Acoustic eigenfrequencies in concentric spheroidal-spherical cavities: Calculation by shape perturbation

The acoustic eigenfrequencies f(nsm) in concentric spheroidal-spherical cavities are determined analytically, for both Dirichlet and Neumann boundary conditions, by a shape perturbation method. Two types of cavities are examined, one with spheroidal outer and spherical inner boundary and inversely for the other. The analytical determination is possible in the case of small h = d/(2R(2)), (h<<1), where d is the interfocal distance of the spheroidal boundary and 2R(2) the length of its rotation axis. In this case exact, closed form expressions are obtained for the expansion coefficients g(nsm)((2)) and g(nsm)((4)) in the resulting relation f(nsm) (h) = f(ns) (0) [1 + h(2)g(nsm)((2)) + h(4)g(nsm)((4)) + O(h(6))]. Analogous expressions are obtained with the use of the parameter v = 1 -(R-2/R-2')(2), (\v\ << 1) where 2R(2)' is the length of the other axis of the spheroidal boundary. There is no need for using any spheroidal wave functions and the expansion formulas connecting them with the concentric spherical ones. The pressure field is expressed in terms of spherical wave functions, while the equation of the spheroidal boundary is given in terms of the spherical co-ordinates. Numerical results are given for various values of the parameters. (C) 1998 Academic Press Limited.