G. C. Gaunaurd
Naval Surface Warfare Center, White Oak Detachment, R-34, Silver Spring, MD 20903-5640
Starting from a phase-matching principle that is the acoustical analog of the Bohr--Somerfeld--Wilson quantization rule of the old ``quantum theory,'' it is analytically shown how to asymptotically obtain the eigenfrequencies of an insonified sphere immersed in a fluid. This technique was first illustrated by J. B. Keller [cf. Ann. Phys. 4, 180--188 (1958)] and it has been extended by many authors, notably L. B. Felsen and J. M. Ho, who have renamed it the ``ray-acoustic algorithm.'' It is shown here how the acoustical counterpart of this quantum principle leads to a resonance condition for the (external) eigenfrequencies of a sphere (rigid, soft, or to some extent, elastic) that exactly coincides with F. W. J. Olver's (1954) classical asymptotic formula for the (complex) zeros of the spherical Hankel functions. The poles of the scattering amplitude of an elastic sphere fall into two great families, one depending on shape, and the other on elastic composition. The asymptotic spacings in between the shape-dependent zeros in the (complex) ka plane are shown to reduce to a uniform value, obtained earlier by numerical means, which manifests itself in all the (RST) ``background'' curves of the sphere. [Work supported by NSWC-DD IR Program.]