Dependence of Spatial and Temporal Correlation of Forward-Scattered Underwater Sound on the Surface Statistics. I. Theory

The correlation of underwater sound forward scattered from a time-varying statistically rough surface has been studied for the low-roughness case. The surface was illuminated by a divergent sound beam. Directionality of the source and receivers was included in a surface-illumination function. As an alternative to solving the Fresnel integral, the illuminated area was divided into subareas, for which the plane-wave Helmholtz-Kirchhoff treatment was valid. We used the criterion that the wavefront should be within λ/8 of the plane tangent to the wavefront over the subarea. The sound signals reflected and scattered at each subarea were separated into coherent and incoherent components, and the coherent components from the subareas were added coherently. The covariance of signals observed at a pair of receivers was computed with the aid of the bivariate Gaussian distribution function. Computations of the sound scattered by a traveling roughness, such as a gravity wave, showed that the covariance of the signals scattered into the first receiver and later into a second receiver can be large at a time delay, which is related to the group velocity of the roughness. The temporal and spatial covariances were also computed for a surface roughness having a damped cosinusoidal dependence upon space and time. The main contributions to the modulations of the spatial or temporal covariances came from the sound that was scattered from the subareas in nonspecular directions.