Calculation of acoustic radiation using equivalent-sphere methods

Among the methods generally used to solve a problem in the domain of acoustic radiation, the equivalent sources method offers an interesting alternative. It consists in replacing the vibrating surface with a distribution of acoustic sources placed inside the structure. The contribution of each source is determined in such a way that the acoustic field radiated by these sources verifies the same boundary conditions on the structure. The number of unknowns in the problem is no longer directly linked to the number of mesh points on the structure, as with boundary elements methods, but to the number of equivalent sources employed in the model. The equivalent source method is therefore of major interest if the acoustic radiation of the structure can be approximated with a sufficiently low number of sources. This paper proposes its application when the equivalent source is a sphere. In this case, the number of unknowns is equal to the number of modes. In contrast to the one-point multipole, the sphere has a surface surrounding a closed volume to express the boundary conditions. Although sphere/multipole equivalence has been demonstrated, the surface of the sphere allows normalization of the functions used, leading to stabilization of the system to be resolved. First, the main acoustic radiation characteristics of a sphere and of the linear system verified by the modal coefficients of an equivalent sphere are presented. The different parameters of the model are then studied: position and radius of the equivalent sphere, truncation of the series, and influence of the spatial sampling (mesh). In the same vein, a second approach is presented. It consists of making each point of the structure correspond to a point of the sphere, and the vibrating field at the surface of the sphere is deduced from that of the structure by simple geometric projection. Results can be obtained very quickly as no matrix inversion is required. The accuracy of the results depends on the distance between the sphere and the structure. Finally, an experimental validation that uses both methods is presented and shows interesting results when the structure is closed, and when its shape is not too far removed from a sphere.