Inverse Acoustic Scattering Problems In A Layered Medium

Acoustic scattering theory is widely used in medicine and nondestructive test-ing, which has caused scientists’more and more attention. Scattering problem contains the direct scattering problem and the inverse scattering problem, the aim of the direct acoustic scattering problem is to solve the Helmholtz equation with some boundary condition, while the aim of the inverse acoustic scattering problem is to find the coefficients and the domain of Helmholtz equation by the knowledge of the solution of the equations or the far-field.In this paper, we will study an inverse acoustic scattering problem in a layered medium. An impenetrable obstacle D is completely covered by two layered medium Do and D1, which are homogeneous and penetrable. Our aim of this paper is to reconstruct the shape of D and the impedance function between Do and D1by the knowledge of the far field. In scattering theory, we are prone to use fields to represent corresponding wave function, so we will use interior and exterior field to describe the corresponding wave functions of the interior and exterior medium’s. Both the interior and exterior fields satisfy corresponding Helmholtz equations, and Neumann boundary condition in the interior boundary, transmission boundary condition in the exterior boundary and Sommerfeld radiation condition in the infinity.Firstly, we combine the integral equation methods and Fedholm theory to prove the existence and uniqueness of the direct scattering problem’s solution. For the cor-responding inverse scattering problem, by applying Tikhonov regularization, New-ton’s method and the principle of the least squares, we give an concrete step to reconstruct the shape of D and the impedance function.