Application of the operator expansion method to scattering from one-dimensional moderately rough Dirichlet random surfaces

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Abstract
A new method for computing wave scattering from rough surfaces, called the operator expansion (OE) method, has been proposed by D. M. Milder [J. Acoust. Soc. Am. 89(2), 529–541 (1991)]. In this paper, the OE method is examined in its application to acoustic scattering from one‐dimensional randomly rough surfaces with Gaussian and Pierson–Moskowitz roughness spectra satisfying the pressure release (Dirichlet) boundary condition. The operator expansion solution, which is expressed in a systematic series, is found to converge rapidly and monotonically for moderately rough surfaces, that is, for surfaces whose slope‐height roughness parameter khs, given by the product of acoustic wave number k, rms surface height h, and rms surface slope s, is less than about 0.25. Through comparison with a numerically exact integral equation solution, the OE method is found to be accurate over a wide range of incident and scattering angles. The method is currently used in a Monte Carlo computation of the scattering cross section, in which scattering is computed from one surface realization at a time and then averaged over 50 realizations. Nevertheless, its efficiency and accuracy in one‐dimensional tests suggest that the operator expansion would be a useful method for computing scattering from two‐dimensional surfaces in roughness regimes encountered in scattering of low‐frequency sound from the ocean surface.

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