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IssuesArchive of Issues2003-4pp.161-168

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V. A. Kovalev, "Synthesis of acoustic pressure scattered by an elastic cylindrical shell on the basis of matched asymptotic approximations," Mech. Solids. 38 (4), 161-168 (2003)
Year 2003 Volume 38 Number 4 Pages 161-168
Title Synthesis of acoustic pressure scattered by an elastic cylindrical shell on the basis of matched asymptotic approximations
Author(s) V. A. Kovalev (Moscow)
Abstract We consider the problem of scattering of a plane stationary acoustic pressure wave by an elastic cylindrical shell. The scattered pressure is synthesized by matching expansions for various asymptotic models representing the interaction between a shell and an acoustic medium. Outside a neighborhood of the zero frequency and the cut-off frequencies we use a model of plane layer type, while in a neighborhood of the zero frequency, the Kirchhoff-Love shell theory or its refinement is used. In a neighborhood of cut-off frequencies, a long-wave high-frequency approximation of the elasticity equations is used. A comparison with the exact solution shows that the approach proposed here gives a good description (with a uniformly small error) of the scattered pressure and the resonance components of partial modes in a fairly wide frequency range for various values of material parameters of the shell.

Asymptotic models representing the interaction of a shell and an acoustic medium are based on asymptotic approximations of three-dimensional elasticity equations. These asymptotic approximations form a basis of the modern dynamical theory of shells [1-4]. The classical Kirchhoff-Love shell theory and its refinements can be regarded as low-frequency approximations. In the high-frequency region, there exist two types of asymptotic approximations. The long-wave high-frequency approximation describes the shell's vibrations at frequencies close to those of thickness resonances for tension or shear. The short-wave high-frequency approximation is characterized by a weak effect of the shell's curvature.

The existence of regions in which different asymptotic approximations are compatible (matching regions) enables one to match these approximations, in order to obtain a uniform approximation in a wide frequency range. For example, for a spherical shell uniform approximations of the scattered pressure and resonance components of partial modes were obtained in [5] by matching solutions corresponding to the refined Kirchhoff-Love theory, the theory of long-wave high-frequency vibrations, and a model of plane layer type.

The aim of the present paper is to show that for cylindrical shells, synthesis of these approximations can be used for the construction of a solution for a fairly wide frequency range and various material parameters of shells. We use asymptotic models developed on the basis of these approximations and describing the interaction of the shell with an acoustic medium. In a neighborhood of zero frequency, an asymptotic model based on the refined Kirchhoff-Love theory [2, 6] is used, and in a neighborhood of thickness resonances, we use the theory of long-wave high-frequency vibrations of shells [3, 7]. Outside these neighborhoods, we use a model of plane layer type [4], which corresponds to a short-wave high-frequency approximation. A comparison with an exact solution [8] confirms high effectiveness of the proposed method for various values of material parameters of shells.
References
1.  J. D. Kaplunov, L. Yu. Kossovich, and E. V. Nolde, Dynamics of Thin Walled Elastic Bodies, Academic Press, New York, 1998.
2.  A. V. Belov, J. D. Kaplunov, and E. V. Nolde, "A refined asymptotic model of fluid-structure interaction in scattering by elastic shells," Flow, Turbulence, and Combustion, Vol. 61, No. 1-4, pp. 255-267, 1999.
3.  J. D. Kaplunov, "High-frequency stress-strain states of small variability in shells immersed in fluid," PMM [Applied Mathematics and Mechanics], Vol. 55, No. 3, pp. 478-485, 1991.
4.  M. V. Vil'de, J. D. Kaplunov, and V. A. Kovalev, "On the approximation of plane layer type in the problem of acoustic wave scattering by a cylindrical shell," Izv. AN. MTT [Mechanics of Solids], No. 3, pp. 180-186, 2002.
5.  V. A. Kovalev, "Matching of asymptotic approximation in the problem of acoustic wave scattering by spherical shells," PMM [Applied Mathematics and Mechanics], Vol. 66, No. 4, pp. 596-606, 2002.
6.  V. A. Kovalev, "Application of a refined asymptotic model to the study of acoustic wave scattering by a spherical shell," Izv. AN. MTT [Mechanics of Solids], No. 2, pp. 155-162, 2002.
7.  J. D. Kaplunov, "High-frequency stress-strain states of small variability in thin elastic shells," Izv. AN SSSR. MTT [Mechanics of Solids], No. 5, pp. 147-157, 1990.
8.  R. D. Doolittle and H. Überall, "Sound scattering by cylindrical shell," J. Acoust. Soc. Amer., Vol. 39, No. 2, pp. 272-275, 1966.
9.  V. A. Kovalev, "Matching of asymptotic models in scattering of acoustic waves by elastic shells", in Proc. Intern. Seminar Day of Diffraction, pp. 144-150, St. Petersburg, 2001.
10.  N. D. Veksler, Acoustic Spectroscopy [in Russian], Valgus, Tallinn, 1989.
Received 10 December 2002
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