The diffraction of sound by an impedance sphere in the vicinity of a ground surface

Abstract

The problem of sound diffraction by an absorbing sphere due to a monopole point source was investigated. The theoretical models were extended to consider the case of sound diffraction by an absorbing sphere with a locally reacting boundary or an extended reaction boundary placed above an outdoor ground surface of finite impedance. The separation of variables techniques and appropriate wave field expansions were used to derive the analytical solutions. By adopting an image method, the solutions could be formulated to account for the multiple scattering of sound between the sphere and its image near a flat acoustically hard or an impedance ground. The effect of ground on the reflected sound fields was incorporated in the theoretical model by employing an approximate analytical solution known as the Weyl–van der Pol formula. An approximation solution was suggested to determine the scattering coefficients from a set of linearly coupled complex equations for an absorbing sphere not too close to the ground. The approximate method substantially reduced the computational time for calculating the sound field. Preliminary measurements were conducted to characterize the acoustical properties of an absorbing sphere made of open cell polyurethane foam. Subsequent experiments were carried out to demonstrate the validity of the proposed theoretical models for various source/receiver configurations around the sphere above an acoustically hard ground and an impedance ground. Satisfactory comparative results were obtained between the theoretical predictions and experimental data. It was found that the theoretical predictions derived from the approximate solution agreed well with the results obtained by using the exact solutions.