Wave scattering from a rough surface: Solution by an iterative method

Two iterative solutions of the Helmholtz equation for a scalar field in above a rough surface that admits the Dirichlet boundary condition are derived. The bases for the two iterative methods are two different boundary integral equations that represent the solution. The first integral equation is classified as a Fredholm integral equation of the first kind. The second is classified as a Fredholm integral equation of the second kind. This classification suggests that it is easier to find stable solution methods to the second equation. In both methods, the boundary integral was separated into a major part which is easy to calculate and a local residual part. The major part is a convolution and thus can be calculated using FFT in complexity O(N log N), where N is the number of surface points in which the surface height and its first derivatives together with the incoming wave and its normal derivative are all known. The residual element of the equations can be approximated efficiently only for surfaces where their amplitude is less than the wavelength of the incoming wave. The iterative schemes were tested numerically against a reference solution in order to examine the applicability range, the error estimation and the stability of the schemes. All tests supported the superiority of the second method. In particular the error estimation and stability tests indicated good performance for surfaces with slope up to 1. Yet, being an equation in the scattered field alone, makes the first method useful as a benchmark solution in its domain of applicability.