Scattering of sound from time-varying irregular free surfaces

The problem of sound scattering from a time-varying irregular free surface is treated anew. The wave equation with its complicated surface boundary condition is transformed to another partial differential equation with a simple, perfectly reflecting, boundary condition. An integral equation for the Green's function G′ of this latter equation is found which admits a Neumann series solution. The Green's function G for the original wave equation is determined from G′ via a simple transformation. The Green's function G is explicitly calculated to second order in the surface wave height. A plane-wave scattering formalism is developed in which scattered-wave amplitudes are expressed in terms of G′. The differential cross section for surface scattering is expressed in terms of the scattered-wave amplitudes and is explicitly calculated to second order in the wave height and expressed in terms of the space-time covariance of the irregular surface. With suitable additional approximations, the autocovariance and power spectrum are computed to second order for an acoustic signal generated from a fixed sinusoidal source. The results are expressed in terms of previously derived plane-wave quantities, the differential cross section, and forwardscattering amplitude. The perturbation calculations are valid provided the rms wave height is small compared to the principal wavelength of sound used.