### ASA 128th Meeting - Austin, Texas - 1994 Nov 28 .. Dec 02

## 4pPAb5. Asymptotic determination of the eigenfrequencies of a sphere in a
fluid.

**G. C. Gaunaurd
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*Naval Surface Warfare Center, White Oak Detachment, R-34, Silver Spring,
MD 20903-5640
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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.]