Acoustic eigenfrequencies in a spheroidal cavity with a concentric penetrable sphere

Abstract

The acoustic eigenfrequencies fnsm in a spheroidal cavity containing a concentric penetrable sphere are determined analytically, for both Dirichlet and Neumann conditions in the spheroidal boundary. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of both spherical and spheroidal wave functions, connected with one another by well-known expansion formulas. In the second, a shape perturbation method, this field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. The analytical determination of the eigenfrequencies is possible when the solution is specialized to small values of h=d/(2R2), (h≪1), with d the interfocal distance of the spheroidal boundary and 2R2 the length of its rotation axis. In this case exact, closed-form expressions are obtained for the expansion coefficients gnsm(2) and gnsm(4) in the resulting relation fnsm(h)=fns(0)[1+h2gnsm(2)+h4gnsm(4)+O(h6)]. Analogous expressions are obtained with the use of the parameter v=1−(R2/R2′)2, (|v|≪1), with 2R2′ the length of the other axis of the spheroidal boundary. Numerical results are given for various values of the parameters.