Reflection on a Rough Surface from an Acoustic Point Source

A simple solution is developed for the reflected waves on a rough surface from a simple harmonic point source. It is assumed that the roughness is represented by a distribution of hemispherical bosses whose size and mutual distance are small relative to the wavelength. It is shown that under these conditions the effect of the roughness is equivalent to a boundary condition for the wave equation. This boundary condition embodies the surface polarization and the mutual interaction of the bosses. If the generating source lies above the reflecting surface the reflected wave is found to be equivalent to that originating from concentrated and distributed image sources on a line situated below the specular image with a magnitude decreasing exponentially with depth. The case of vanishingly small roughness is discussed along with the field intensity at large distance and grazing incidence. The effect of fluid viscosity is also evaluated.