Coherence function for the stochastic scattering by a time-varying, slightly rough, acoustically soft surface

Approximations for the coherence function of the random acoustic field scattered by a moving rough surface are extracted from a second-order perturbation expansion. The statistically rough surface is characterized by a simple, wide-sense stationary autocorrelation function that exhibits temporal and spatial dispersive behavior in terms of three parameters: the acoustically-small mean-square surface height, a correlation length, and a group velocity. Excitation is provided by the quintessential obliquely-incident time-harmonic plane-wave. Asymptotic evaluation of Fourier integral representations of the stochastic field yields expressions for the coherence function that explicitly display the dependence upon the space and time coordinates and their interaction via the geometrical parameters of the rough surface.