The collocation multipole method for solving multiple scattering problems with circular boundaries

Abstract

This paper presents a semi-analytical approach to solving the multiple scattering problems with circular boundaries. To satisfy the Helmholtz equation in polar coordinates, the multipole expansion for the scattered acoustic field is formulated in terms of the Hankel functions which also satisfy the radiation condition at infinity. Rather than using the addition theorem, the multipole method and directional derivative are both combined to propose a collocation multipole method in which the acoustic field and its normal derivative with respect to non-local polar coordinates can be calculated without any truncated error. The boundary conditions are satisfied by uniformly collocating points on the boundaries. By truncating the multipole expansion, a finite linear algebraic system is acquired and the scattered field can then be determined according to 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 along the scatterers and the far field scattering pattern can be determined. For the acoustic scattering of one circular cylinder, the proposed results match well with the analytical solutions. The proposed scattered fields induced by two and five circular–cylindrical scatterers are critically compared with those provided by the boundary element method and ones reported in the literature to validate the present method. Finally, the effects of the separation between scatterers and the incident wave number on the near and far field of acoustic scattering are investigated.