Energy conservation in the Kirchhoff approximation

The scattering of sound from rough interfaces is frequently modeled using the Kirchhoff approximation. As has been shown by Lynch and Wagner [J. Acoust. Soc. Am. 47(3)] and others, for the case of a pressure-release surface, the Kirchhoff approximation fails to conserve energy. In particular, Lynch and Wagner derive an analytical expression for the proportion of incident energy conserved for a surface with a Gaussian roughness spectrum. They demonstrate that energy is not conserved near normal incidence due to the failure of the Kirchhoff approximation to multiply scatter rays back into the upper half-space. In this work, a Monte Carlo technique is used to quantify the degree to which energy is not conserved in the three-dimensional Kirchhoff approximation; these results are compared with theoretical prediction of Lynch and Wagner for the Gaussian spectrum. A similar Monte Carlo analysis is undertaken for other roughness types. Finally, it is shown that the integral solution, a model that accounts for multiple scattering and shadowing, conserves energy in the pressure-release case. [Work supported by ONR, Ocean Acoustics.]