incidence,whileinthecaseofSH-incidence, thehighest|U y |amplitudesareonthexz plane. • The coefficients of scattered waves and refracted waves can be solved analytically by the continuityboundaryconditionsontheinteractionalsurfacebetweenthevalleyandthehalf- space. For specificm andn, the unique solution of (A (s) mn ,C (s) mn ,A (R) mn ,C (R) mn ) can be obtained by the 4⇥ 4 matrix in (4.56), and the unique solution of (B (s) mn ,B (R) mn ) can be achieved by the 2⇥ 2matrixin(4.57). Howevertheaccuracyofresultforextremelyhighfrequency⌘ is limitedbecauseoftheroundofferrorbecauseoffinite-bitrepresentationofeachrealnumber. Thehighestfrequency⌘ presentedinthisanalysisis⌘ =5. • For the study of the SH-wave incidence, it is observed that at the high frequency (⌘ =5), amplificationsoftheresultant|U y |and|U z |forthealmosthorizontalincidencesof✓ =30 and15 arelessthanthatofverticalandalmostverticalincidences✓ =90 and60 . ForP- andSV-waveincidence,thisisnotthecase. 188 4.6 Appendix to Chapter 4: The Application of Odd-Term- Only Series Expansion to An Alluvial Valley on the Half- space Alloftheboundaryconditionsforthehemisphericalalluvialvalleyonthehalf-spaceareshownin thefollowing: I. Thezero-stressboundaryconditionsonthehalf-spacesurface,where✓ =⇡/ 2andra: (ff) ✓✓ + (s) ✓✓ =0, ⌧ (ff) ✓r +⌧ (s) ✓r =0, ⌧ (ff) ✓ +⌧ (s) ✓ =0. (4.15) II. Thezero-stressboundaryconditionsonthevalleysurface,where✓ =⇡/ 2andr< a: (R) ✓✓ =0, ⌧ (R) ✓r =0, ⌧ (R) ✓ =0. (4.16) III. Thecontinuityboundaryconditionsontheinteractingsurfaceofthehalf-spaceandthevalley, wherer = a: (ff) rr + (s) rr = (R) rr ; ⌧ (ff) r✓ +⌧ (s) r✓ = ⌧ (R) r✓ ; ⌧ (ff) r +⌧ (s) r = ⌧ (R) r ; U (ff) r +U (s) r =U (R) r ; U (ff) ✓ +U (s) ✓ =U (R) ✓ ; U (ff) +U (s) =U (R) . (4.17) 189 Thestress-freeboundaryconditionsinCategoriesIandIIonthehalf-spaceorthevalleysurface canbeprovedautomaticallysatisfiedbythesimilarstepsfortheboundaryconditionsonthehalf- space in the canyon problem in Chapter 3. Thus, only the six continuity boundary conditions in CategoryIIIwillbeanalyzedasfollows. 1. Thecontinuityboundaryconditionofnormalstressonthesoil-foundationinteractional surface (ff) rr | r=a + (s) rr | r=a = (R) rr | r=a . Thisboundaryconditionisequivalentto (ff) rr | r=a + (s) rr | r=a (R) rr | r=a = ✓ 2 a 2 ◆ 1 X m=0 ⇥ µ (ff) rr (m)+µ (s) rr (m)µ f (R) rr (m) ⇤ cosm sinm =0. (4.18) where (ff) rr (m), (s) rr (m) and (R) rr (m) are the normal stress due to the free-field waves, the scattered waves, and the refracted waves, respectively, with orderm=0,1,2,..., atr = a. The top trigonometric function (cosm ) is for the in-plane P- or SV-incidence, while the bottom trigonometric function (sinm ) is for the out-of-plane SH-waves. This notation will be carried on through this section. According to Pao and Mow in Appendix B.2, their expressionsareintheformof: (ff) rr (m)= 1 X n m h a (ff) mn E (1) 11 (n,k ↵ a)+c (ff) mn E (1) 13 (n,k a) i P m n (u), (s) rr (m)= 1 X n m m+n=odd h A (s) mn E (3) 11 (n,k ↵ a)+C (s) mn E (3) 13 (n,k a) i P m n (u), (R) rr (m)= 1 X n m m+n=odd h A (R) mn E (1) 11 (n,k ↵f a)+C (R) mn E (1) 13 (n,k f a) i P m n (u). (4.19) Note that E (i) jk (n,k ↵ a) and E (i) jk (n,k a) are the expression of stress components in term of thepotentialfunctionsforthehalf-spacemedium,whileE (i) jk (n,k ↵f a)andE (i) jk (n,k f a)are 190 potential-stress relationships for the valley medium. When the superscript i =1 the E function contains the first kind of Bessel function j n (.), and when i=3, the E consists of thefirstkindofHankel’sfunctionh (1) n (.). Applytheorthogonalityoftrigonometricfunctionof inthefull-rangefrom⇡ to⇡ ,(4.18) canbesimplified,form=0,1,2,...,tobe: µ (ff) rr (m)+µ (s) rr (m)µ f (R) rr (m)=0. (4.20) To plug the equations of (4.19) into (4.20), one can obtain the relationship among the coef- ficientsA (s) mn ,C (s) mn ,A (R) mn . andC (R) mn suchthat: 1 X n m m+n=odd 2 4 A (s) mn E (3) 11 (n,k ↵ a)+C (s) mn E (3) 13 (n,k a) ( µ f µ )A (R) mn E (1) 11 (n,k ↵f a)( µ f µ )C (R) mn E (1) 13 (n,k f a) 3 5 P m n (u) = 1 X l m h a (ff) ml E (1) 11 (l,k ↵ a)+c (ff) ml E (1) 13 (l,k a) i P m l (u). (4.21) Moreover, from the orthogonality of the Legendre Polynomials, to multiply P m n (u) (m + n =odd)onbothsidesandintegrateu=cos✓ from 0to 1,(4.21)thenbecomes A (s) mn E (3) 11 (n,k ↵ a)+C (s) mn E (3) 13 (n,k ↵ a) ( µ f µ )A (R) mn E (1) 11 (n,k ↵f a)( µ f µ )C (R) mn E (1) 13 (n,k f a) = R 1 0 (ff) rr (m)P m n (u)du R 1 0 P m n (u) 2 du . (4.22) Notethattherighthandsideof(4.22)onlyincludestheknowncoefficients. 2. Thecontinuityboundaryconditionofshearpolarstressonthesoil-foundationinterac- tionalsurface⌧ (ff) r✓ | r=a +⌧ (s) r✓ | r=a = ⌧ (R) r✓ | r=a . 191 Thisboundaryconditionisequivalentto ⌧ (ff) r✓ | r=a +⌧ (s) r✓ | r=a ⌧ (R) r✓ | r=a = ✓ 2 a 2 ◆ 1 X m=0 h µ⌧ (ff) r✓ (m)+µ⌧ (s) r✓ (m)µ f ⌧ (R) r✓ (m) i cosm sinm =0. (4.23) Similartoboundarycondition1,form=0,1,2,... µ⌧ (ff) r✓ (m)+µ⌧ (s) r✓ (m)µ f ⌧ (R) r✓ (m)=0. (4.24) where the shear stress ⌧ r✓ due to the free-field waves, the scattered waves, and the refracted wavecanbeexpressedinthefollowingform: ⌧ (ff) r✓ (m)= 1 X n m ✓ ⌧ (ff)+ mn dP m n (u) d✓ ±⌧ (ff) mn maP m n (u) sin✓ ◆ , ⌧ (s) r✓ (m)= 1 X n m m+n=odd ✓ ⌧ (s)+ mn dP m n (u) d✓ ±⌧ (s) mn maP m n (u) sin✓ ◆ , ⌧ (R) r✓ (m)= 1 X n m m+n=odd ✓ ⌧ (R)+ mn dP m n (u) d✓ ±⌧ (R) mn maP m n (u) sin✓ ◆ , (4.25) withtheabovesymbolsdefinedas,fornm: ⌧ (ff)+ mn =a (ff) mn E (1) 41 (n,k ↵ a)+c (ff) mn E (1) 43 (n,k a), ⌧ (ff) mn =k b (ff) mn E (1) 42 (n,k a); (4.26) ⌧ (s)+ mn =A (s) mn E (3) 41 (n,k ↵ a)+C (s) mn E (3) 43 (n,k a), ⌧ (s) mn =k B (s) mn E (3) 42 (n,k a); (4.27) 192 and ⌧ (R)+ mn =A (R) mn E (1) 41 (n,k ↵f a)+C (R) mn E (1) 43 (n,k f a), ⌧ (R) mn =k f B (R) mn E (1) 42 (n,k f a). (4.28) Toplug(4.25)into(4.24),itbecomes 1 X n m m+n=odd ✓ ⌧ (s)+ mn µ f µ ⌧ (R)+ mn ◆ dP m n (u) d✓ + 1 X n m ⌧ (ff)+ mn dP m n (u) d✓ ± 1 X n m m+n=odd ✓ ⌧ (s) mn µ f µ ⌧ (R) mn ◆ maP m n (u) sin✓ ± 1 X n m ⌧ (ff) mn maP m n (u) sin✓ =0. (4.29) Equation(4.29)willbesolvedtogetherwithboundarycondition3inthenextpart. 3. The continuity boundary condition of shear azimuthal stress on the soil-foundation interactionalsurface⌧ (ff) r | r=a +⌧ (s) r | r=a = ⌧ (R) r | r=a . Let ⌧ (ff) r | r=a +⌧ (s) r | r=a ⌧ (R) r | r=a = ✓ 2 a 2 ◆ 1 X m=0 h µ⌧ (ff) r (m)+µ⌧ (s) r (m)µ f ⌧ (R) r (m) i sinm cosm =0. (4.30) Thenform=0,1,2,...,thereis µ⌧ (ff) r (m)+µ⌧ (s) r (m)µ f ⌧ (R) r (m)=0, (4.31) 193 where⌧ r duetothedifferenttypesofwavescanbeexpressedas ⌧ (ff) r (m)= 1 X n m ✓ ⌥ ⌧ (ff)+ mn mP m n (u) sin✓ +⌧ (ff) mn adP m n (u) d✓ ◆ , ⌧ (s) r (m)= 1 X n m m+n=odd ✓ ⌥ ⌧ (s)+ mn mP m n (u) sin✓ +⌧ (s) mn adP m n (u) d✓ ◆ , ⌧ (R) r (m)= 1 X n m m+n=odd ✓ ⌥ ⌧ (R)+ mn mP m n (u) sin✓ +⌧ (R) mn adP m n (u) d✓ ◆ . (4.32) with ⌧ (ff)+ mn ,⌧ (ff) mn ,⌧ (s)+ mn ,⌧ (s) mn ,⌧ (R)+ mn and ⌧ (R) mn are already defined in equations (4.26) to (4.28). Afterrearrangingtheexpressionof(4.31),itisnowgivenas ⌥ 1 X n m m+n=odd ✓ ⌧ (s)+ mn µ f µ ⌧ (R)+ mn ◆ mP m n (u) sin✓ ⌥ 1 X n m ⌧ (ff)+ mn mP m n (u) sin✓ + 1 X n m m+n=odd ✓ ⌧ (s) mn µ f µ ⌧ (R) mn ◆ adP m n (u) d✓ + 1 X n m ⌧ (ff) mn adP m n (u) d✓ =0. (4.33) According to (Morse and Feshbach, 1953), dP m n (u) d✓ and P m n (u) sin✓ are independent in the half- spacewhereu=cos✓ =[0,1],whichissimplybecause dP m n (u) d✓ and P m n (u) sin✓ areorthogonalin the full space whereu is from -1 to 1. Combining (4.29) and (4.33), the following relation- shipsareproposed: 1 X n m m+n=odd ✓ ⌧ (s)+ mn µ f µ ⌧ (R)+ mn ◆ P m n (u) sin✓ = 1 X n m ⌧ (ff)+ mn P m n (u) sin✓ , (4.34a) 1 X n m m+n=odd ✓ ⌧ (s) mn µ f µ ⌧ (R) mn ◆ P m n (u) sin✓ = 1 X n m ⌧ (ff) mn P m n (u) sin✓ . (4.34b) 194 ToapplytheorthogonalityoftheassociatedLegendrepolynomials,thefinalformof(4.34a) is A (s) mn E (3) 41 (n,k ↵ a)+C (s) mn E (3) 43 (n,k a) ( µ f µ )A (R) mn E (1) 41 (n,k ↵f a)( µ f µ )C (R) mn E (1) 43 (n,k f a) = 1 X l m R 1 0 ⌧ (ff)+ ml P m l (u)P m n (u)du R 1 0 P m n (u) 2 du , (4.35) andthefinalformof(4.34b)is B (s) mn E (3) 42 (n,k a) µ f µ k f k B (R) mn E (1) 42 (n,k f a) = 1 X l m R 1 0 b (ff) ml E (1) 42 (l,k a)P m l (u)P m n (u)du R 1 0 P m n (u) 2 du . (4.36) 4. The continuity boundary condition of normal displacement on the soil-foundation interactionalsurfaceU (ff) r | r=a +U (s) r | r=a =U (R) r | r=a . The boundary conditions of displacements are very similar to the boundary conditions of stresses, so the same strategy will be used for the displacement. To restate this boundary condition: U (ff) r | r=a +U (s) r | r=a U (R) r | r=a = ✓ 1 a ◆ 1 X m=0 ⇥ U (ff) r (m)+U (s) r (m)U (R) r (m) ⇤ cosm sinm =0, (4.37) whereU (ff) r (m), U (s) r (m) andU (R) r (m) are the terms ofU r due to the free-field waves, the scatteredwavesandtherefractedwaves,respectively,withtheorderm=0,1,2,...atr =a. Becauseoftheorthogonalityofthetrigonometricfunctions cosm or sinm ,onecanget U (ff) r (m)+U (s) r (m)U (R) r (m)=0. (4.38) 195 From the potential-displacement relationships given in Appendix B.1 , the detailed expres- sionsaregivenas U (ff) r (m)= 1 X n m h a (ff) mn D (1) 11 (n,k ↵ a)+c (ff) mn D (1) 13 (n,k a) i P m n (u), U (s) r (m)= 1 X n m m+n=odd h A (s) mn D (3) 11 (n,k ↵ a)+C (s) mn D (3) 13 (n,k a) i P m n (u), U (R) r (m)= 1 X n m m+n=odd h A (R) mn D (1) 11 (n,k ↵f a)+C (R) mn D (1) 13 (n,k f a) i P m n (u). (4.39) Similarlytothestressfunctions,D (i) jk (n,k ↵ a)andD (i) jk (n,k a)arethepotential-displacement functionsforthehalf-spacemedium,whileD (i) jk (n,k ↵f a)andD (i) jk (n,k f a)arethepotential- displacementfunctionsforthemediumofthevalley. Still,thei=1denotesthefirstkindof Besselfunctionj n (.),andthei=3referstothefirstkindofHankel’sfunctionh (1) n (.). Consequently(4.38)canbeexpressedas: 1 X n m m+n=odd 2 4 A (s) mn D (3) 11 (n,k ↵ a)+C (s) mn D (3) 13 (n,k a) A (R) mn D (1) 11 (n,k ↵f a)C (R) mn D (1) 13 (n,k f a) 3 5 P m n (u) = 1 X l m h a (ff) ml D (1) 11 (l,k ↵ a)+c (ff) ml D (1) 13 (l,k a) i P m l (u). (4.40) ToapplytheorthogonalityoftheassociatedLegendrePolynomials,(4.40)thennowhasthe form A (s) mn D (3) 11 (n,k ↵ a)+C (s) mn D (3) 13 (n,k ↵ a) A (R) mn D (1) 11 (n,k ↵f a)C (R) mn D (1) 13 (n,k f a) = R 1 0 U (ff) r (m)P m n (u)du R 1 0 P m n (u) 2 du (4.41) 196 5. The continuity boundary condition of normal displacement on the soil-foundation interactionalsurfaceU (ff) ✓ | r=a +U (s) ✓ | r=a =U (R) ✓ | r=a . Thisboundaryconditionis U (ff) ✓ | r=a +U (s) ✓ | r=a U (R) ✓ | r=a = ✓ 1 a ◆ 1 X m=0 h U (ff) ✓ (m)+U (s) ✓ (m)U (R) ✓ (m) i cosm sinm =0, (4.42) that,form=0,1,2,... U (ff) ✓ (m)+U (s) ✓ (m)U (R) ✓ (m)=0, (4.43) wheretheU ✓ duetothefree-fieldwaves,thescatteredwaves,andtherefractedwavescanbe expressedas: U (ff) ✓ (m)= 1 X n m ✓ U (ff)+ mn dP m n (u) d✓ ±U (ff) mn maP m n (u) sin✓ ◆ , U (s) ✓ (m)= 1 X n m m+n=odd ✓ U (s)+ mn dP m n (u) d✓ ±U (s) mn maP m n (u) sin✓ ◆ , U (R) ✓ (m)= 1 X n m m+n=odd ✓ U (R)+ mn dP m n (u) d✓ ±U (R) mn maP m n (u) sin✓ ◆ , (4.44) withtheabovefunctionsdefinedinthefollowing,fornm, U (ff)+ mn =a (ff) mn D (1) 21 (n,k ↵ a)+c (ff) mn D (1) 23 (n,k a), U (ff) mn =k b (ff) mn D (1) 22 (n,k a); (4.45) 197 U (s)+ mn =A (s) mn D (3) 21 (n,k ↵ a)+C (s) mn D (3) 23 (n,k a), U (s) mn =k B (s) mn D (3) 22 (n,k a; (4.46) and U (R)+ mn =A (R) mn D (1) 21 (n,k ↵f a)+C (R) mn D (1) 23 (n,k f a), U (R) mn =k f B (R) mn D (1) 22 (n,k f a). (4.47) (4.43)canbealsorewrittenas 1 X n m m+n=odd U (s)+ mn U (R)+ mn dP m n (u) d✓ + 1 X n m U (ff)+ mn dP m n (u) d✓ ± 1 X n m m+n=odd U (s) mn U (R) mn maP m n (u) sin✓ ± 1 X n m U (ff) mn maP m n (u) sin✓ =0. (4.48) Thisboundaryconditionwillbesolvedtogetherwithboundarycondition6inthefollowing. 6. The continuity boundary condition of normal displacement on the soil-foundation interactionalsurfaceU (ff) | r=a +U (s) | r=a =U (R) | r=a . Let U (ff) | r=a +U (s) | r=a U (R) | r=a = ✓ 1 a ◆ 1 X m=0 h U (ff) (m)+U (s) (m)U (R) (m) i sinm cosm =0, (4.49) sothatform=0,1,2,... U (ff) (m)+U (s) (m)U (R) (m)=0, (4.50) 198 where U (ff) (m)= 1 X n m ✓ ⌥ U (ff)+ mn mP m n (u) sin✓ +U (ff) mn adP m n (u) d✓ ◆ , U (s) (m)= 1 X n m m+n=odd ✓ ⌥ U (s)+ mn mP m n (u) sin✓ +U (s) mn adP m n (u) d✓ ◆ , U (R) (m)= 1 X n m m+n=odd ✓ ⌥ U (R)+ mn mP m n (u) sin✓ +U (R) mn adP m n (u) d✓ ◆ . (4.51) and the functions U (ff)+ mn ,U (ff) mn ,U (s)+ mn ,U (s) mn ,U (R)+ mn , and U (R) mn are already defined in (4.45)to(4.47). Torewrite(4.50),thereis ⌥ 1 X n m m+n=odd U (s)+ mn U (R)+ mn mP m n (u) sin✓ ⌥ 1 X n m U (ff)+ mn mP m n (u) sin✓ + 1 X n m m+n=odd U (s) mn U (R) mn adP m n (u) d✓ + 1 X n m U (ff) mn adP m n (u) d✓ =0. (4.52) Combining (4.48) and (4.52), with the independency of dP m n (u) d✓ and P m n (u) sin✓ in the half-space, thefollowingrelationshipscanbeobtained: 1 X n m m+n=odd U (s)+ mn U (R)+ mn P m n (u) sin✓ = 1 X n m U (ff)+ mn P m n (u) sin✓ , (4.53a) 1 X n m m+n=odd U (s) mn U (R) mn P m n (u) sin✓ = 1 X n m U (ff) mn P m n (u) sin✓ . (4.53b) 199 ToapplytheorthogonalityoftheassociatedLegendrepolynomials,thefinalformof(4.53a) is A (s) mn D (3) 21 (n,k ↵ a)+C (s) mn D (3) 23 (n,k ↵ a) A (R) mn D (1) 21 (n,k ↵f a)C (R) mn D (1) 23 (n,k f a) = 1 X l m R 1 0 U (ff)+ ml P m l (u)P m n (u)du R 1 0 P m n (u) 2 du , (4.54) andthefinalformof(4.53b)is B (s) mn D (3) 22 (n,k a) k f k B (R) mn D (1) 22 (n,k f a) = 1 X l m R 1 0 b (ff) ml D (1) 22 (l,k a)P m l (u)P m n (u)du R 1 0 P m n (u) 2 du . (4.55) Therefore, from boundary conditions 1 to 6, the combination of (4.22), (4.35), (4.41) and (4.54) givesthefollowingexpressionofmatrix 2 6 6 6 6 6 6 6 4 E (3) 11 (n,k ↵ a) E (3) 13 (n,k a) µ f µ E (1) 11 (n, k ↵f k↵ k ↵ a) µ f µ E (1) 13 (n, k f k k a) E (3) 41 (n,k ↵ a) E (3) 43 (n,k a) µ f µ E (1) 41 (n, k ↵f k↵ k ↵ a) µ f µ E (1) 43 (n, k f k k a) D (3) 11 (n,k ↵ a) D (3) 13 (n,k a) D (1) 11 (n, k ↵f k↵ k ↵ a) D (1) 13 (n, k f k k a) D (3) 21 (n,k ↵ a) D (3) 23 (n,k a) D (1) 21 (n, k ↵f k↵ k ↵ a) D (1) 23 (n, k f k k a) 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 A (s) mn C (s) mn A (R) mn C (R) mn 3 7 7 7 7 7 7 7 5 = 1 X l m m ln 2 6 6 6 6 6 6 6 4 E (1) 11 (l,k ↵ a) E (1) 13 (l,k a) E (1) 41 (l,k ↵ a) E (1) 43 (l,k a) D (1) 11 (l,k ↵ a) D (1) 13 (l,k a) D (1) 21 (l,k ↵ a) D (1) 23 (l,k a) 3 7 7 7 7 7 7 7 5 2 4 a (ff) ml c (ff) ml 3 5 , (4.56) from which the complete and unique solution of (A (s) mn ,C (s) mn ,A (R) mn , C (R) mn ) can be obtained for any m0andnm. 200 Similarly, the unique solution ofB (s) mn andB (R) mn can be obtained by (4.36) and (4.55) given in thematrixformsuchthat 2 4 E (3) 42 (n,k a) µ f µ k f k E (1) 42 (n, k f k k a) D (3) 22 (n,k a) k f k D (1) 22 (n, k f k k a) 3 5 2 4 B (s) mn B (R) mn 3 5 = 1 X l m m ln 2 4 b (ff) ml E (1) 42 (l,k a) b (ff) ml D (1) 22 (l,k a) 3 5 . (4.57) Comparisonbetweenvalleyandcanyon. Apparently, the results will depend on the ratio of the wave numbers k f k or k ↵f k↵ , and ratio of the shear modulus µ f µ . Numerically, when µ f µ ! 0, C ↵f C↵ ! 0 so that k ↵f k↵ !1 , and it becomes the caseofthecanyon,andonlythescatteredwavesoutsideremain,andthe 4⇥ 4matrixreducestoa 2⇥ 2matrix: 2 4 E (3) 11 (n,k ↵ a) E (3) 13 (n,k a) E (3) 41 (n,k ↵ a) E (3) 43 (n,k a) 3 5 2 4 A (s) mn C (s) mn 3 5 = 1 X l m m ln 2 4 E (1) 11 (l,k ↵ a) E (1) 13 (l,k a) E (1) 41 (l,k ↵ a) E (1) 43 (l,k a) 3 5 2 4 a (ff) ml c (ff) ml 3 5 , (4.58) and E (3) 42 (n,k a)B (s) mn = 1 X l m m ln b (ff) ml E (1) 42 (l,k a). (4.59) 201 Chapter5 TheSoil-structure-interaction(SSI) problem 5.1 Introduction Modern research on SSI in earthquake engineering is already 80 years old. Sezawa and Kanai (Sezawa and Kanai, 1935a) (Sezawa and Kanai, 1935b) (Sezawa and Kanai, 1935c) are the pio- neers who started to explore the SSI interactional effects in early 1930s. Earthquake-resistant designevolvedaroundtheconceptofresponsespectrummethod(RSM)(Trifunac,2009a),which uses the linear single-degree-freedom (SDOF) model (Udwadia and Trifunac, 1974) and only the single largest peak of response (Gupta and Trifunac, 1988). The formulation of the RSM is based on the vibrational solution of the dynamic equilibrium equations and ignores the wave propa- gation in the structure (Todorovska and Trifunac, 1990) (Trifunac et al., 1999) (Todorovska and Lee, 1990) considering only translational components of strong motion (Trifunac, 2009b). As it is used in engineering design, the RSM also ignores the consequences of SSI (Luco et al., 1987) (Wong and Trifunac, 1975). However, this also ignores the benefits of soils (Trifunac and Todor- ovska,1997b)(TrifunacandTodorovska,1997a)(TrifunacandTodorovska,1999)(Trifunacetal., 2001a)(Trifunacetal.,2001b)(TrifunacandTodorovska,1998)andshouldbeintroducedintothe engineering design procedures. This will become possible with the full development of 3-D SSI method which is capable of considering non-linear response of both soil and structure. For this developmentitwillbenecessarytounderstandthe3-DSSIphenomenainconsiderabledetailand 202 tocalibratethenumericalmethodfortheiranalysis. Inthispaper,weintroduceonesuchanalytical model,whichshouldalsoprovideusefulinsightinto3-DSSIresponse. In the 1960s and 1970s, with the advances in digital computation, numerical methods became available for SSI studies (Fleming et al., 1965), and analyses could be formulated based on finite- element analysis (FEA) (Lysmer et al., 1975) (Hadjian et al., 1975) (Awojobi and Grootenhuis, 1965), finite-difference method (FDM) (Gicev and Trifunac, 2007b) (Gicev and Trifunac, 2007a) (Giˇ cev and Trifunac, 2009) (Gicev and Trifunac, 2009) (Giˇ cev and Trifunac, 2012) and direct or indirect boundary-element methods (BEM or IBEM)(Liang et al., 2013a)(Liang et al., 2013b). However,allnumericalmethods“shouldbequalifiedbymakingacomparisonwithanexactsolu- tionofasimplecontinuumproblem. Onlythencantheextensionstosolveproblemswithcompli- catedgeometriesandconstitutiverelationshipsbejustified”(Hadjianetal.,1975). The analytic solution of the SSI problem of rigid disk placing on elastic semi-infinite wave was presented by Luco and Westmann in 1971 (Luco and Westmann, 1971). Based on Luco’s (Luco, 1969) 2-D study of interaction of shear wall and rigid circular foundation with vertical- plane SH-wave incidence, Trifunac (Trifunac, 1972) extended this solution to arbitrary angle of plane SH-wave incidence. Then Wong and Trifunac (Wong and Trifunac, 1974b) generalized the solution of a model with rigid elliptical foundation. However, neither 2-D in-plane nor 3-D SSI problems has been thoroughly solved because of the mode conversion between longitudinal and shear waves and mixed boundary conditions. Lee (Lee, 1979) presented an analytic approach of 3-D SSI by simplifying the upper building by a single-degree-freedom (SDOF) oscillator. In 2014, Lee and Liu (Lee and Liu, 2013) and Lee and Zhu (Lee and Zhu, 2013) revisited the 2-D and3-Dboundary-valuedelasticproblemsandimprovedtheanalyticmethodtohighfrequencies. ThispaperfollowsLeeandZhu(2014)(LeeandZhu,2013)andTrifunac’sclassicalworkof2-D shear wall problem (Trifunac, 1972) to use Bessel-Fourier half-range expansion to investigate the interaction of cylindrical building and 3-D hemispherical rigid foundation placed on an infinite half-space. 203 Figure5.1. 3-DSSIproblemwithvertical-planeP-waveincidence. 5.2 TheModel In Fig. 5.1 the half-space has density ⇢ , Lam´ e constants and µ. A cylindrical building with density ⇢ b and Lam´ e constants b and µ b is attached to a hemispherical rigid foundation which has density ⇢ f and Lam´ e constants f andµ f . There is no slippage between the building and the foundation. Allmediaarehomogeneous,isotropicandelastic. Anxyz Cartesiancoordinate system is adopted at the center of the foundation, which has the downward z-direction. Another Cartesiancoordinatesystem ¯ x¯ y¯ z islocatedattopofthebuilding. BothCartesiancoordinate systems follow the right-hand rule. The radius of the spherical foundation is a and the building is a circular cylinder with radiusa and height H. The bottom of the building, the cylindrical base, is resting on top of the rigid hemispherical foundation. Due to the shape of foundation, a spherical coordinate system is also adopted, as in Fig. 5.2. Excitation is given by the vertical P-waveincidence. 204 Figure5.2. Thesphericalcoordinatesystem. 5.2.1 Thegroundmotion Thenormal-planeP-waveincidentpotential' (i) withfrequency! isdescribedas: ' (i) = 1 k ↵ exp(ik ↵ z) = 1 k ↵ 1 X n=0 (2n+1)(i) n j n (k ↵ r)P n (u), (5.1) wherej n (k ↵ r)isthesphericalBesselfunctionofthe 1 st kindofordern;P n (u)isthe(0 th -degree) Legendre polynomial of degree n, withu=cos✓ , 0 u 1 for 0 ✓  ⇡/ 2 in the half-space; k is the wave number, such that k = k ↵ = !/C ↵ for P-waves and k = k = !/C for S-waves, whereC ↵ = p ( +2µ)/⇢ andC = p µ/⇢ arethewavespeedofP-waveandS-waveinthehalf- spacerespectively. 1/k ↵ isthescalefactorofP-wavepotential,sothatitwillhaveadisplacement 205 amplitudeof“1”atthehalf-spacesurface. Thepresenceofthehalf-spacesurfacez=0resultsin reflectedplaneP-wavewithpotential' (r) ,propagatingverticallydownwardsandoftheform ' (r) = 1 k ↵ exp(+ik ↵ z) = 1 k ↵ 1 X n=0 (2n+1)(+i) n j n (k ↵ r)P n (u). (5.2) TherearenoreflectedS-waves. Togethertheincidentandreflectedplanewavesformtheinput free-fieldP-wavepotential ' (ff) = ' (i) +' (r) = 1 k ↵ [exp(ik ↵ z)exp(+ik ↵ z)], (5.3) which,insphericalcoordinates,willsimplifyto ' (ff) = 1 X n=0 a 2n+1 j 2n+1 (k ↵ r)P 2n+1 (u) with a 2n+1 = 1 k ↵ (8n+6)i 2n+1 , so a 1 =6i/k ↵ . (5.4) Duetotheexistenceofthefoundation,therearealsoscatteredwavesinthehalf-spacemedium refracted from the foundation. The methodology adopted will follow Lee and Zhu (Lee and Zhu, 2013),whichisthatthescatteredwavesandrefractedwavesaretoberepresentedbyanexpansion ofsolelyodd-Legendrepolynomials. Thedetailedexpressionsare: thescatteredwavepotentialin thehalf-spacemediumare ' (s) = 1 X n=0 A 2n+1 h (1) 2n+1 (k ↵ r)P 2n+1 (u) (s) = 1 X n=0 C 2n+1 h (1) 2n+1 (k r)P 2n+1 (u) (5.5) 206 wherethesphericalHankelfunctionsareusedtorepresentoutgoingwaves. Therearenorefracted waves in the foundation because the foundation is assumed rigid. The relation between odd- and even-Legendre Polynomials in the half-range, 0  u=cos✓  1, is introduced (Lee and Liu, 2013)(LeeandZhu,2013)(Leeetal.,2004): P 2m+1 (u)= 1 X n=0 mn P 2n (u), (5.6) where mn aretheexpansioncoefficients. Thewavepotentialin(5.5)canalsobeexpandedintermsofeven-degreeLegendrepolynomials as ' (s) = 1 X n=0 1 X m=0 A 2m+1 h (1) 2m+1 (k ↵ r) mn ! P 2n (u) (s) = 1 X n=0 1 X m=0 C 2m+1 h (1) 2m+1 (k r) mn ! P 2n (u) (5.7) where A 2n+1 ,C 2n+1 are unknown coefficients to be solved by applying the appropriate boundary conditions. 5.2.2 Motionofthebuilding Similar to (Trifunac, 1972) and (Wong and Trifunac, 1974b), the displacementU z of the building will satisfy the following conditions, in the ¯ x¯ y¯ z coordinate system with ¯ o on top of the building: (a) Thedifferentialequation @ 2 U z @ ¯ z 2 = ⇢ b E b @ 2 U z @t 2 (5.8) whereE b isyoung’smodulusand⇢ b themassdensityofthebuilding,givingthebuildingwave speedofc b = p E b /⇢ b . 207 (b) Zerostressesontopofthebuildingat ¯ z=0,usingone-dimensional(1-D)Hooke’slaw: ¯ z¯ z =E b @U z @ ¯ z =0 (5.9) (c) Thedisplacementattheconnectionwithfoundationwhere ¯ z =H hastheform U z = e i!t (5.10) Fromconditiona,bandc U z (¯ z)= e i!t cos( ¯ k¯ z) cos( ¯ kH) (5.11) where ¯ k =!/c b =!/ p E b /⇢ b (5.12) isthewavenumberofthebuilding. Alsoforabuildingwithoneendfreeoneendfixed,itsnaturalfrequenciesisgivenby ¯ kH=(2n+1) ⇡ 2 ; n=0,1,2,... (5.13) 5.3 TheBoundaryConditions So this problem has been represented with two parts, the ground motion and the building motion. The link between these two parts is that they must have the same motion at the connecting area where is the rigid-foundation flat surface at z =0 and r  a. Other than the displacement continuity boundary conditions, there are also zero-stress boundary conditions for the waves in the half-space outside the foundation. Boundary conditions and the corresponding analysis are as follows: 208 1. Thezeronormal-stressboundaryconditiononthesurfaceofhalf-space ✓✓ | z=0 =0,r a. Since the free-field incident and reflected plane P-waves already satisfy the zero-stress boundary conditions the normal stress will only be computed for the scattered P- and S- wavewiththeexpressionofonlyodd-degreeseriesinu⌘ cos✓ whichhastheform ✓✓ | ✓ =⇡/ 2 = 2µ r 2 1 X n=1,3,5... h A n E (3) 21 (n,k ↵ r)+C n E (3) 23 (n,k r) i P n (0). (5.14) where E (i) jk are the stress-potential terms given in Appendix B.2. The superscript i is used to denote the type of spherical Bessel functions and/or Hankel functions used. Here (i = 1,2,3,4) are respectively for the functions j n ,y n ,h (1) n and h (2) n . The subscript j is used to denote the particular type of stress. Here the subscripts j =1,2,4 are for rr , ✓✓ and ⌧ r✓ . Stressesarecalculatedfromthewaves' or respectivelywhenk=1or3. Equation (5.14) is summed over all the odd integers,n=1,3,5..., whereP n (0) = 0 for all oddn-degreeLegendrepolynomialsandthusforallra, zz | z=0 = ✓✓ | ✓ =⇡/ 2 =0. 2. The zero-shear-stress boundary condition on the surface of half-space ⌧ ✓r | z=0 =0,r a. Here the shear stress will be calculated with the even-Legendre expansion as given in (5.7) whichtakestheform ⌧ ✓r | ✓ =⇡/ 2 = 2µ r 2 1 X n=0 " 1 X m=0 ⇣ A 2m+1 E (3) 41 (2m+1,k ↵ r)+C 2m+1 E (3) 43 (2m+1,k r) ⌘ mn # P 1 2n (0). (5.15) Equation (5.15) is summed over all the even terms 2n=0,2,4,..., where P 1 2n (0) = 0 for all 1 st order Legendre polynomials of even degrees in n, and thus for all r a, ⌧ zx | z=0 = ⌧ zy | z=0 = ⌧ ✓r | ✓ =⇡/ 2 =0issatisfied. 209 3. ThehorizontaldisplacementamplitudesoftherigidfoundationU x = U y =0,withthe vertical-displacement amplitudeU z = which keeps the continuity with the building motion Transforming the displacement components from Cartesian coordinate system to spherical coordinatesystem,therelationshipisgivenby 2 6 6 6 4 U r U ✓ U 3 7 7 7 5 = 2 6 6 6 4 sin✓ cos sin✓ sin cos✓ cos✓ cos cos✓ sin sin✓ sin cos 0 3 7 7 7 5 2 6 6 6 4 U x U y U z 3 7 7 7 5 (5.16) [U x ,U y ,U z ] 0 is [0,0,] 0 ,becausetherearenodrivingforcesinxandy direction,and U r =cos✓ ; U ✓ =sin✓ ; U =0 (5.17) withU ⌘ 0becausethisproblemisaxisymmetric. Forarigidfoundationthedisplacementontheboundaryr =aisbasedonthemotionofthe entirebody. Thoseare,fromtheodd-Legendre-onlyformasin(5.5) U r = 1 a 1 X n=1,3,5,..., h a n D (1) 11 (n,k ↵ a)+A n D (3) 11 (n,k ↵ a)+C n D (3) 13 (n,k a) i P n (u) U ✓ = 1 a 1 X n=1,3,5,..., h a n D (1) 21 (n,k ↵ a)+A n D (3) 21 (n,k ↵ a)+C n D (3) 23 (n,k a) i dP n (u) d✓ (5.18) whereD (i) jk arethedisplacement-potentialequationsgiveninAppendixB.1. Thesuperscript iisusedtodenotethetypeofsphericalBesselfunctionsand/orHankelfunctionsused. Here i=1,2,3,4 are respectively for the functions j n ,y n ,h (1) n and h (2) n . The subscript j is used 210 to denote the particular type of displacement functions. Here the subscripts j =1,2 are the spherical displacement components for U r and U ✓ . Displacements are due to ' or respectivelywhenk=1or3. Combing(5.17)and(5.18)gives 1 a 1 X n=1,3,5,..., h a n D (1) 11 (n,k ↵ a)+A n D (3) 11 (n,k ↵ a)+C n D (3) 13 (n,k a) i P n (u)=cos✓ 1 a 1 X n=1,3,5,..., h a n D (1) 21 (n,k ↵ a)+A n D (3) 21 (n,k ↵ a)+C n D (3) 23 (n,k a) i dP n (u) d✓ =sin✓ (5.19) whereu=cos✓ . NoticethatP 1 (cos✓ )=cos✓ and dP 1 (cos✓ )/d✓ =sin✓ ,thenapplying theorthogonalityofoddorevenLegendrepolynomialsto(5.19)gives: forn=1 a 1 D (1) 11 (1,k ↵ a)+A 1 D (3) 11 (1,k ↵ a)+C 1 D (3) 13 (1,k a)= a; a 1 D (1) 21 (1,k ↵ a)+A 1 D (3) 21 (1,k ↵ a)+C 1 D (3) 23 (1,k a)= a; (5.20) forn=3,5,7,..., a n D (1) 11 (n,k ↵ a)+A n D (3) 11 (n,k ↵ a)+C n D (3) 13 (n,k a)=0; a n D (1) 21 (n,k ↵ a)+A n D (3) 21 (n,k ↵ a)+C n D (3) 23 (n,k a)=0. (5.21) Therefore A n and C n (n=3,5,7,...,) can be solved in pairs with (5.21). The solution of A 1 ,C 1 and willbefoundinthenextboundarycondition. 4. TogeneratetheimpedancefunctionusingNewton’s 2 nd law. Newton’s 2 nd lawforthisproblemis F z +F b =! 2 M f = ! 2 M f . (5.22) 211 where M f = 2 3 ⇡ a 3 ⇢ f is the mass of the hemispherical foundation, ⇢ f its mass density, F z is the vertical load from the soil to the foundation andF b is the axial loading, which is also vertical,fromthebuildingtothefoundation. The derivation of F z is first presented. The normal stress rr on the spherical foundation surfacer = ahasthefollowingform,duetofree-fieldwaveandscatteredwavesrespectively: (ff) rr | r=a = 2µ a 2 1 X n=1,3,5,..., a n E (1) 11 (n,k ↵ a)P n (u) (s) rr | r=a = 2µ a 2 1 X n=1,3,5,..., ⇣ A n E (3) 11 (n,k ↵ a)+C n E (3) 13 (n,k a) ⌘ P n (u); (5.23) Similarly, the shear stresses ⌧ r✓ on the foundation surfacer = a due to free-field wave and scatteredwavesare ⌧ (ff) r✓ | r=a = 2µ a 2 1 X n=1,3,5,..., a n E (1) 41 (n,k ↵ a)P 1 n (u) ⌧ (s) r✓ | r=a = 2µ a 2 1 X n=1,3,5,..., ⇣ A n E (3) 41 (n,k ↵ a)+C n E (3) 43 (n,k a) ⌘ P 1 n (u) (5.24) Thetransformationmatrixwillapplytoconvertstresscomponentsfromsphericalcoordinate systemtoCartesiancoordinatesystematr = a: 2 6 6 6 4 ⌧ rx ⌧ ry ⌧ rz 3 7 7 7 5 | r=a = 2 6 6 6 4 sin✓ cos cos✓ cos sin sin✓ sin cos✓ sin cos cos✓ sin✓ 0 3 7 7 7 5 2 6 6 6 4 rr ⌧ r✓ ⌧ r 3 7 7 7 5 | r=a (5.25) 212 After the stress expression is obtained, the interacting forcesF x ,F y andF z from half-space medium to foundation are double-integrals of the Cartesian stress components with respect to✓ and : F x = Z ⇡ = ⇡ Z ⇡/ 2 ✓ =0 ⌧ rx | r=a a 2 sin✓ d✓ d (5.26a) F y = Z ⇡ = ⇡ Z ⇡/ 2 ✓ =0 ⌧ ry | r=a a 2 sin✓ d✓ d (5.26b) F z = Z ⇡ = ⇡ Z ⇡/ 2 ✓ =0 ⌧ rz | r=a a 2 sin✓ d✓ d (5.26c) Substituting⌧ rx ,⌧ ry and⌧ )rz withtherelationshipgivenin(5.25),andlearningthat⌧ r ⌘ 0 because of axisymmetry, the following expressions with spherical stress components are obtained: F x = Z ⇡ = ⇡ Z ⇡/ 2 ✓ =0 (sin✓ rr +cos✓⌧ r✓ ) | r=a a 2 sin✓ cos d✓ d ; (5.27a) F y = Z ⇡ = ⇡ Z ⇡/ 2 ✓ =0 (sin✓ rr +cos✓⌧ r✓ ) | r=a a 2 sin✓ sin d✓ d ; (5.27b) F z = Z ⇡ = ⇡ Z ⇡/ 2 ✓ =0 (cos✓ rr sin✓⌧ r✓ ) | r=a a 2 sin✓ d✓ d. (5.27c) Notethat Z ⇡ = ⇡ cos d = Z ⇡ = ⇡ sin d =0 so(5.27a)and(5.27b)implythatF x =F y =0andthereforebothhorizontalcomponentsof forcesarezero. 213 Substituting(5.23)and(5.24)into(5.27c) F z =4⇡µ 1 X n=1,3,5,..., h a n E (1) 11 (n)+A n E (3) 11 (n)+C n E (3) 13 (n) i Z ⇡/ 2 ✓ =0 P n (cos✓ )cos✓ sin✓ d✓ 4⇡µ 1 X n=1,3,5,..., h a n E (1) 41 (n)+A n E (3) 41 (n)+C n E (3) 43 (n) i Z ⇡/ 2 ✓ =0 dP n (cos✓ ) d✓ (cos✓ )sin 2 ✓ d✓ (5.28) witha n giveninequation(5.4). NotethatE (i) jk (n)hereistheshorthandnotationforE (i) j1 (n,k ↵ a)orE (i) j3 (n,k a). Theintegralsinequation(5.28)canbeevaluated(AbramowitzandStegun,1972): Z ⇡/ 2 ✓ =0 P n (cos✓ )cos✓ sin✓ d✓ = 8 < : 1 3 n=1 0 n=3,5,7,..., (5.29) and Z ⇡/ 2 ✓ =0 dP n (cos✓ ) d✓ sin✓ 2 d✓ =P n (0)2 Z ⇡/ 2 ✓ =0 P n (cos✓ )cos✓ sin✓ d✓ = 8 < : 2 3 n=1 0 n=3,5,7,..., (5.30) To integrate (5.28) of (5.29) and (5.30), the final expression of F z is only with coefficients A 1 andC 1 : F z = 4⇡µ 3 h a 1 ⇣ E (1) 11 (1)+2E (1) 41 (1) ⌘ +A 1 ⇣ E (3) 11 (1)+2E (3) 41 (1) ⌘ +C 1 ⇣ E (3) 13 (1)+2E (3) 43 (1) ⌘i (5.31) witha 1 =6i/k ↵ (equation(5.4)). 214 The total force at the base given by the action of the building onto the foundation involves thenormalstressandconnectingarea,whichis,from(5.11) F b = ⇡ a 2 E b @U z @ ¯ z ¯ z=H =⇡ a 2 E b ¯ ktan( ¯ kH). (5.32) Equation (5.22), (5.31) and (5.32) together give the third equation in addition to (5.20) to solveforA 1 ,C 1 and . 5.4 ResponseandSurfaceDisplacement 5.4.1 Interaction From(5.20),A 1 andC 1 canbeexpressedinthematrixformas 2 4 A 1 C 1 3 5 = 2 4 D (3) 11 (1,k ↵ a) D (3) 13 (1,k a) D (3) 21 (1,k ↵ a) D (3) 23 (1,k a) 3 5 1 2 4 a a 1 D (1) 11 (1,k ↵ a) a a 1 D (1) 21 (1,k ↵ a) 3 5 (5.33) orwitha 1 =6i/k ↵ (equation(5.4)) 2 4 A 1 C 1 3 5 = 2 4 ↵ 11 ↵ 12 ↵ 21 ↵ 22 3 5 2 4 a 6i/k ↵ 3 5 (5.34) 215 where ↵ 11 = D (3) 23 (1)D (3) 13 (1) D (3) 11 (1)D (3) 23 (1)D (3) 21 (1)D (3) 13 (1) ; ↵ 12 = D (1) 21 (1)D (3) 13 (1)D (1) 11 (1)D (3) 23 (1) D (3) 11 (1)D (3) 23 (1)D (3) 21 (1)D (3) 13 (1) ; ↵ 21 = D (3) 11 (1)D (3) 21 (1) D (3) 11 (1)D (3) 23 (1)D (3) 21 (1)D (3) 13 (1) ; ↵ 22 = D (1) 11 (1)D (3) 21 (1)D (1) 21 (1)D (3) 11 (1) D (3) 11 (1)D (3) 23 (1)D (3) 21 (1)D (3) 13 (1) , andD (i) jk aredefinedasbefore. With(5.34),(5.31)canbeexpressedas F z = 4⇡µ 3  6i k ↵ E 1 +(a) E 2 (5.35) where E 1 = ⇣ E (1) 11 (1)+2E (1) 41 (1) ⌘ + ⇣ E (3) 11 (1)+2E (3) 41 (1) ⌘ ↵ 12 + ⇣ E (3) 13 (1)+2E (3) 43 (1) ⌘ ↵ 22 ; E 2 = ⇣ E (3) 11 (1)+2E (3) 41 (1) ⌘ ↵ 11 + ⇣ E (3) 13 (1)+2E (3) 43 (1) ⌘ ↵ 21 . Fromthedefinitionof!,itgivesthat ! 2 =k 2 C 2 = µk 2 ⇢ = µk 2 (2/3⇡ a 3 ) ⇢ (2/3⇡ a 3 ) = (2/3)µ⇡ (k a) 2 a M s (5.36) whereM s isthemassofsoilthatisreplacedbythefoundation. Assuming the Poisson’s ratio⌫ =0.25, so thatµ b = G b = E b 2(1+⌫ ) according to the property of homogeneousisotropicmaterials,therefollows E b =2(1+⌫ ) ⇢ b ⇡ a 2 H ⇡ a 2 H ! 2 ¯ k 2 =2(1+⌫ ) M b ⇡ ( ¯ ka) 2 H (2/3)µ⇡ (k a) 2 a M s =2(1+⌫ ) ✓ M b M s ◆ (2/3)µ(k a) 2 a ( ¯ ka) 2 H (5.37) 216 whereM b ismassofthebuilding. Byinserting(5.37)into(5.32) F b =⇡ a 2  2(1+⌫ ) ✓ M b M s ◆ (2/3)µ(k a) 2 a ( ¯ ka) 2 H ¯ ktan( ¯ kH) = 4(1+⌫ ) 3 µ⇡ ✓ M b M s ◆ (k a) 2 tan( ¯ kH) ( ¯ kH) (a) . (5.38) andcombing(5.34),(5.35)and(5.38),therefollows 4⇡µ 3  6i k ↵ E 1 +(a) E 2 4(1+⌫ ) 3 µ⇡ ✓ M b M s ◆ (k a) 2 tan(( ¯ kH) ( ¯ kH) (a) =M f ! 2 =M f (2/3)µ⇡ (k a) 2 a M s =(2/3)µ⇡ (k a) 2 ✓ M f M s ◆ (a) (5.39) Then isgivenby: = 6iE 1 /(k ↵ a) (k a) 2 2 ⇣ M f Ms 2(1+⌫ ) M b Ms tan ¯ kH ¯ kH ⌘ +E 2 (5.40) Similarto(Trifunac,1972),somedimensionlessvariablesareintroduced. M f /M s isthemass ratioaswellasthedensityratiooffoundationmaterialandsoil;M b /M s isthemassratioofbuild- ingandthecorrespondingmassofsoilreplacedbythefoundation;and✏ = ¯ kH k a isadimensionless ratioalsousedinthe2-DSH-incidentcase. By the assumption that  = k /k ↵ = C ↵ /C is fixed to be p 3 all factors above are of k a. | | isplotted versusk a from0 to⇡ withM b /M s ,M f /M s and✏ fixed inFigs. 5.3to 5.5. Unlike (Trifunac,1972),itisdifficulttogiveananalyticexpressionofenvelopeof| |containingthepeak values,so“splineinterpolation”isusedtoplotthecurve. Eachfigurecontainsa✏=0case. From the definition of ✏, ✏=0 may refer to the condition that either the shear wall is rigid (k ¯ k) or the building mass is condensed at z=0 so that H ⇡ 0. Therefore the same conclusion holds as 217 in(WongandTrifunac,1974b). Atpointsgivenby(5.13),thenaturalfrequenciesofthebuilding, thereisnomotionofthefoundationexceptfor✏=0. Asthevalueof✏increases,thezerosof| | become more distributed and the peak values of | | always occurs in low frequency range. The peak values of | | are gradually decreasing with the increase ofM b /M s , as can be seen from the comparisonofFig.5.3andFig.5.5. Figure5.3. EffectofinteractionwithM b /M s =1,M f /M s =1,✏ = ¯ kH k a =0,2,4. 218 Figure5.4. EffectofinteractionwithM b /M s =1,M f /M s =2,✏ = ¯ kH k a =0,2,4. Figure5.5. EffectofinteractionwithM b /M s =4,M f /M s =1,✏ = ¯ kH k a =0,2,4. 219 5.4.2 RelativeResponse It is also important to assess the severity of relative oscillation on top of the building. Similar to (Trifunac,1972),therelativeresponseisdefinedas | R u z |= |U z | ¯ z=0 |= | | 1 cos ¯ kH 1 = 6i(1cos( ¯ kH))E 1 /(k ↵ a) (ka) 2 2 ⇣ M f Ms cos( ¯ kH)(2.5) M b Ms sin( ¯ kH) ¯ kH ⌘ +E 2 , (5.41) andtheinteraction,when| |=2,thescaledfree-fieldamplitude,is | R u z |=2 1 cos ¯ kH 1 . (5.42) In Figs. 5.6 and 5.7, the curves when | |=2 are given by (5.42) and they blow up to infinity at ¯ kH =(2n+1) ⇡ 2 , n=0,1,2,..., the natural frequencies of the building as given in (5.13). However | R u z | for M b /M s =1,2,4 are finite in (5.41) because is 0 for ¯ kH =(2n+1) ⇡ 2 . This shows that the soil-structure interaction effect is functioning as a “damper” of the building response. It can also be seen that M b /M s is playing a pronounced role in the relative response. The hump-shaped peaks of | R u z | of caseM b /M s =1 disappear with increase ofM b /M s and the peak values also drop in amplitudes. Scales of vertical axes in Fig. 5.6 and Fig. 5.7 are different, because the increase of ✏ has a strong effect on | R u z |, such that | R u z | is narrowly banded around naturalfrequenciesinthecaseof✏=4andthecorrespondingamplitudesareabout7timeslarger thaninthecaseof✏=2. 220 Figure5.6. Effectofinteractiononrelativeresponsewith M f /M s =1,✏=2,M b /M s =1,2,4and=2 . Figure5.7. Effectofinteractiononrelativeresponsewith M f /M s =1,✏=4,M b /M s =1,2,4and=2 . 221 Figure5.8. Shadingareaofcross-sectionalplane. 5.4.3 DisplacementsontheVerticalCross-sectionalPlane Displacement on a vertical cross-sectional plane can be used to indicate the underground defor- mation. The pair |U v | and |U h | is introduced due to symmetry of this model where |U v | is the amplitude of vertical displacement (same as |U z |) and |U h | represents amplitude of the horizontal displacementparalleltothehalf-spacesurface. Figs. 5.9,5.11and5.13indicateamplitudesofU v and U h in the shaded (striped) square zone in Fig.5.8 for 0  x/a  5 and 0  z/a  5 with M b /M s =1,M f /M s =1 and ✏=2, but the dimensionless incident-wave frequency ⌘ = k a/⇡ equals 0.5, 1.0 and 5.0 separately. Figs. 5.10, 5.12 and 5.14 show the same results but at specific depthswherez/a=0,0.5, 1,3and5. U v andU h ofthesoilmedium(r> a)canbecomputedby U r andU ✓ givenin(5.18)withthefollowingtransformation: U v =cos✓U r sin✓U ✓ ; U h =sin✓U r +cos✓U ✓ . (5.43) 222 Itisnotsurprisingtoseethatcurvesofz/a=0.0inFigs. 5.10(a),5.12(a)and5.14(a)areequal to zero which means |U h |⌘ 0 at the half-space surface because both the free-field and scattered waveshavenoverticalmotionsatthehalf-spacesurface. ThisisbecausetheLegendrepolynomial of odd order is zero at z =0, where u=cos✓ =0 when ✓ = ⇡/ 2 (Lee and Zhu, 2013). An increasing⌘ can result in higher amplitudes of eitherU h orU v , at the same time peaks of |U h | are usuallylocatedaroundthefoundationwhile|U v |haspeaksdistributedevenlythroughouttheentire vertical plane. Highest values of U h and U v always appear at z=0 as shown in Figs. 5.9, 5.11 and 5.13. |U v | will eventually converge to “2” whenx/a is large while |U h | will approach zero as inthefree-fieldstatewhenthepointisfarfromthestructure. Notethat|U v |withinthefoundationarea(r a)areequalto ,ascalculatedin(5.40). 223 (a) Verticaldisplacement|U h | (b) Horizontaldisplacement|U v | Figure5.9. DisplacementonaradialverticalplanearoundarigidfoundationwithM b /M s =1, M f /M s =1,✏=2,⌘ =0.5at 0 x/a 5, 0 z/a 5alongfixed . 224 (a) Verticaldisplacement|U h | (b) Horizontaldisplacement|U v | Figure5.10. DisplacementswithM b /M s =1,M f /M s =1,✏=2,⌘ =0.5at 0 x/a 5, z/a=0.5,1.0,3.0,5.0alongfixed . 225 (a) Verticaldisplacement|U h | (b) Horizontaldisplacement|U v | Figure5.11. DisplacementonaradialverticalplanearoundarigidfoundationwithM b /M s =1, M f /M s =1,✏=2,⌘ =1.0at 0 x/a 5, 0 z/a 5alongfixed . 226 (a) Verticaldisplacement|U h | (b) Horizontaldisplacement|U v | Figure5.12. DisplacementswithM b /M s =1,M f /M s =1,✏=2,⌘ =1at 0 x/a 5, z/a=0.5,1.0,3.0,5.0alongfixed . 227 (a) Verticaldisplacement|U h | (b) Horizontaldisplacement|U v | Figure5.13. DisplacementonaradialverticalplanearoundarigidfoundationwithM b /M s =1, M f /M s =1,✏=2,⌘ =5.0at 0 x/a 5, 0 z/a 5alongfixed . 228 (a) Verticaldisplacement|U h | (b) Horizontaldisplacement|U v | Figure5.14. DisplacementwithM b /M s =1,M f /M s =1,✏=2,⌘ =5at 0 x/a 5, z/a=0.5,1.0,3.0,5.0alongfixed . 229 5.5 Conclusions Inthispaper,thethree-dimensionalboundary-valueproblemofthesoil-structureinteraction(SSI) of a building on rigid hemispherical foundation subjected to vertical-plane P-wave incidence is discussed. Over forty years ago, in the early 1970s, the two-dimensional analytic solution of SSI of a shear wall on a semi-circular rigid foundation excited by incident out-of-plane SH waves with arbitrary angle was presented in (Trifunac, 1972). This was followed by (Wirgin and Bard, 1996),whichcomparedtheanalyticresultswithnumericalsimulationofthegroundmotionduring the 1986 Mexico City earthquake; and by (Gueguen et al., 2000) which compared their results of experimentaldatawithrespecttotheanalyticsolution. However, during the last forty years, very few analytic solutions have been obtained for such SSIproblem,neitherin2-Dnor3-D.Theonlyotherwell-knownanalyticsolutionforsoil-structure interactionistheSSIproblemofashearwallontwo-dimensionalsemi-ellipticrigidfoundationby (Wong and Trifunac, 1974b). Note further that these analytic solutions are only for out-of-plane SH-waves and are for 2-D foundations that are non-flexible in nature. For other types of incident waves, suchas P-, SV-, surfaceRayleighwaves orout-of-planeLovewaves, the solutionsaredif- ficult to obtain in close-form. This is because for in-plane P- and SV-wave, the presence of both longitudinal (P-) and shear (SV- and SH-) incident and scattered waves results in the difficulty of boundaryconditionsofwhichnormalstressesandshearstressesshouldbesatisfiedsimultaneously on the infinite half-space. Soil-structure interaction problems involving non-rigid (flexible) foun- dations are another challenging barrier in the development of close-form solution in earthquake engineering. This paper presents close-form analytic wave function solution on 3-D hemispherical rigid foundations excited by in-plane longitudinal P-waves. The difficulty in satisfying the 3-D stress- free boundary conditions is overcome analytically. Here, the technique of half-range expansion which has been proposed by problem of wave scattering and diffraction around hemispherical canyon by (Lee and Zhu, 2013) is adopted. This is an extension of the same technique used for 230 the solution for the corresponding 2-D scattering and diffraction around in-plane P-waves round a semi-circular canyon proposed by (Lee and Liu, 2013). The advantage of this new technique is that the new expansions automatically satisfy all stress-free boundary conditions at the infinite largehalf-spacesurfaceandcangeneratestrictlyorthogonaltermsatthefiniteboundaryconditions aroundthehemisphericalfoundationsurface. Sincetheboundaryconditionsonthehalf-spaceare alreadynaturallysatisfiedandtakencareof,problemresultingfromthemodeconversionbetween longitudinal and shear waves can be avoided. Similar to the canyon problem, results can be com- puted up to much higher dimensionless frequency (⌘ = k a/⇡ ). This is a great improvement relativetothepreviousresultsonthesamemodel(Lee,1978)andisenoughtocovertherangeof seismicwaveinvestigatedbycivilengineersandstrong-motionseismologists. The approach presented in this paper is effective on 3-D rigid foundation with normal-plane P-wave incidence, and this new method can and will next be extended to plane shear wave with arbitraryanglesofincidenceorsurfacewavessuchasLoveandRayleighwaves. Flexiblefounda- tioncanlaterbetakenintoconsideration. 231 AppendixA CoefficientsofSeriesExpansionofthe Free-fieldWaves A.1 RotationofCoordinates: theWignersmall-dFunction In section 1.3, the series expansions of a vertical P-incident wave potential and the correspond- ing reflected wave potential were used to complete the problem. However, the series expansion for a non-vertical wave potential is difficult to derive directly by the orthogonality of the Bessel- Legendrefunctions,particularlyfortheshearwavepotential. Therefore,themethodofcoordinate rotation from the vertical waves will be presented in this chapter, to obtain the coefficients of the incident and reflected wave potentials to satisfy the stress-free boundary conditions on the half- spacesurface. FigureA.1demonstratestherelationshipbetweentheCartesiancoordinatesandthe sphericalcoordinatesthathavebeenused. FigureA.1. Sphericalcoordinates. 232 TheleftpictureinFig. A.2showstheEulerangleoftherotation inthexz planefromthe systemxyz tothesystemx 0 y 0 z 0 . Inaddition,intherightpictureofFig. A.2thespherical coordinates (r,✓ 0 , 0 )iscomingwiththerotatedx 0 y 0 z 0 system. FigureA.2. Eulerangleofrotationbyy-axis. The Wigner small-d function will be used in the following but the function itself will not be introducedhere. Becausetheangleofrotation✓ ↵ or✓ istherotationinthexzplane,theWigner small-dfunctionthenonlyhas (inFig. A.2)asitsparameter. AccordingtoStein(1961),theWignersmall-dfunctionconnectstheoriginalsphericalcoordi- natesandtherotatedsphericalcoordinatesinthefollowingway: Y 1 n (✓ 0 , 0 )= n X m= n Y m n (✓, )d (n) m1 ( ), (A.1) wheretheY functionsaredefinedas Y m n (✓, )=(1) m  (n+1)!(nm)! 4⇡ (n+m)! 1 2 P m n (cos✓ )e im , 233 andthedetailedexpressionoftheWignersmall-dfunctionis d (n) m1 ( )=  (n+m)!(nm)! (n+1)!(n1)! 1 2 X 0 @ n+1 nm 1 A 0 @ n1 1 A (1) n m ✓ cos( 2 ) ◆ 2 +m+1 ✓ sin( 2 ) ◆ 2n 2 m 1 . Therefore,fortheLegendrepolynomialswithorder 1,thefollowingrelationshipthatissimplified from(A.1)holds: P 1 n (cos✓ 0 )e i 0 = n X m= n (1) m+1  (n+1)!(nm)! (n1)!(n+m)! 1/2 d (n) m1 ( )P m n (cos✓ )e im = n X m= n K nm ( )P m n (cos✓ )e im , (A.2) where K nm ( )=(1) m+1  (n+1)!(nm)! (n1)!(n+m)! 1/2 d (n) m1 ( ). Equation(A.2)willserveasthekeyrelationshipinthederivationoftheincidentandreflected coefficientsforalltheP-,SV-andSH-incidenceswitharbitraryanglesinthehalf-space. Notethat the vertical shear waves contain only m=1, so that the left hand side of (A.2) includes the 1 st orderofassociatedLegendrepolynomialswitharbitrarydegreen. 234 A.2 PlaneP-incidence Figure A.3 shows a incident plane P-wave in the half-space without any topographies. The angle of incidence is ✓ ↵ with respect to the horizontal xaxis, and the potential of this P-incidence is denotedby' (i) . Thetimefactore i!t isomittedbecauseofthesteadystateassumption. FigureA.3. Thefree-fieldwaveofP-incidence. TheincidentP-wavepotentialhasthefollowingexpressionintheCartesiancoordinates: ' (i) =exp(ik ↵ (xcos✓ ↵ zsin✓ ↵ )), (A.3) whoseexpressioninthesphericalcoordinateis ' (i) = 1 X m=0 1 X n m a (i) mn j n (k ↵ r)P m n (u)cosm, (A.4) where u=cos✓ and the coefficients a (i) mn of this scalar wave potential can be easily obtained in MorseandFeshbach(1953): a (i) mn = mn P m n (sin✓ ↵ )= ✏ m i n (2n+1)(nm)!/(n+m)!(1) m+n P m n (sin✓ ↵ ), (A.5) 235 with ✏ 0 =1,✏ m =2form> 0;m=0,1,2,....,m n. Thepresenceofthehalf-spaceboundaryconditions zz | z=0 = ⌧ yz | z=0 = ⌧ xz | z=0 =0resultsinthe reflected plane P-wave potential ' (r) and reflected plane shear potential (r) . By Snell’s law, the reflectedangleofP-waveisequaltotheincidentangle✓ ↵ ,andthereflectedangleofshearwaveis ✓ inFig. A.3. Thefollowingrelationshipholdsfor✓ ↵ and✓ : cos✓ ↵ C ↵ = cos✓ C , (A.6) whereC ↵ andC arethespeedsoftheP-andS-waves,respectively. IntheCartesiansystem ' (r) =K 1 exp(ik ↵ (xcos✓ ↵ +zsin✓ ↵ )), (A.7a) (r) =K 2 exp(ik (xcos✓ +zsin✓ )), (A.7b) where K 1 = sin2✓ ↵ sin2✓ (C ↵ /C ) 2 cos 2 2✓ sin2✓ ↵ sin2✓ +(C ↵ /C ) 2 cos 2 2✓ , K 2 = 2sin2✓ ↵ cos2✓ sin2✓ ↵ sin2✓ +(C ↵ /C ) 2 cos 2 2✓ , (A.8) are the coefficients of reflections determined by the half-space boundary conditions. k ↵ = !/C ↵ and k = !/C are the wave numbers of the longitudinal wave and the transverse wave respec- tively. Aswas statedin section 1.2, (r) canbe polarizedinto twopotential (r) and (r) inmutually orthogonaldirections. ThereflectedP-andS-wavesareexpandedinthesphericalcoordinatesas ' (r) = 1 X m=0 1 X n m a (r) mn j n (k ↵ r)P m n (u)cosm, (A.9a) 236 (r) = 1 X m=0 1 X n m k b (r) mn j n (k r)P m n (u)sinm, (A.9b) (r) = 1 X m=0 1 X n m c (r) mn j n (k r)P m n (u)cosm. (A.9c) Because ' (r) is a scalar wave, the a (r) mn in (A.9a) can be derived using (A.8) and according to (A.5)that a (r) mn =K 1 mn P m n (sin✓ ↵ )=K 1 ✏ m i n (2n+1)(nm)!/(n+m)!P m n (sin✓ ↵ ), (A.10) where ✏ 0 =1,✏ m =2form> 0;m=0,1,2,....,m n. However, the coefficients b (r) mn and c (r) mn in (A.9b) and (A.9c) for the vectorial shear waves cannot be obtained with the same method for a (r) mn , so they will be derived by the method of rotation introducedinSectionA. AccordingtoKnopoff(1959)andÁvila-CarreraandSánchez-Sesma(2006),theverticalshear wavepotentialcanbeorthogonalizedandexpandedbytheBessel-Legendrepolynomialsas = 1 X n=1 k b 1n j n (k r)P 1 n (u)sin, = 1 X n=1 c 1n j n (k r)P 1 n (u)cos, (A.11) whereonlythem=1seriesisincludedandb 1n andc 1n (n0)havethefollowingexpressions: b 1n =i n 1 2n+1 n(n+1) , c 1n =i n 1 2n+1 n(n+1) (i). (A.12) 237 The reflected shear wave can be understood as a vertical downward shear wave in a rotated coordinates with the Euler angle =(⇡/ 2✓ ) by the yaxis. This reflected shear potential isscaledbythereflectioncoefficientK 2 ,soitscoefficientsofexpansionintherotatedcoordinates are b (r) 0 n =K 2 i n 1 2n+1 n(n+1) , c (r) 0 n =K 2 i n 1 2n+1 n(n+1) (i). (A.13) Thewavepotentialisascalarsothatitcanbeexpressedinarotatedcoordinatesystem,according toAppendixA.1,as: (r) = (r) 0 = 1 X n=1 b (r) 0 n j n (k r)P 1 n (cos✓ 0 )sin 0 = 1 X n=0 b (r) 0 n j n (k r) n X m= n K nm ( )P m n (cos✓ )sinm = 1 X n=0 n X m=0 b (r) 0 n  K nm ( )+K n( m) ( )(1) m+1 (nm)! (n+m)! j n (k r)P m n (cos✓ )sinm. (A.14) Sothatthefollowingexpression: b (r) mn = 8 < : b (r) 0 n h K nm ( )+K n( m) ( )(1) m+1 (n m)! (n+m)! i m6=0,nm; b (r) 0 n K nm ( ) m=0,nm. (A.15) arethecoefficientsof (r) in(A.9b). Similarly,c (r) mn for (r) in(A.9c)canbeprovedtobe c (r) mn = 8 < : c (r) 0 n h K nm ( )+K n( m) ( )(1) m (n m)! (n+m)! i m6=0,nm; c (r) 0 n K nm ( ) m=0,nm. (A.16) 238 A.3 PlaneSV-incidence InFig. A.4,theplaneSV-incidencewiththeangleofincidence✓ andthefrequency! ispresented in the half-space, and the terme i!t continues to be omitted. On the boundary condition surface, thereexistthezero-stressconditions,andtheincidentandreflectedwavepotentialsareintheform of (i) = 1 X m=0 1 X n m k b (i) mn j n (k r)P m n (u)sinm, (A.17a) (i) = 1 X m=0 1 X n m c (i) mn j n (k r)P m n (u)cosm, (A.17b) ' (r) = 1 X m=0 1 X n m a (r) mn j n (k ↵ r)P m n (u)cosm, (A.17c) (r) = 1 X m=0 1 X n m k b (r) mn j n (k r)P m n (u)sinm, (A.17d) (r) = 1 X m=0 1 X n m c (r) mn j n (k r)P m n (u)cosm. (A.17e) FigureA.4. Thefree-fieldwaveofSV-incidence. 239 Thereflectedwavepotentialsarecanbedividedtwocasesdependingon✓ : • Theangleofincidenceisbeyondorequaltothecriticalangle✓ ✓ cr ;and • Theangleofincidenceislessthanthecriticalangle✓ <✓ cr , where,accordingtotheSnell’slawin(A.6),thecriticalangleisgivenby✓ cr =cos 1 (C /C ↵ ). The coefficients of the Bessel-Legendre series expansion of the incident potentials are firstly focusedon. ForaverticallyupwardnormalizedSV-wavepotential,itsexpressionintheCartesian coordinatesystemis (i) = 0 B B B @ (i) x (i) y (i) z 1 C C C A = 0 B B B @ 0 0 e ik z 0 1 C C C A . (A.18) From Knopoff (1959) and Ávila-Carrera and Sánchez-Sesma (2006), the vertical incident wave potentialsareexpressedas (i) 0 = 1 X n=1 b (i) 0 1n j n (k r)P 1 n (u 0 )sin 0 , (A.19a) (i) 0 = 1 X n=1 c (i) 0 1n j n (k r)P 1 n (u 0 )cos 0 , (A.19b) with b (i) 1n =(i) n 2n+1 n(n+1)i , c (i) 1n =(i) n+1 2n+1 n(n+1)i , (A.20) whereu 0 =cos✓ 0 intherotatedsystem. 240 Now take the incident SV-wave with ✓ as a vertical SV-wave travels along thez direction, thatthex 0 y 0 z 0 systemisrotatedfromthexyz systembyanEulerangle =⇡/ 2✓ . Because (i) and (i) 0 sharethesamevalue,then (i) = (i) 0 = 1 X n=1 b (i) 1n j n (k r)P 1 n (u 0 )sin 0 = 1 X n=0 n X m=0 b (i) 1n  K nm ( )+K n( m) ( )(1) m+1 (nm)! (n+m)! j n (k r)P m n (cos✓ )sinm. (A.21) Sothat, b (i) mn = 8 < : b (i) 1n h K nm ( )+K n( m) ( )(1) m+1 (n m)! (n+m)! i m6=0,nm; b (i) 1n K nm ( ) m=0,nm, (A.22) arethecoefficientsrequiredin(A.17a). Similarly,c (i) nm for (i) in(A.17b)aredefinedas c (i) mn = 8 < : c (i) 1n h K nm ( )+K n( m) ( )(1) m (n m)! (n+m)! i m6=0,nm; c (i) 1n K nm ( ) m=0,nm. (A.23) The second part is find out the coefficients for the reflected wave potentials. There are two cases. CaseI. isascenarioinwhichtheincidentangle✓ >✓ cr ,andthenresultsinareflectedP-wave potential with angle ✓ ↵ and the reflected shear waves with angle ✓ in (A.4). ✓ ↵ and ✓ areheldbytheSnell’slawin(A.6). 241 Duetothestress-freeconditions,thereflectioncoefficientsK 1 andK 2 aregivenby K 1 = (C ↵ /C ) 2 sin4✓ sin2✓ ↵ sin2✓ +(C ↵ /C ) 2 cos 2 2✓ , K 2 = sin2✓ ↵ sin2✓ (C ↵ /C ) 2 cos 2 2✓ sin2✓ ↵ sin2✓ +(C ↵ /C ) 2 cos 2 2✓ . (A.24) Theformulationofcoefficientsa (r) mn ,b (r) mn andc (r) mn in(A.17c)to(A.17e)areofsameform with (A.10), (A.15) and (A.16) in the previous section with the reflection coefficients substitutedwiththeonesgivenin(A.24). CaseII. is a scenario in which ✓  ✓ cr , and then results in the same reflected shear wave potentialsasinCaseI. However,theP-waveherebecomesasurfacewave. ThereflectioncoefficientsK 1 andK 2 aregivenby K 1 = (C ↵ /C ) 2 sin4✓ 2s c sin2✓ +(C ↵ /C ) 2 sin 2 ✓ ; K 2 =exp(2i⇠ ); (A.25) where s =(C ↵ /C )cos✓ ; c =(1s 2 ) 1/2 =i(s 2 1) 1/2 ; ⌫ = ((C ↵ /C ) 2 cos 2 ✓ 1) 1/2 C ↵ /C ; tan⇠ = 2⌫ sin2✓ cos✓ cos 2 2✓ ConsequentlythereflectedP-wavepotentialisgivenas ' (r) =K 1 expik ↵ (c z +s x) = 1 X m=0 1 X n m a (r) mn j n (k ↵ r)P m n (u)cosm, (A.26) 242 where,fromLee(1978),thecoefficientsofthisscalarwavepotentialtakesformof a (r) mn =K 1 ✏ m i n (2n+1) (nm)! (n+m)! P m n (c ) (A.27) where✏ m isdefinedsameas(A.5). Ashasbeenstated,thecoefficientsof (r) and (r) takethesameformof(A.17d)and(A.17e) withthecoefficientsK 1 andK 2 in(A.25). 243 A.4 PlaneSH-incidence The incidence is an SH-wave with angle ✓ as shown in Fig. A.5, and the reflected waves only consistshearwaveswiththereflectedangle✓ . FigureA.5. Thefree-fieldwaveofSH-incidence. Because the incidence is an out-of-plane wave, the positions of sinm and cosm are switched,andtheincidentandreflectedwavepotentialarethengivenas (i) = 1 X m=0 1 X n m k b (i) mn j n (k r)P m n (u)cosm, (A.28a) (i) = 1 X m=0 1 X n m c (i) mn j n (k r)P m n (u)sinm, (A.28b) (r) = 1 X m=0 1 X n m k b (r) mn j n (k r)P m n (u)cosm, (A.28c) (r) = 1 X m=0 1 X n m c (r) mn j n (k r)P m n (u)sinm. (A.28d) In order to apply the rotational transformation, the coefficients of a normal incidence of SH- wave should be firstly derived. The vertical SV-incidence and the vertical SH-incidence have similar properties except that the motions of particles of SV- and SH-waves are perpendicular to 244 each other. Therefore, a transformation connecting the SV- and SH-waves can be used to obtain thecoefficientsofaverticalSH-wavepotentialfromthatoftheverticalSV-wavein(A.20). Thex 0 y 0 z 0 inFig. A.6isthecoordinatesforaverticalSV-waveandthexyz system, whichis 90 rotatedinthex 0 y 0 plane,isthesystemfortheverticalSH-wave. FigureA.6. ThetransformationfromSV-wavetoSH-wave. Obviouslythefollowingrelationshipswillbegenerated: x =y 0 y =x 0 . followedby u 0 =cos✓ 0 =cos✓ =u; (A.29a) 0 = +90 . (A.29b) 245 Nowtotransform (i) and (i) in(A.19a)and(A.19b)intothecoordinatesofSH.Thereare: (i) = 1 X n 1 b (i) 1n j n (k r)P 1 n (u)sin( +⇡/ 2) = 1 X n 1 b (i) 1n j n (k r)P 1 n (u)cos, (A.30) (i) = 1 X n 1 c (i) 1n j n (k r)P 1 n (u)cos( +⇡/ 2) = 1 X n 1 (c (i) 1n )j n (k r)P 1 n (u)sin, (A.31) where b 1n and c 1n have been given in (A.20). So that the coefficients of the Bessel-Legendre expansionforaverticalSH-potentialisintheformthat (i) = 1 X n=1 b (i) 1n j n (k r)P 1 n (u)cos, (A.32a) (i) = 1 X n=1 c (i) 1n j n (k r)P 1 n (u)sin, (A.32b) withcoefficients b (i) 1n =(i) n 2n+1 n(n+1)i , c (i) 1n =(i) n+1 2n+1 n(n+1)i . (A.33) With the same method applied to (A.32a) as in (A.21), the coefficient of (i) with ✓ can be derivedas b (i) mn = 8 < : b (i) 1n h K nm ( )+K n( m) ( )(1) m (n m)! (n+m)! i m6=0,nm; b (i) 1n K nm ( ) m=0,nm, (A.34) where =⇡/ 2✓ . 246 Similarly,theexpressionofc (i) mn canbeobtainedas c (i) mn = 8 < : c (i) 1n h K nm ( )+K n( m) ( )(1) m+1 (n m)! (n+m)! i m6=0,nm; c (i) 1n K nm ( ) m=0,nm. (A.35) Uptohere,thederivationofcoefficientsoftheincidentpotentialsof(A.28a)and(A.28b)iscom- pleted. In a similar way, the coefficients b (r) mn and c (r) mn that are of the reflected wave potentials in (A.28c)and(A.28d)aretaketheformof b (r) mn = 8 < : b (r) 1n h K nm ( )+K n( m) ( )(1) m (n m)! (n+m)! i m6=0,nm; b (r) 1n K nm ( ) m=0,nm, (A.36) and c (r) mn = 8 < : c (r) 1n h K nm ( )+K n( m) ( )(1) m+1 (n m)! (n+m)! i m6=0,nm; c (r) 1n K nm ( ) m=0,nm, (A.37) where b (r) 1n =(i) n 1 2n+1 n(n+1) , c (r) 1n =(i) n 2n+1 n(n+1) , (A.38) and =(⇡/ 2✓ ). 247 AppendixB DisplacementVectorandStressTensor (MowandPao,1971) In a spherical coordinate system, (U r ,U ✓ ,U ) are the displacement components, and ( rr , ✓✓ , ,⌧ r✓ ,⌧ r ,⌧ ✓ ) consist the stress tensor. Given (', , ) as the arbitrary P-, SV-, and SH-wavepotentials,respectively,theirexpressionsareinthefollowingform(withe i!t omitted): ' =z n (k ↵ r)P m n (u) cosm sinm , =z n (k r)P m n (u) sinm cosm , =z n (k r)P m n (u) cosm sinm , (B.1) where z n = z (i) n is the spherical Bessel function of the 1 st kind (j n ) or 2 nd kind (y n ) or Hankel functionof1 st kind(h (1) n )or2 nd kind(h (2) n )wheni=1,2,3,4,respectively,P m n (u)istheLegendre polynomial of degree n and order m, and u =cos✓ . The first rows are used in the in-plane incidence—the P- or SV-waves, while the second rows are for the case of the out-of-plane SH- incidence. 248 B.1 FunctionsofDisplacementfromWavePotential Thedetailedexpressionsofthepotential-displacementrelationshipsaregiveninthefollowing: U r = 1 r h D (i) 11 +lD (i) 13 i P m n (u) cosm sinm , U ✓ = 1 r " ⇣ D (i) 21 +lD (i) 23 ⌘ dP m n (u) d✓ ± mrD (i) 22 sin✓ P m n (u) # cosm sinm . U = 1 r  ⌥ m sin✓ ⇣ D (i) 21 +lD (i) 23 ⌘ P m n (u)rD (i) 22 dP m n (u) d✓ sinm cosm , (B.2) where D (i) 11 =nz (i) n (k ↵ r)k ↵ rz (i) n+1 (k ↵ r), D (i) 13 =n(n+1)z (i) n (k r), D (i) 21 =z (i) n (k ↵ r), D (i) 22 =z (i) n (k r), D (i) 23 =(n+1)z (i) n (k r)k rz (i) n+1 (k r). (B.3) 249 B.2 FunctionsofStressfromWavePotential FromLee(1979),thepotential-stressrelationshipsare: rr dueto: ' : ✓ 2µ r 2 ◆ E (i) 11 cosm sinm where E (i) 11 =E (i) 11 P m n (u), : none where E (i) 12 =0, : ✓ 2µl r 2 ◆ E (i) 13 cosm sinm where E (i) 13 =E (i) 13 P m n (u), E (i) 11 = ✓ n 2 n k 2 r 2 2 ◆ z n (k ↵ r)+2k ↵ rz n+1 (k ↵ r), E (i) 12 =0, E (i) 13 =n(n+1)[(n1)z n (k r)k rz n+1 (k r)]; (B.4) ✓✓ dueto: ' : ✓ 2µ r 2 ◆ E (i) 21 cosm sinm where E (i) 21 =E (i) 21 P m n (u)+ ˆ E (i) 21 ˆ P m n (u), : ✓ 2µ r ◆ E (i) 22 cosm sinm , : ✓ 2µl r 2 ◆ E (i) 23 cosm sinm where E (i) 23 =E (i) 23 P m n (u)+ ˆ E (i) 23 ˆ P m n (u), E (i) 21 = ✓ n 2 k 2 r 2 2 +k 2 ↵ r 2 ◆ z n (k ↵ r)k ↵ rz n+1 (k ↵ r), ˆ E (i) 21 =z n (k ↵ r), E (i) 22 = ±m 1u 2 z n (k r) ⇥ (n1)uP m n (u)(n+m)P m n 1 (u) ⇤ , (B.5) 250 E (i) 23 =(n 2 +n)[nz n (k r)k rz n+1 (k r)], ˆ E (i) 23 =(n+1)z n (k r)k rz n+1 (k r), ˆ P m n (u)= 1 1u 2 ⇥ (m 2 u 2 n)P m n (u)+(m+n)uP m n 1 (u) ⇤ ; dueto: ' : ✓ 2µ r 2 ◆ E (i) 31 cosm sinm where E (i) 31 =E (i) 31 P m n (u) ˆ E (i) 31 ˆ P m n (u), : ✓ 2µ r ◆ E (i) 32 cosm sinm , : ✓ 2µl r 2 ◆ E (i) 33 cosm sinm where E (i) 33 =E (i) 33 P m n (u) ˆ E (i) 33 ˆ P m n (u), E (i) 31 = ✓ n k 2 r 2 2 +k 2 ↵ r 2 ◆ z n (k ↵ r)k ↵ rz n+1 (k ↵ r), ˆ E (i) 31 =z n (k r), E (i) 32 = ±m 1u 2 z n (k r) ⇥ (n1)uP m n (u)+(n+m)P m n 1 (u) ⇤ , E (i) 33 =n(n+1)z n (k r), ˆ E (i) 33 =(n+1)z n (k r)k rz n+1 (k r), ˆ P m n (u)= 1 1u 2 ⇥ (m 2 u 2 n)P m n (u)+(m+n)uP m n 1 (u) ⇤ ; (B.6) ⌧ r✓ dueto: ' : ✓ 2µ r 2 ◆ E (i) 41 cosm sinm where E (i) 41 =E (i) 41 dP m n (u) d✓ , : ✓ 2µ r ◆ E (i) 42 cosm sinm where E (i) 42 = ±m sin✓ E (i) 42 P m n (u), 251 : ✓ 2µl r 2 ◆ E (i) 43 cosm sinm where E (i) 43 =E (i) 43 dP m n (u) d✓ , E (i) 41 =(n1)z n (k ↵ r)k ↵ rz n+1 (k ↵ r), E (i) 42 = 1 2 [(n1)z n (k r)k rz n+1 (k r)], E (i) 43 = n 2 1k 2 r 2 /2 z n (k r)+k rz n+1 (k r); (B.7) ⌧ r dueto: ' : ✓ 2µ r 2 ◆ E (i) 51 sinm cosm where E (i) 51 = ⌥ m sin✓ E (i) 51 P m n (u), : ✓ 2µ r ◆ E (i) 52 sinm cosm where E (i) 52 =E (i) 52 dP m n (u) d✓ , : ✓ 2µl r 2 ◆ E (i) 53 sinm cosm where E (i) 53 = ⌥ m sin✓ E (i) 53 P m n (u), E (i) 51 =E (i) 41 , E (i) 52 =E (i) 42 , E (i) 53 =E (i) 43 ; (B.8) ⌧ ✓ dueto: ' : ✓ 2µ r 2 ◆ E (i) 61 sinm cosm where E (i) 61 = ⌥ m sin✓ E (i) 61 ✓ dP m n (u) d✓ cot✓P m n (u) ◆ , : ✓ 2µ r ◆ E (i) 62 sinm cosm where E (i) 62 = E (i) 62 sin 2 ✓ ✓ n 2 n 2 sin 2 ✓ +nm 2 ◆ P m n (u)(n+m)cos✓P m n 1 (u) , 252 : ✓ 2µl r 2 ◆ E (i) 53 sinm cosm where E (i) 63 = ⌥ m sin✓ E (i) 63 ✓ dP m n (u) d✓ cot✓P m n (u) ◆ , E (i) 61 =z n (k ↵ r), E (i) 62 =z n (k r), E (i) 63 =(n+1)z n (k r)k rz n+1 (k r). (B.9) 253 AppendixC Three-dimensionalGraphsofDisplacement AmplitudesaroundtheCanyon C.1 The Diffraction around a Hemispherical Canyon for the PlaneP-incidence FigureC.1. |U x |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. 254 FigureC.2. |U y |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. FigureC.3. |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. 255 FigureC.4. |U x |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. FigureC.5. |U y |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. 256 FigureC.6. |U z |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. FigureC.7. |U x |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. 257 FigureC.8. |U y |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. FigureC.9. |U z |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthecanyon. 258 C.2 The Diffraction around a Hemispherical Canyon for the PlaneSV-incidence FigureC.10. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. 259 FigureC.11. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. FigureC.12. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. 260 FigureC.13. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. FigureC.14. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. 261 FigureC.15. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. FigureC.16. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. 262 FigureC.17. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. FigureC.18. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthecanyon. 263 C.3 The Diffraction around a Hemispherical Canyon for the PlaneSH-incidence FigureC.19. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. 264 FigureC.20. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. FigureC.21. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. 265 FigureC.22. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. FigureC.23. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. 266 FigureC.24. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. FigureC.25. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. 267 FigureC.26. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. FigureC.27. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthecanyon. 268 AppendixD Three-dimensionalGraphsofDisplacement AmplitudesaroundtheAlluvialValley D.1 The Diffraction around a Hemispherical Alluvial Valley forthePlaneP-incidence D.1.1 CaseI:µ f /µ=0.25,C ↵f /C ↵ =0.50 FigureD.1. |U x |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 269 FigureD.2. |U y |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 FigureD.3. |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 270 FigureD.4. |U x |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 FigureD.5. |U y |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 271 FigureD.6. |U z |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 FigureD.7. |U x |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 272 FigureD.8. |U y |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 FigureD.9. |U z |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.25,C ↵f /C ↵ =0.50 273 D.1.2 CaseII:µ f /µ=0.30,C ↵f /C ↵ =0.60 FigureD.10. |U x |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 274 FigureD.11. |U y |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 FigureD.12. |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 275 FigureD.13. |U x |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 FigureD.14. |U y |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 276 FigureD.15. |U z |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 FigureD.16. |U x |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 277 FigureD.17. |U y |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 FigureD.18. |U z |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.30,C ↵f /C ↵ =0.60 278 D.1.3 CaseIII:µ f /µ=0.40,C ↵f /C ↵ =0.60 FigureD.19. |U x |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 279 FigureD.20. |U y |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 FigureD.21. |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 280 FigureD.22. |U x |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 FigureD.23. |U y |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 281 FigureD.24. |U z |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 FigureD.25. |U x |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 282 FigureD.26. |U y |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 FigureD.27. |U z |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=0.40,C ↵f /C ↵ =0.60 283 D.1.4 CaseIV:µ f /µ=3.00,C ↵f /C ↵ =1.50 FigureD.28. |U x |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 284 FigureD.29. |U y |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 FigureD.30. |U z |with⌘ =1,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 285 FigureD.31. |U x |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 FigureD.32. |U y |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 286 FigureD.33. |U z |with⌘ =3,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 FigureD.34. |U x |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 287 FigureD.35. |U y |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 FigureD.36. |U z |with⌘ =5,✓ ↵ =90 ,60 ,30 ,and 15 forP-incidencearoundthevalley: µ f /µ=3.00,C ↵f /C ↵ =1.50 288 D.2 The Diffraction around a Hemispherical Alluvial Valley forthePlaneSV-incidence D.2.1 CaseI:µ f /µ=0.25,C f /C =0.50 FigureD.37. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 289 FigureD.38. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.39. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 290 FigureD.40. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.41. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 291 FigureD.42. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.43. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 292 FigureD.44. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.45. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 293 D.2.2 CaseII:µ f /µ=0.30,C f /C =0.60 FigureD.46. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 294 FigureD.47. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.48. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 295 FigureD.49. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.50. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 296 FigureD.51. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.52. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 297 FigureD.53. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.54. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 298 D.2.3 CaseIII:µ f /µ=0.40,C f /C =0.60 FigureD.55. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 299 FigureD.56. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.57. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 300 FigureD.58. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.59. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 301 FigureD.60. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.61. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 302 FigureD.62. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.63. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 303 D.2.4 CaseIV:µ f /µ=3.00,C f /C =1.50 FigureD.64. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 304 FigureD.65. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.66. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 305 FigureD.67. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.68. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 306 FigureD.69. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.70. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 307 FigureD.71. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.72. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSV-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 308 D.3 The Diffraction around a Hemispherical Alluvial Valley forthePlaneSH-incidence D.3.1 CaseI:µ f /µ=0.25,C f /C =0.50 FigureD.73. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 309 FigureD.74. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.75. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 310 FigureD.76. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.77. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 311 FigureD.78. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.79. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 312 FigureD.80. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 FigureD.81. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.25,C f /C =0.50 313 D.3.2 CaseII:µ f /µ=0.30,C f /C =0.60 FigureD.82. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 314 FigureD.83. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.84. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 315 FigureD.85. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.86. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 316 FigureD.87. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.88. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 317 FigureD.89. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 FigureD.90. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.30,C f /C =0.60 318 D.3.3 CaseIII:µ f /µ=0.40,C f /C =0.60 FigureD.91. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 319 FigureD.92. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.93. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 320 FigureD.94. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.95. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 321 FigureD.96. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.97. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 322 FigureD.98. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 FigureD.99. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=0.40,C f /C =0.60 323 D.3.4 CaseIV:µ f /µ=3.00,C f /C =1.50 FigureD.100. |U x |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 324 FigureD.101. |U y |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.102. |U z |with⌘ =1,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 325 FigureD.103. |U x |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.104. |U y |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 326 FigureD.105. |U z |with⌘ =3,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.106. |U x |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 327 FigureD.107. |U y |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 FigureD.108. |U z |with⌘ =5,✓ =90 ,60 ,30 ,and 15 forSH-incidencearoundthevalley: µ f /µ=3.00,C f /C =1.50 328 Bibliography Abramowitz, M. and Stegun, I. 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Abstract (if available)

Abstract The three-dimensional (3-D) diffraction and scattering by a hemispherical canyon or valley subjected to seismic plane and spherical waves in an elastic half-space, since the development of the theory of elastic wave propagation, have long been a challenging boundary-valued problem in the field of applied mathematics and applied mechanics. In particular, strong-motion seismologists and earthquake engineers used this problem to study and understand the amplification and de-amplification effects induced by surface topography. With in-plane wave excitation, the scattered waves will in all cases, in both two dimensional (2-D) and three-dimensional (3-D) problems, consist of both longitudinal (P-) and shear (S-) waves. In 3-D problems, the shear (S-) waves will be both in-plane (SV-) and out-of-plane (SH-) waves. Together at the half-space flat surface, they must satisfy both the zero-normal and zero-shear stress boundary conditions. The P- and S-waves together, however, are not orthogonal nor independent at the half-space surface, and yet they have to simultaneously satisfy the half-space boundary conditions. This is what makes this boundary-valued problem a very challenging problem to solve. In the last fifty years, attempts to solve this problem after involved numerical approximation to the geometry and/or the wave functions have been made. In this dissertation the proper forms of the orthogonal spherical wave functions for both the P- and S-waves are re-defined, so that they can simultaneously satisfy the zero-stress boundary conditions at the half-space surface. This dissertation presents the general cases of plane P- and S-wave incidences with arbitrary angles from the bottom of the half-space. With the 3-D zero-stress boundary conditions defined and satisfied at the half-space surface, the challenging part solved, the remaining boundary conditions at the hemispherical canyon or valley surface can next be applied and satisfied.

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Asset Metadata

Creator Zhu, Guanying (author)

Core Title Three-dimensional diffraction and scattering of elastic waves around hemispherical surface topographies

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 07/26/2016

Defense Date 08/09/2016

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag arbitrary angle,OAI-PMH Harvest,plane P-wave,plane S-waves,stress elastic half-space,three-dimensional

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lee, Vincent W. (committee chair), Lee, Jiin-Jen (committee member), Trifunac, Mihailo D. (committee member), Udwadis, Firdaus E. (committee member), Wellford, L. Carter (committee member)

Creator Email guanying@usc.edu,judyz0503@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-279833

Unique identifier UC11280269

Identifier etd-ZhuGuanyin-4618.pdf (filename),usctheses-c40-279833 (legacy record id)

Legacy Identifier etd-ZhuGuanyin-4618.pdf

Dmrecord 279833

Document Type Dissertation

Format application/pdf (imt)

Rights Zhu, Guanying

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA