Three dimensional analyses of scattering by pressure-release sinusoidal surfaces

Scattering by pressure-release sinusoidal surfaces in three dimensions is analyzed using the Fresnel phase approximation and realistic source and receiver directivity approximations. Geometrical shadowing and second-order scattering are explicitly included to explore the validity of the Kirchhoff approximation. No restrictions on the surface heights and slopes are made. The "goodness" of the resulting expressions is verified by requiring the pressure scattered by a sinusoidal surface to reduce to the image solution as the surface amplitude goes to zero. The first-order scattered pressure achieves a very good approximation to the image solution, and the second-order scattered pressure goes to zero, as expected, under this requirement. The theory is compared with available experimental scattering measurements, and the agreement is good. Because the slopes on the experimental surface are very steep, shadowing corrections are indispensible to achieving accurate first and second order scattering results. Shadowing has a greater impact on the scattering prediction than the second-order scattering contribution. This suggests that the Kirchhoff approximation may be more robust when incorporated into a theory including a detailed shadowing treatment as well as the Fresnel phase approximation and a good directivity approximation.