 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
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| In English (Mech. Solids): | | 3212 |
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| M. V. Vil'de, Yu. D. Kaplunov, and V. A. Kovalev, "Approximate description of resonances of whispering gallery type waves in the problem of acoustic wave scattering by elastic circular cylinders and spheres," Mech. Solids. 37 (4), 147-158 (2002) |
| Year |
2002 |
Volume |
37 |
Number |
4 |
Pages |
147-158 |
| Title |
Approximate description of resonances of whispering gallery type waves in the problem of acoustic wave scattering by elastic circular cylinders and spheres |
| Author(s) |
M. V. Vil'de (Saratov)
Yu. D. Kaplunov (Moscow)
V. A. Kovalev (Moscow) |
| Abstract |
We study resonances of higher order partial modes in
the problem of scattering of stationary acoustic waves by elastic cylinders and
spheres. Our approach relies on short-wave asymptotic models,
which allow us to describe resonances of waves similar to those
arising in a whispering gallery.
On the basis of these models, we obtain approximate formulas for
the resonance component of partial modes, as well as elementary local
approximations of resonance curves in a neighborhood of resonance frequencies.
High efficiency of this approach is demonstrated by
comparing the approximate solution with the exact solution.
The study of resonances of partial modes is one of the basic
elements of the resonance theory of scattering [1]. In this connection,
simple approximate formulas describing resonance curves are very important
for the identification of qualitative laws of the scattering process.
Thus, approximate methods for bodies of small thickness
allow us to study acoustic wave scattering by elastic shells,
in the cases of both short-wave and long-wave vibrations
(see, for instance, [2]). For thick-walled and continuous bodies,
asymptotic analysis is usually possible only for partial modes with
high wavenumbers which correspond to short-wave vibrations
of the body.
In [3], resonances of Rayleigh waves in the problems of scattering
by elastic cylinders and spheres were studied in the short-wave approximation
on the basis of an approximate model in which the elasticity equations
preserve only the highest order derivatives, while the radial component is
fixed on the contact surface, i.e., the description of motion is
reduced to the problem of plane elasticity in Cartesian coordinates.
This model was used in [3] to obtain estimates for resonance frequencies
and construct approximations for resonance curves.
In the present paper, a similar approach is applied to
an approximate description of waves, like those in a whispering gallery,
which appear
in the process of acoustic wave scattering by elastic cylinders and
spheres. Such waves arise in a cylinder or a sphere because of their
finite curvature and are analogues of acoustic waves in a whispering gallery [4].
For a partial mode with a fixed number, the corresponding resonance
frequencies lie above the resonance frequency of the Rayleigh wave.
In this case, the radial component cannot be fixed on the contact surface,
since there is no exponential decay of vibrations away from the boundary.
However, on the basis of asymptotic properties of whispering gallery waves [4],
it is possible to simplify the original equations and construct short-wave
models describing vibrations of the body. These models are used to
obtain approximate formulas for resonance components of partial modes
and construct elementary local approximations for resonance curves
in a neighborhood of resonance frequencies. The efficiency of the method is
verified by comparing the formulas proposed here with the exact solution of
the problem. |
| References |
| 1. | N. D. Veksler, Acoustic Spectrography [in Russian], Valgus, Tallinn, 1989. |
| 2. | J. D. Kaplunov, L. Yu. Kossovich, and E. V. Nolde,
Dynamics of Thin-Walled Elastic Bodies, Academic Press, New York, 1998. |
| 3. | Yu. D. Kaplunov and V. A. Kovalev,
"An approximate description of the Rayleigh wave resonances
in problems of acoustic wave scattering by elastic cylinders and
spheres," Izv. AN. MTT [Mechanics of Solids], No. 4, pp. 180-186,
2000. |
| 4. | V. M. Babich and V. S. Buldyrev,
Asymptotic Methods in Problems of Diffraction of Short Waves [in Russian], Nauka,
Moscow, 1972. |
| 5. | M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Tables [Russian translation],
Nauka, Moscow, 1979. |
| 6. | J. Heading, An Introduction to Phase-integral Methods (the
WKB method) [Russian translation], Mir, Moscow, 1965. |
|
| Received |
12 February 2002 |
| << Previous article | Volume 37, Issue 4 / 2002 | Next article >> |
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