Pt. Chen, VARIATIONAL FORMULATION OF INTERIOR CAVITY FREQUENCIES FOR SPHEROIDALBODIES, The Journal of the Acoustical Society of America, 100(5), 1996, pp. 2980-2988
This paper presents interior cavity frequencies for spheroidal bodies
in cases of Neumann and Dirichlet homogenous boundary conditions. The
Helmholtz wave equation is expressed in terms of prolate spheroidal co
ordinates from which a variational formulation is derived for the dete
rmination of the interior cavity frequencies. The search of stationari
ty is performed by means of a Rayleigh-Ritz type expansion of trial fu
nctions. The trial functions are expressed as a double summation of ba
sis functions in radial and angular coordinates. A variable transforma
tion is applied to the variational form in order to have homogenous bo
undary conditions, which are essential for establishing a complete fun
ction space of the associated boundary conditions. The present analysi
s is verified by comparisons of interior frequencies of a spherical bo
dy. of prolate spheroidal bodies, and of the interior characteristic f
unctions with the radial and angular functions obtained from prolate s
pheroidal differential equations. Interior frequencies are tabulated f
or various aspect ratios of spheroidal bodies in accordance with harmo
nic orders in azimuthal directions, for both Neumann and Dirichlet con
ditions. (C) 1996 Acoustical Society of America.