The Conservation Laws and Stability of Fluid Waves of Permanent Form

Abstract

The solution of Nekrasov’s integral equation is described. By means of this solution the wave kinetic, potential, and full mechanical energies are defined as functions of fluid depth and wavelength. The wave obeys the laws of mass and energy conservation. It is found that for any constant depth of fluid the wavelength is bounded from above by a value denoted as maximal wavelength. At maximal wavelength 1) the maximum slope of the free surface of the wave exceeds 38^{o} and the value 45^{o} is supposed attainable,2) the wave kinetic energy vanishes. The stability of a steady wave considered as a compound pendulum is analyzed.

Cite this paper

T. Bodnar, "The Conservation Laws and Stability of Fluid Waves of Permanent Form,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 486-490. doi: 10.4236/am.2013.43072.

T. Bodnar, "The Conservation Laws and Stability of Fluid Waves of Permanent Form,"

References

[1] T. A. Bodnar’, “One Approximate Solution of the Nekrasov Problem,” Journal of Applied Mechanics and Technical Physics, Vol. 48, No. 6, 2007, pp. 818-823.
doi:10.1007/s10808-007-0105-9

[2] T. A. Bodnar’, “On Steady Periodic Waves on the Surface of a Fluid of Finite Depth,” Journal of Applied Mechanics and Technical Physics, Vol. 52, No. 3, 2011, pp. 378-384.
doi:10.1134/S0021894411030072

[3] T. A. Bodnar’, “On Steady Waves on the Surface of a Finite-Depth Fluid,” Free Boundary Problems: Theory, Experiment, and Applications, 3rd All-Russian Conference with International Participation, Biisk, 28 June-3 July 2008, pp. 25-26.

[4] A. I. Nekrasov, “Exact Theory of Steady Waves on the Surface of a Heavy Fluid,” Izd.Akad.Nauk SSSR, Moskow, 1951. (In Russian)

[5] R. Courant and D. Hilbert, “Methods of Mathematical Physics,” Interscience, New York, 1953.

[6] G. A. Chandler and I. G. Graham, “The Computation of Water Waves Modelled by Nekrasov’s Equation,” SIAM Journal on Numerical Analysis, Vol. 30, 1993, pp. 1041-1065. doi:10.1137/0730054

[7] R. Courant, “Differential and Integral Calculus,” Interscience, New York,1936.

[8] L. N. Sretenskii, “Theory of Fluid Wave Motion,” Nauka, Moscow, 1977. (In Russian)

[9] T. A. Bodnar’, “Conservation Law of the Full Mechanical Energy and Stability of the Steady-State Waves on the Surface of a Fluid of Finite Depth,” In: IV All-Russian Conference with foreign participation on Free Boundary Problems: Theory, Experiment, and Applications, Biisk, 5-10 July 2011, pp. 18-19.