A MODEL FOR SCATTERING FROM AN ACOUSTICALLY HARD SURFACE WITH NONDIFFERENTIABLE PERIODIC ROUGHNESS

A model is described and numerical results are shown for the far-field scatter of a plane wave from an acoustically hard surface with a periodic roughness profile function that is continuous everywhere, but is nondifferentiable at a finite number of points within a period, and that may have a large maximum amplitude or slope. Scattering from these types of surfaces is modeled by approximating the roughness profile by truncating its Fourier series representation, and using the integral equation scattering formalism of Holford [J. Acoust. Soc. Am.70, (1981) 1116–1128]. To determine the effects of this approximation on the far-field scatter, the exact profile function and the scattering formalism of DeSanto [Radio Sci.16, (1981) 1315–1326], which is not an integral equation formalism, is also used to calculate the far-field scatter. Both formalisms are used to calculate plane wave scattering from sinusoidal surface roughness and from nondifferentiable periodic surface roughness constructed from (1) a trapezoid, (2) a rectified sinusoid, (3) concave upward and concave downward semi-ellipses, and (4) a power law spectrum. It is shown that the amplitudes of the plane wave modes scattered from a continuous and nondifferentiable surface and from its truncated Fourier series representation are essentially identical for all surfaces considered and over a wide range of incident grazing angles, wavenumbers, and surface roughness. In addition, it is shown that the Holford formalism may be used to calculate scattering from surfaces considerably rougher and at grazing angles lower than the DeSanto formalism, but that the Holford formalism is more computationally intensive than the DeSanto formalism, and that in the regions where both theories are numerically stable, the DeSanto formalism is preferable.