二維垂直橫向等向性彈性半平面頻率域Lamb問題之解題方法研究

Abstract

本研究的目的是為了延伸其原先提出解决各向同性介質中的散射問题，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.