Numerical Methods In Acoustic Scattering Problems With Damping Boundary Conditions

The inverse acoustic scattering problem is a typical Math-matical Physics inverse problem. A novel method for recovering the impedance coefficient from the far field pattern of the scattered wave for time-harmonic acoustic wave is given in this paper. The direct scattering problem for time-harmonic acoustic wave with impedance boundary condition is to find a solution of the Helmholtz equation. Firstly, the solution of the Helmholtz equation is approximated by the single-layer. The Nystrom method, Galerkin method and collocation method are presented to solve the corresponding second kind boundary integral equation and the theoretic analysis and the numerical examples are both given showing that the Nystrom method is the best to apply. Secondly, a method for recovering the impedance coefficient is given employing the results of the direct scattering problem. The inverse scatting problem in this paper is both nonlinear and improperly posed. Tikhonov regularization methods are used to transform this problem into an optimization problem and solving it by applying Quasi-Newton method then an approximated results are computed. The convergence is rigorously proven and the accurate graph and the approximated graph of the numerical examples are given in the same coordinate system showing that this method is both accurate and simple to use.