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1. Carl Eckart, “ Scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 566570 (1953).
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8. P. J. Welton, “ Three-dimensional analysis of scattering by pressure-release plane surfaces and the validity of the image solution,” J. Acoust. Soc. Am. 131, 6979 (2012).

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Two widely used surface slope approximations are compared to an initially exact method that treats the slopes via a differential operator acting on the characteristic function. The differential operator treatment ceases to be exact when the integrand in the scattering integrals is approximated using a Gaussian directivity approximation and Fresnel phase approximation. Analysis is restricted to the Kirchhoff approximation (single scattering). One of the simpler slope approximations agrees with the more comprehensive differential operator approximation for all backscattering geometries, as well as for specular scattering geometries down to grazing angles comparable to the source beamwidth.


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