Acoustic eigenvalues of a quasispherical resonator: Second order shape perturbation theory for arbitrary modes

Abstract

The boundary-shape formalism of Morse and Ingard is applied to the acoustic modes of a deformed spherical resonator (quasisphere) with rigid boundaries. For boundary shapes described by r = a [1 - ∈ℱ(θ, φ)], where ∈ is a small scale parameter and ℱ is a function of order unity, the frequency perturbation is calculated to order ∈2. The formal results apply to acoustic modes whose angular dependence is designated by the indices ℓ and m. Specific examples are worked out for the radial (ℓ = 0) and triplet (ℓ = 1) modes, for prolate and oblate spheroids, and for triaxial ellipsoids. The exact eigenvalues for the spheroids, and eigenvalue determined with finite-element calculations, are shown to agree with perturbation theory through terms of order ∈2. This work is an extension of the author's previous papers on the acoustic eigenfrequencies of deformed spherical resonators, which were limited to the second-order perturbation for radial modes [J. Acoust. Soc. Am. 71, 1109-1113(1982)] and the first order-perturbation for arbitrary modes [J. Acoust. Soc. Am. 79, 278-285 (1986)].