ACOUSTIC EIGENFREQUENCIES IN CONCENTRIC SPHEROIDAL–SPHERICAL CAVITIES: CALCULATION BY SHAPE PERTURBATION

Abstract

The acoustic eigenfrequenciesfnsmin 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 smallh=d/(2R2), (h⪡1), wheredis the interfocal distance of the spheroidal boundary and 2R2the length of its rotation axis. In this case exact, closed form expressions are obtained for the expansion coefficientsgnsm(2)andgnsm(4)in the resulting relationfnsm(h)=fns(0) [1+h2gnsm(2)+h4gnsm(4)+Oh6)]. Analogous expressions are obtained with the use of the parameterv=1−(R2/R2′)2, (∣v∣⪡1) where 2R2′ 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.