A Computational Method to Calculate the Exact Solution for Acoustic Scattering by Fluid Spheroids

Abstract

The problem of scattering of harmonic plane acoustic waves by fluid spheroids (prolate and oblate) is addressed from an analytical approach. Mathematically, it consists in solving the Helmholtz equation in an unbounded domain with Sommerfeld radiation condition at infinity. The domain where propagation takes place is characterised by density and sound speed values ρ 0 and c 0 , respectively, while ρ 1 and c 1 are the corresponding density and sound speed values of an immersed object that is responsible of the scattered field. Since Helmholtz equation is separable in prolate/oblate spheroidal coordinates, its exact solution for the scattered field can be expressed as an expansion on prolate/oblate spheroidal functions multiplied by coefficients whose values depend upon the boundary conditions verified at the medium-immersed fluid obstacle interface. The general case (c 0 = c 1 ) is cumbersome because it requires to solve successive matrix systems that are ill-conditioned when c 1 /c 0 is far from unity. In this paper, a numerical implementation of the general exact solution that is valid for any range of eccentricity values and for c 0 = c 1 , is provided. The high level solver code has been written in the Julia programming language while a software package recently released in the literature has been used to compute the spheroidal functions. Several limit cases (Dirichlet and Neumann boundary conditions, spheroid tending to sphere) have been satisfactorily verified using the implemented code. The corresponding example scripts can be downloaded from the authors' web (GitHub) site. The numerical implementation of the exact solution leads to results that are in agreement with reported results obtained through approximate solutions for far-field and nearfield regimes. Additionally, the new code has been used to extend results reported in the literature.

Axisymmetric scattering of a fluid spheroid illuminated by an acoustical Bessel beam

“…Otherwise, two different methods for a plane acoustic wave from a penetrable prolate or oblate spheroid are used for the evaluation [7]. In addition, Edmundo Lavia et al derived a computational method to calculate the exact solution for liquid spheroid which agrees with reported predicted results obtained through approximated solution for far-field and near-field regimes [8]. Based on the discussions of the scattering of the fluid spheroid, the acoustical radiation force of a fluid spheroid or a liquid-filled spherical shell could be obtained, which could benefit the application of the biomedical, drug deliver, biochemistry and other areas [9].…”

Acoustic scattering by two fluid confocal prolate spheroids

“…In the following paragraphs the continuity equations will be transformed into a matrix system which will allow to solve the scattering problem for the spheroidal shell. The notation to be used closely follows the previous work [30].…”

“…Afterwards, different works dealing with numerical calculations were restricted to: certain particular cases of sound speed or density contrasts [22,23], the low frequency regime [22,[24][25][26][27][28] or low eccentricity spheroids [29]. A numerical evaluation of the exact solution without any limitation, based on [21] and using a computational code [20] for spheroidal wave function calculation in arbitrary precision was presented in [30].…”

“…This work presents a numerical evaluation of the exact, in terms of a series, solution for the acoustic scattering of two fluid prolate confocal spheroids (from now on this setup will be called an spheroidal shell) valid for any value of eccentricity and arbitrary fluid properties of the three involved physical mediums. The oblate case can be worked out following the same lines with only slight modifications, see for details [30]. In view of that, this work is devoted to the prolate spheroid.…”

Acoustic scattering theory without large-distance asymptotics

“…Acoustic scattering deals with the propagation of sounds. There are many researches on acoustic scattering, for example, acoustic waves scattering on different targets such as fluid spheroids [7] and finite rigid plates [8].…”