Prediction of three-dimensional scattering by pressure-release sinusoidal surfaces

Acoustic scattering by pressure-release sinusoidal surfaces is analyzed in three dimensions using the potential integral formula. Boundary values for sinusoidal surfaces are rigorously determined using a Fredholm boundary value integral equation. No restrictions on the surface heights and slopes are made. An incident field composed of spherical waves produced by a beamed source is used as this conforms to the reported acoustic experiments. Spherical waves provide a general solution because they transition to plane waves in the limit as the range to the surface becomes large with respect to the surface dimensions. In this limit, the Fraunhofer phase approximation is valid, and the solution mirrors the published "exact" solutions based on plane waves. This solution is, thus, applicable to both acoustical and scalar optical experiments. A periodic solution is assumed for the unknown boundary values. This approach produces a compact, computationally efficient solution in the form of a periodic Green's function. Predictions by the potential integral formula are compared to scattering measurements made on three different surfaces, and the agreement is good. A key finding is that acoustic experiments must be conducted using narrow beam widths to avoid interference in the measurement of grating order locations, amplitudes, and widths.