Effective boundary conditions for a semi-infinite elastic medium bounded by a rough planar stress-free surface

Abstract

The problem of determining the static or dynamic elastic displacement field in a semi-infinite medium bounded by a rough, planar, stress-free surface is a random problem due to the stochastic nature of the surface-profile function. What is usually solved for in such problems is the displacement field in the medium averaged over the ensemble of realizations of the surface roughness. With the aid of Green's theorem we have replaced the true boundary conditions on the displacement field at the actual surface of the medium by effective boundary conditions satisfied by the average displacement field at the nominal, flat surface. The average field can then be obtained by solving an effective (nonstochastic) problem that is formally similar to the flat-surface problem. We apply this method to obtain the attenuation length and frequency shift of a Rayleigh wave due to surface roughness on an isotropic medium. The results are found to be in agreement with those of a calculation based on the boundary-matching method (Rayleigh's method).