Abstract
Several problems concerned with small oscillations of an ideal liquid, taking account of the surface-tension forces, have been considered in [1–3] (as a rule, these are cases when the equilibrium liquid surface is spherical, plane, or differs only slightly from plane). Below we formulate the problem of the natural frequencies of small oscillations of a liquid for the general case of an equilibrium liquid surface in a weak potential mass force field. It is shown that the natural frequencies and the corresponding eigenfunctions of this problem may be found by the Ritz method. We note that analogous results in a somewhat different formulation have been obtained in the recently published [3].
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References
H. Lamb, Hydrodynamics [Russian translation], 1947.
L. D. Landau and E. M. Lifshitz, Mechanics of Continuous Media [in Russian], Gostekhizdat, 1953.
N. N. Moiseev and F. L. Chernous'ko, “Problems of oscillations of a liquid subjected to surface-tension forces,” Zh. vychislit. matem. i matem. fiz., vol. 5, no. 6, 1965.
V. V. Rumyantsev, “On the stability of the motion of a solid body containing a liquid having surface tension,” PMM, vol. 28, no. 4, pp. 746–753, 1964.
W. Blaschke, Differential Geometry [Russian translation], ONTI, 1935.
A. D. Tyuptsov, “Hydrostatics in weak force fields. Stability of equilibrium forms of a liquid surface,” Izv. AN SSSR, Mekhanika zhidkosti i gaza [Fluid Dynamics], no. 2, 1966.
S. L. Sobolev, Some Applications of Functional Analysis to Mathematical Physics [in Russian], Izd. LGU, 1950.
A. J. McConnell, Introduction to Tensor Analysis with Applications to Geometry, Mechanics, and Physics [Russian translation], Fizmatgiz, 1963.
S. G. Mikhlin, Minimum Problem for a Quadratic Functional [in Russian], Gostekhizdat, 1952.
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The author wishes to thank A. D. Myshkis and A. D. Tyuptsov for several helpful discussions.
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Kopachevskii, N.D. Hydrodynamics in weak force fields. Small oscillations of an ideal liquid. Fluid Dyn 1, 96–98 (1966). https://doi.org/10.1007/BF01013829
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DOI: https://doi.org/10.1007/BF01013829