Finite series approach for the calculation of beam shape coefficients in ultrasonic and other acoustic scattering

Abstract

Partial wave expansions with spherical wave functions are usually employed to describe complex velocity potentials or pressure fields in ultrasonic and acoustic scattering by spherical particles. The expansion coefficients – known as the beam shape coefficients (BSCs) – can be calculated using different approaches which have been largely unexplored in the acoustical literature. Here, we formally present the Finite Series method for the evaluation of BSCs of ultrasonic and acoustic arbitrary-shaped beams. Such a series, which relies on Neumann expansion theorem, is an alternative to quadratures whenever analytical solutions cannot be obtained from direct integration over polar and azimuthal spherical angles. As examples of application, we consider the calculation of BSCs of Bessel, Laguerre–Gauss and Gaussian beams under on-axis configuration, including a discussion on a modified version of the finite series method recently presented in the context of optical fields.