Three-dimensional acoustic scattering by multiple spheres using collocation multipole method

Abstract

This paper presents a semi-analytical approach to solve the three-dimensional acoustic scattering problems with multiple spheres subjected to a plane sound wave. To satisfy the three-dimensional Helmholtz equation in a spherical coordinate system, the multipole expansion for the scattered acoustic field is formulated in terms of the associated Legendre functions and the spherical Hankel functions that also satisfy the radiation condition at infinity. The multipole method, the directional derivative and the collocation technique are combined to propose a collocation multipole method in which the acoustic field and its normal derivative with respect to the non-local spherical coordinate system can be calculated without any truncated error, frequently occurred when using the addition theorem. The boundary conditions are satisfied by collocating points on the surface of each sphere. By truncating the higher order terms of the multipole expansion, a finite linear algebraic system is acquired. The scattered field can then be determined according to the given incident sound wave. The convergence analysis considering the specified error, the separation of spheres and the wave number of an incident wave is first carried out to provide guide lines for the proposed method. Then the proposed results for acoustic scattering by one, two and three spheres are validated by using the available analytical method and numerical methods such as boundary element method. Finally, the effects of the separation between scatterers, the incident wave number and the incident angle on the acoustic scattering are investigated extensively.