The validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum

The validity of the perturbation approximation for rough surface scattering is examined (1) by comparison with exact results obtained by solving an integral equation and (2) through comparison of low-order perturbation predictions with higher-order predictions. The pressure release boundary condition is assumed, and the field quantity calculated is the bistatic scattering cross section. A Gaussian roughness spectrum is used, and the surfaces have height variations in only one direction. It is found, in general, that the condition kh≪1 (k is the acoustic wavenumber, h is the rms surface height) is insufficient to guarantee the accuracy of first-order (or higher-order) perturbation theory. When the surface correlation length l becomes too large or too small with h held fixed, higher-order perturbation terms can make larger contributions to the scattering cross section than lower-order terms. An explanation for this result is given. The regions of validity for low-order perturbation theory are also given. These results resolve (for the Gaussian roughness spectrum) the long-standing discrepancy between the first-order perturbation cross section and the cross section obtained with the Kirchhoff approximation for conditions when both approximations have generally been considered accurate. It is shown explicitly that, for some cases, the Kirchhoff prediction is the correct one. Finally, it is found that the perturbation expansion for the scattering cross section reduces to a power series expansion in the rms slope s when kl≪1. This observation provides additional insight into the connection between the small slope assumption and the validity of low-order perturbation theory.