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The reflection of a plane wave at a rough interface separating two fluid media is examined in the approximation of “small roughness,” such that second-order terms in the magnitude of the roughness may be neglected. Exact solutions are obtained when the roughness is harmonic, while asymptotic results are obtained for arbitrary distributions. The analysis deals principally with an incident wave that is harmonic in time, but the problem of the reflection of a pulse from a perfectly reflecting, sinusoidal boundary is solved. It is found that such a boundary acts as a band-pass filter of the nonspecular components of the reflected wave. Outisde of this pass band the reflection is not only specular but distortionless. Rather less generality is possible when the boundary is not perfectly reflecting, but the pass band is found to be independent of the properties of the reflecting medium.
John W. Miles (1954) studied this question.