Construction of dynamic Green's function for an infinite acoustic field with multiple prolate spheroids

Abstract

The acoustic pressure of an unbounded acoustic field with multiple prolate spheroids with the Robin boundary conditions subjected to a time-harmonic point source located at an arbitrary location is solved semi-analytically in this work. This resultant solution is the so-called dynamic Green's function, which is important for acoustic problems such as sound scattering and noise control. It can be obtained by combining the fundamental solution with a homogenous solution, which is determined by using the collocation multipole procedure to satisfy the required Robin boundary conditions. To consider the geometries as described herein, the regular solution is expanded with angular and radial prolate spheroidal wave functions. As an alternate to the complex addition theorem applied to problems in multiply connected domains, by the directional derivative, the multipole expansion is computed in a straightforward manner among different local prolate spheroidal coordinate systems. By taking the finite terms of the multipole expansion at all collocating points, an algebraic system is acquired, and then the unknown coefficients are determined to complete the proposed dynamic Green's function by the Robin boundary conditions. The present results of one spheroid agree with the available analytical solutions. For the case of more than one spheroid, the proposed results are verified by comparison with the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. In the end, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheroids, and the aspect ratio of spheroid on the dynamic Green's functions. The proposed results can be applied to solve the time-harmonic problems for an unbounded acoustic field containing multiple spheroids. In the form of numerical Green's functions, they can improve the computational efficiency and increase the application of the boundary integral equation method.