A mean‐field solution of the reflection of a spherical acoustic wave from a rough interface

We use mean‐field methods to calculate the reflection by a rough interface of incident acoustic waves emanating from a point source. The calculation is accurate to second order in roughness height. For the special cases of very long and very short roughness wavelengths, we find closed‐form expressions for the reflected field. We give special attention to the head‐wave arrival and find the roughness can attenuate or enhance head‐wave arrival amplitude depending on the velocities and densities of the media. The roughness can also cause a delay of the head‐wave arrival though the apparent velocity is not changed (to second‐order accuracy). As a prerequisite to the rough‐interface calculation, we consider a smooth interface and find an asymptotic method of calculating the reflected field which avoids severe distortions of the path of integration.