本研究的目的是為了延伸其原先提出解&;#20915;各向同性介質中的散射問&;#39064;，解決表面散射波在頻率域之問題是利用不同的平面波撞擊入射，其可以分解成自由場以及散射場，而自由場的入射包含入射以及反射波，而散射場則是由未知強度的水平及垂直波源的廣義Lamb問題和邊界條件所組成。橫向等向性材料可以根據三岔角的存在與否以及三岔角相對於坐標軸的位置來做分類，其可以分成五類。垂直或是水平線性載重之Lamb問題可以利用傅立葉轉換，將其轉換成水平波速以及垂直波速。這兩種波場分別存在於四個黎曼面的其中兩個上，其中分支點和相連的分支切割必須小心的選取，以確保讓多值函數變成單值。為了從Lamb積分式裡找出停駐相，將位在各個黎曼面上的原始積分的路徑，扭曲成最速徒降路徑，讓積分的收斂大幅加快。

The objective of this research is to extend the approach which is originally proposed for solving scattering problem of isotropic medium to solving the same problem for vertically transversely isotropic medium. The strategy of solving the surface scattering problem in frequency domain impinging by the incidence of plane wave of different kind is to decompose the total wave fields into the known free field which consisted of incident and reflective waves as well as the scattering field which consisted of generalized Lamb’s solution with unknown amplitude determined by boundary condition of scatter itself. Both of the determination of reflected coefficients of a specific plane waves at free surface and the representation of Lamb’s solution can be work out through the concept of horizontal and vertical slowness surfaces pertinent to the material under consideration. It can be shown that the transversely isotropic materials under consideration can be classified into five distinct categories according to the existence or nonexistence of cuspidal edges and the orientation of cusps relative to the coordinate axes. The solutions of the Lamb problem for vertical as well as horizontal surface line forces can be represented by two kind of Fourier synthesis of plane wave, namely, quasi-longitudinal and quasi-transverse wave which expressed in terms of horizontal wave-number. These two kinds of wave field are defined on two sheets of the four Riemann surface in which the branch points and the associated branch cuts are carefully chosen to ensure the single value of multi-value radical function. In order to extract the stationary phase from the Lamb’s integral, the original integration path in each Riemann surface is then distorted to the so called steepest descent path from which the integration converges rapidly.

中文摘要 i
Abstract ii
目錄 iii
圖目錄 vi
表目錄 ix
第1章 導論 1
1.1 研究動機 1
1.2 文獻回顧 2
1.3 研究方法與論文架構 4
第2章 基本原理與材料性質 6
2.1 橫向等向性材料之性質與正定性 6
2.2 基本理論推導 10
2.3 波數 13
2.4 相位角和射線角、相速度和群速度 17
2.5 質點振動方向 19
第2章附圖 22
第3章 二維半空間之散射場(Lamb問題) 26
3.1 純量勢能函數解 26
3.2 直接傅立葉轉換解 30
3.3 位移場積分之無因次化 37
第4章 分支切割及材料之分類 41
4.1 共同分支點 41
4.2 材料分類之討論 48
4.2.1 Case1 N2>0 54
4.2.2 Case2 N2<0 58
4.3 雷利極點之討論 62
4.4 原始積分路徑 65
4.4.1 Zinc之原始積分路徑 66
4.4.2 Beryl及Magnesium之原始積分路徑 67
第4章附圖 68
第5章 自由場之組成及反射係數之計算 75
5.1 q-P波入射之反射係數與自由場 75
5.1.1 Zinc材料 75
5.1.2 Beryl &; Magnesium材料 79
5.2 q-SV波入射之反射係數與自由場 80
5.2.1 Zinc材料 80
5.2.2 Beryl &; Magnesium材料 83
第5章附圖 85
第6章 積分路徑之變換與傅立葉積分之求值 88
6.1 鞍點求法 88
6.2 Steepest descent path路徑 90
6.3 三種材料γ=0之最速陡降路徑 95
第6章附圖 98
第7章 結論與未來展望 106
7.1 結論 106
7.2 未來展望 107
參考文獻 108
附錄A 純量勢能函數F之推導 112
附錄B 判別式N5之表示式 119

[1]Lamb, H. “On the propagation tremors over the surface the surface of an elastic solid.” Philos. Trans. R. Soc. London, Ser. A, A203, 1–42. 1904
[2]Achenbach, J. D. “Wave propagation in elastic solids”, North- Holland, Amsterdam, The Netherlands. (1973)
[3]Aki, K., and Richards, P. G. “Quantitative seismology theory and methods”, W. H. Freeman and Co., New York. (1980)
[4]Apsel, R. J., and Luco, J. E. “On the Green’s functions for a layered half space part II.” Bull. Seismol. Soc. Am., 734, 931–951. (1983)
[5]Pak, R. Y. S. “Asymmetric wave propagation in an elastic half-space by a method of potentials.” J. Appl. Mech., 54(1), 121–126. (1987)
[6]Pak, R. Y. S., and Guzina, B. B. “Three-dimensional Green’s functions for a multilayered half-space in displacement potentials.” J. Eng. Mech., 128(4), 449–461. (2002)
[7]V.A. Sveklo, “Plane waves and Rayleigh waves in anisotropic media”, Doklady Akademii Nauk SSSR 59 (1948) 871–874
[8]R. Stoneley, “The propagation of surface elastic waves in a cubic crystal,Proceedings of the Royal Society of London”, United Kingdom A232 (1955)447–458.
[9]L. Gold, “Rayleigh wave propagation on anisotropic (cubic) media”, PhysicalReview 104 (6) (1956) 1532–1536.
[10]H. Deresiewicz, R.D. Mindlin, “Waves on the surface of a crystal”, Journal of Applied Physics 28 (6) (1957) 669–671.
[11]J.L. Synge, “Elastic waves in anisotropic media”, Journal of Mathematical Physics 35 (1957) 323–335.
[12]D.C. Gazis, R. Herman, R.F. Wallis, “Surface elastic waves in cubic crystals”, Physical Review 119 (2) (1960) 533–544.
[13]V.T. Buchwald, “Rayleigh waves in anisotropic media”, Quarterly Journal of Mechanics and Applied Mathematics 14 (4) (1961) 461–468.
[14]V.T. Buchwald, A. Davis, “Surface waves in elastic media with cubic symmetry”, Quarterly Journal of Mechanics and Applied Mathematics 16 (1963) 283–293.
[15]L. Cagniard, “Reflexion et Refraction des Ondes Sismiques Progressives”, Gauthiers-Villars, Paris, 1939.
[16]A.T. de Hoop, “A modification of Cagniard’s method for solving seismic pulse problems”, Applied Science Research B8 (1960) 349–356.
[17]J.H.M.T. van der Hijden, “Propagation of Transient Elastic Waves in Stratified Anisotropic Media”, North-Holland, Amsterdam, The Netherlands, 1987.
[18]E.A. Kraut, “Advances in the theory of anisotropic elastic wave propagation”, Reviews of Geophysics 1 (3) (1963) 401–448.
[19]R. Burridge, “Lamb’s problem for an anisotropic half-space”, Quarterly Journal of Mechanics and Applied Mathematics 24 (1) (1971) 81–98.
[20]R.L. Ryan, “Pulse propagation in a transversely isotropic half-space”, Journal of Sound and Vibration 14 (4) (1971) 511–524.
[21]R.G. Payton, “Elastic Wave Propagation in Transversely Isotropic Media”, Martinus Nijhoff, The Hague, The Netherlands, 1983
[22]D.M. Barnett, J. Lothe, K. Nishioka, R.J. Asaro, “Elastic surface waves in anisotropic crystals: a simplified method for calculating Rayleigh velocities using dislocation theory”, Journal of Physics F: Metal Physics 3 (1973) 1083–1096.
[23]D.M. Barnett, J. Lothe, “Consideration of the existence of surface wave (Rayleigh wave) solutions in anisotropic elastic crystals”, Journal of Physics F: Metal Physics 4 (1974) 671–686.
[24]J. Lothe, D.M. Barnett, “On the existence of surface-wave solutions for anisotropic elastic half-spaces with free surface”, Journal of Applied Physics 47 (1976) 428–433.
[25]Wu, K.-C. “Extension of Stroh’s formalism to self-similar problems in two-dimensional elastodynamics.” Proceedings of the Royal Society of London (2000). A456, 869–890.
[26]Sveklo, V. A., Elastic vibrations of anisotropic bodies. Uchenye zapiski Leningr. univ. ser. matem. n. (Scientific Memoirs of Leningrad Univ. , Mathematical Sci- ences Series). No17, 1949.
[27]Rank, P. H. and von Mises. R., Differential and Integral Equations of Mathematical Physics, Part 2, Moscow-Leningrad, ONTI, 1937.
[28]Lighthill, M. J., 1960, “Study of magneto-hydrodynamic waves and other anisotropic wave motions”, Philosophical Transactions of the Royal Society of London,A, 252, 397–430.
[29]廖文義, “含缺陷半無限域程受彈性波之反應”, 國立台灣大學土木工程研究所博士論文 (1997)
[30]ZHAO, Ai-Hua, ZHANG, Mei-Gen and DING, Zhi-Feng, “Seismic travel time computation for transversely isotropic media”, CHINESE JOURNAL OF GEOPHYSICS Vol.49, No.6, 2006, pp: 1603-1612
[31]C. G. Caracostis and A. R. Robinson and, “Wave propagation problem in certain elastic anisotropic half-spaces”, University of Illinois Engineering Experiment Station. College of Engineering. University of Illinois at Urbana-Champaign Technical Report :Civil Engineering Studies SRS-487,1980
[32]K.A. Anagnostopoulos, A. Charalambopoulos, C.V. Massalas. “On the investigation of elasticity equation for orthotropic materials and the solution of the associated scattering problem.”, Int. J. of Solids and Structures, 42 (2005), pp.6376–6408.&;#8195;