Limitations of the operator expansion method

Abstract

Preliminary studies of the operator expansion method applied to scattering from rough surfaces satisfying the Dirichlet boundary condition [P. J. Kaczkowski and E. I. Thorsos, J. Acoust. Soc. Am. 90, 2258 (A) (1991)] have indicated that this relatively new method [D. M. Milder, J. Acoust. Soc. Am. 89, 529–541 (1991)] has a broad range of validity. For moderate rms surface slopes, the method is accurate over almost all scattering angles, and represents a vast improvement over the Kirchhoff approximation and small perturbation methods. Further study of the operator expansion method has led to new insights that establish a link between the formal validity of the method and the validity of the Rayleigh hypothesis. While the Rayleigh hypothesis appears to place a strict limit on the operator expansion, numerical examples will be presented that illustrate that the accuracy of the scattering cross section computed by the operator expansion method degrades only gradually as the rms slope is increased beyond that limit. For scattering from surfaces rough in one dimension, the accuracy of the operator expansion solution is established through comparison with the solution to an integral equation. Studies of the convergence of the terms in the operator expansion series indicate how the convergence rate can be used to infer the accuracy of the solution at any given order. This property will be useful when applying the operator expansion method to scattering from surfaces rough in two dimensions, for which exact solutions are still very costly. [Work supported by ONR.]