In a previous paper [J. Math. Phys. 13, 1903 (1972)], Feynman diagram methods were used to construct the Green's function and its first two moments for scalar wave scattering from a random rough surface with a Neumann boundary condition. This paper extends these formal diagram methods to the calculation of the Green's function and its first two moments for an elastic half-space bounded by a random rough surface considered as a free boundary. The surface height is a single valued bounded function with Gaussian statistics. The Dyson and Bethe-Salpeter equations, for the mean and second moment respectively, of the Green's function, are derived. Some simplifications of these integral equations and some examples are presented.
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John A. DeSanto (1973) studied this question.