Analyses of scattreing waves due to a scattering object embedded deep site of the layered medium by means of the spectral representaion of Green's function

本文研究了层状介质中格林函数的谱表示及其在散射问题中的应用。对于层状标量波场，得到谱表示的过程相当简单。另一方面，对于三维层状介质，由于无法建立瑞利波模态的正交关系，推导格林函数的步骤不是很简单。然而，发现超函数的概念适用于导出格林函数的谱表示。由于散射波是由层状介质中散射物体与平面波相互作用产生的，因此将边界积分方程法和格林函数的谱表示引入散射问题。格林函数的谱表示使我们能够在边界积分方程方法中引入特征值问题的观点。通过交换边界积分和谱积分或求和的阶数，将散射波分解为点谱和连续谱的本征函数。因此，利用层状介质的本征函数可以理解散射波。

This research deals with the spectral representation of Green's function for a layered medium and its application to the scattering problem. Regarding the layered scalar wave field, the procedure for obtaining the spectral representation is rather simple. On the other hand, as for a 3-D layered medium, the precedure of deriving Green's function is not very simple due to the inability of establishing the orthogonality relations of the Rayleigh wave modes. Nevertheless, the concept of the Hyperfunction is found to be applicable to derive the spectral representation of Green's function.The boundary integral equation method as well as the spectral representation of Green's function are introduced to the scattering problem in that the scattering waves are caused by the interaction between a scattering object in a layered medium and a plane wave. The spectral representation of Green's function enables us to introduce a viewpoint of eigenvalue problems into the boundary integral equation method. The scattering waves are decomposed into oeigenfunctions for the point and continuous spectra via interchanging the order of the boundary integral and the spectral integral or summation. As a result, an understanding of scattering waves becomes possible by means of the eigenfunctions for a layered medium.