J.-M. Chesneaux et A. Wirgin
Reflection from a corrugated surface revisited
J. Acoust. Soc. Am., 96 (2), Pt. 1, 1994, 1116-1129.
The problem of scattering of a plane sonic wave from a soft surface with periodic (sinusoidal) uneveness along one direction is examined by means of the Rayleigh plane wave expansion and the Waterman extinction methods, numerically implemented by Fourier projection and expansion respectively. The computations are done with real, double-precision, stochastic arithmetic instead of the usual complex, double-precision floating-point arithmetic in order to precisely evaluate the numerical accuracy of the results conditioned by round-off errors. In this arithmetic two results are identical if they have the same number of significant digits and their difference is a number with no significant digits. It is shown that the low-order plane-wave coefficients obtained by the Rayleigh and Waterman methods are identical when obtained from matrix systems that are large enough to give "convergence" of these coefficients. For the same matrix size, the higher-order coefficients differ all the more the higher the diffraction order, but increasing the size of the Rayleigh method matrix makes the higher-order coefficients more nearly approach those obtained by the Waterman method. It is also shown that the Waterman (Fourier-series) computation of the near-field is generally meaningful, whereas that of Rayleigh, involving summation of the plane waves is generally meaningless except near the points of the scattering surface first encountered by the incident wave (i.e., those in the valleys when the incident wave comes from below). Thus, the two methods are numerically equivalent as concerns the part of the far-field covered by the low-order plane waves, but non-equivalent as regards the near-field (which involves all the plane waves in the Rayleigh method). The only one of the two methods which can account for the near-field is that of Waterman. Low-order plane-wave scattering coefficients, with at least two-to-three-digit accuracy, and which are identical (to this precision) to the plane wave coefficients computed by the rigorous integral equation method, are obtained by both the Rayleigh and Waterman methods for scattering surfaces with slopes as large as 2.26 when the number of non-evanescent waves is five. This slope is five times the threshold value beyond which the Rayleigh plane-wave series (before projection) is mathematically non-convergent. The number of significant digits increases as the slope decreases.