Scattering from a random surface

This report gives a formulation of the problem of propagation of scalar waves over a random surface. By a judicious choice of variables it is able to show that this situation is equivalent to propagation of these waves through a medium of random fluctuations with fluctuating source and receiver. The wave equations in the new coordinates has an additional term, the fluctuation operator, which depends on derivatives of the surface in space and time. An expansion in the fluctuation operator is given which guarantees the desired boundary conditions at every order. This report treats both the cases where the surface is time dependent, such as the sea or surface, or fixed in time. Also discussed is the situation where the source and receiver lie between the random surface and another, possibly also random, surface. In detail, this report considers acoustic waves for which the surfaces are pressure released. The method is directly applicable to electromagnetic waves and other boundary conditions.