Abstract
An equation is derived describing small-amplitude vibrations of an arbitrary curved diaphragm, whose surface is considered as a two-dimensional Riemannian space. The derivation is based on the variational principle, from which the motion equation and conservation law follow in a form invariant with respect to arbitrary transformations of coordinates on the diaphragm surface. It has been shown that the wave equation, along with the two-dimensional Laplace-Beltrami operator, includes an additional term proportional to the scalar curvature of the diaphragm surface. As an example, the equations are considered for a spherical diaphragm and a catenoid-shaped diaphragm with a minimal surface of revolution.
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References
P. K. Rashevskii, Riemannian Geometry and Tensor Analysis (Nauka, Moscow, 1967).
L. D. Landau and E. M. Lifshits, Hydrodynamics (Nauka, Moscow, 1986).
L. D. Landau and E. M. Lifshits, Field Theory (Nauka, Moscow, 1973).
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Translated from Zhurnal Tekhnichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Fiziki, Vol. 70, No. 1, 2000, pp. 10–15.
Original Russian Text Copyright © 2000 by Glinski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \).
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Glinskii, G.F. On the theory of small-amplitude vibrations of curved-surface diaphragms. Tech. Phys. 45, 8–13 (2000). https://doi.org/10.1134/1.1259560
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DOI: https://doi.org/10.1134/1.1259560