在非均勻物質中重建散射解的問題

Abstract

在一些實際的物理應用中, 如地質探測, 又如身體的 X 光照射,我們會遇到所謂的散射問題及其反問題.這類問題是要利用探測波及所觀察到的散射波,來推測所撞擊到的未知物體的形狀和性質.而散射波在無窮遠處的行為, 我們稱之為 far field.由這 far field 我們通常可以去做兩件事, 一件事是求出散射波的完整型態,另一件事就是求出未知障礙物的形狀等性質,在此我們將專門地去考慮如何在非均勻介質中重建散射波,並將討論侷限於在三維空間中的聲波中.我們假設在某個半徑之內是非均勻介質, 而之外的仍是均勻介質,如此一來, 我們就可以用處理反問題的手法來處理均勻介質的部分,得到均勻介質中的散射波, 另外再用解柯西問題的方式,求到非均勻介質中的散射波.

In problems dealing with the scattering of acoustic waves, we encounter functions known as scattered fields and far fields . In this article, I will derive the scattered field from a given far field pattern where the total field is governed by the equation Lu+k^2 u=0 outside a domain D. Here, L is an elliptic operator which coincides with the Laplace operator outside a ball of radius R centered at the origin. The derivation of the scattered field is divided into two parts. First, according to known theory in the scattering of acoustic waves, we can derive the scattered field in the region outside of the ball. Then with the Cauchy data of the scattered field on the boundary of the ball, we will solve the Cauchy problem for the scattered field in the region inside the ball omitting D, and hence we get the complete scattered field.