Theoretical Determination of the Random-Incidence Scattering Coefficients of Infinite Rigid Surfaces with a Periodic Rectangular Roughness Profile

An exact solution is developed for the sound field reflected by an infinite rigid rough surface with a periodic rectangular profile. The random-incidence scattering coefficient of this surface is derived from the numerical calculation of this solution for several directions of the incident plane wave. Scattering coefficients' values are given in this paper for a great number of configurations by varying the geometrical parameters of the periodic profile, leading to a very detailed description of the scattering properties of this kind of profile. In particular, it is shown that, when the ratio H/L of the depth to the spatial period of the profile is small, the key parameter is H/lambda, the ratio of the depth of the profile to the wavelength. The random-incidence scattering coefficient tends to increase with this ratio, as long as H/lambda is less than 0.3. This evolution is similar to Gaussian and sine-shaped profiles analysed in previous studies. For H/lambda > 0.3, the coefficient's value oscillates while converging to an asymptotic value which depends on the width of the wells. Resonance effects have also been highlighted for periodic rectangular profiles, in particular by comparing the computed scattering coefficients with the few measured values published in the scientific literature. The scattering coefficients' values published in this paper can be introduced in room acoustics' models to characterize periodic rectangular surfaces, provided that their dimensions are much greater than the wavelength. Additional measurements and BEM computations are required to deeply analyse the influence of the finite size and the acoustical absorption of real surfaces having this profile.