A new spectral method for numerical solution of the unbounded rough surface scattering problem

This paper is concerned with a numerical solution for the scattering problem by an unbounded rough surface, which is referred to as a non-local perturbation of an infinite plane surface such that the whole surface lies within a finite distance of the original plane. A new and innovative spectral method is proposed to solve the unbounded rough surface scattering problem. The method uses a transformed field expansion to reduce the boundary value problem with a complex scattering surface into a successive sequence of transmission problems of the Helmholtz equation with a plane surface. Hermite orthonormal basis functions are used to further simply the transmission problems to fully decoupled one-dimensional two-point boundary value problems with piecewise constant wavenumbers, which can be solved efficiently by a Legendre-Galerkin method. Numerical examples are presented for both the rough surface scattering and the plane surface scattering, where the analytic solution is available. Ample numerical results presented in the paper indicate that the new spectral method is efficient, accurate, and well suited to solve the scattering problem by unbounded rough surfaces.