Scattering from a random interface
The problem of scattering from a random interface separating two fluids with different densities and sound speeds is considered. The method is to write coupled integral equations in coordinate space connecting the surface and volume values of the Green's function for the deterministic problem. In Fourier transform space the equations simplify, and it is possible to write a single integral equation for the Fourier transform of the surface value of the Green's function. Feynman-diagram methods can be used to aid the construction of both the Dyson equation for the mean of this Green's function and the Bethe-Salpeter equation for the mean of its second moment. These are derived assuming a Gaussian distribution of surface heights and using the accompanying cluster decomposition. As an example, a simple integral equation for the scattering amplitude corresponding to multiple scattering using the Kirchhoff approximation is also derived. It is analogous to the smoothing approximation used in random volume scattering theory. Its numerical solution for the special case of a Neumann surface is presented and, for large values of the Rayleigh roughness parameter, yields more coherent specular intensity than the Kirchhoff approximation. Other examples and the relation of our formal ism to other methods are also discussed. In the limiting cases the general formalism reduces to the standard results. In particular, in the flat surface limit we get the result in Officer's book.
DeSanto, John Anthony