Exact boundary integral equations for scattering of scalar waves from infinite rough interfaces

This is a companion paper to a previously published result [4] which treated the formulation of the exact integral equations for scalar scattering from perfectly reflecting infinite rough surfaces. Here we develop the corresponding integral equations for an infinite rough interface separating two different homogeneous media. In the upper medium we follow the development of the previous paper specifically using the incident and perfectly reflected plane waves as the Born term, and the image Green’s function. This development enabled us to evaluate the Green’s theorem integral exactly on the upper hemisphere. The remaining scattered field satisfied the standard radiation condition. Here the transmitted field also satisfies a standard radiation condition. Green’s theorem in both regions plus continuity conditions at the interface yield the exact equations for the infinite rough interface scattering problem. In the flat surface limit we solve the equations exactly and derive the standard Fresnel reflection and transmission coefficients. It is possible to do this latter derivation since the surface is infinite and we can use exact Fourier transforms.