This paper treats plane wave scattering from a rough interface separating two semi-infinite fluids of different density occupying regions V1 and V2. The interface is initially considered to be deterministic. A coordinate-space integral equation is derived for the surface value of the Green's function using Green's theorem in both regions and continuity conditions at the interface. For a source in V1, Green's theorem in V2 when evaluated at a field point in V1 is a nonlocal impedance-type boundary condition. It is also called an extended or extinction boundary condition. Since there is only one free-space Green's function, the results of Green's theorem in both regions combine to yield a single integral equation. An integral relation is then used to find the scattered or transmitted field values. In Fourier transform space the T-matrix for the scattered field satisfies a Lippmann-Schwinger (LS) integral equation familiar from quantum potential scattering theory. Here the ‘potential’ is noncentral and complex. Several examples and a discussion of the generalization to the electromagnetic problem are listed. When the surface is considered to be a homogeneously distributed Gaussian random variable, this LS equation, Feynman diagram methods, and cluster decomposition methods from statistical mechanics are used to derive an integral equation (Dyson equation) satisfied by the coherent part of T. This is a one-dimensional equation, and an approximation to it, whose Born term is the result due to Ament, is solved numerically. The result is a multiple scatter theory for coherent specular return which, when compared with experimental measurements, yields good results even for large roughness.
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