Computation of scattering of a plane wave from multiple prolate spheroids using the collocation multipole method

Abstract

The collocation multipole method is presented to solve three-dimensional acoustic scattering problems with multiple prolate spheroids subjected to a plane sound wave. To satisfy the three-dimensional Helmholtz equation in prolate spheroidal coordinates and the radiation condition at infinity, the scattered field is formulated in terms of radial and angular prolate spheroidal wave functions. Instead of using the complicated addition theorem of prolate spheroidal wave functions, the multipole method, the directional derivative, and the collocation technique are combined to solve multiple scattering problems semi-analytically. For the sound-hard or Neumann conditions, the normal derivative of the acoustic pressure with respect to a non-local prolate spheroidal coordinate system is developed without any truncation error for multiply connected domain problems. By truncating the higher order terms of the multipole expansion, a finite linear algebraic system is obtained and the scattered field is determined from the given incident acoustic wave. Once the total field is calculated as the sum of the incident field and the scattered field, the near field acoustic pressure and the far field scattering pattern are determined. Numerical experiments for convergence are performed to provide the guide lines for the proposed method. The proposed results of acoustic scattering by one, two, and three prolate spheroids are compared with those of an available analytical method and the boundary element method to validate the proposed method. Finally, the effects of the eccentricity of a prolate spheroidal scatterer, the separation between scatterers and the incident wave number on the near-field acoustic pressure and the far-field scattering pattern are investigated.