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Finite amplitude surface waves in a liquid layer

Published online by Cambridge University Press:  29 March 2006

Ali Hasan Nayfeh
Ali Hasan Nayfeh
Affiliation:
Aerotherm Corporation, Mountain View, California
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Abstract

An analysis is presented for the interaction of capillary and gravity waves in a liquid layer of finite depth. The method of multiple scales is used to obtain a third-order expansion uniformly valid for all times. Although this expansion is valid for a wide range of wave-numbers, it breaks down at two critical wave-numbers if the liquid depth is larger than √3/kc, kc = (ρg/T)½, where g is the gravitational acceleration, and ρ and T are the liquid density and surface tension, respectively. For a deep liquid, the singularities are at kc/√2 and kc/√3 respectively, as found by Wilton (1915), and Pierson & Fife (1961).

A second-order expansion valid for wave-numbers near the first critical value (corresponding to a wavelength of 2·44 cm in deep water) is obtained. This expansion shows that two different wave profiles could exist at or near the first critical wave-number. One of these profiles is gravity-like while the other is capillary-like.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 40 , Issue 4 , 9 March 1970 , pp. 671 - 684
Copyright
© 1970 Cambridge University Press

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References

Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532.Google Scholar
Kinsman, B. 1965 Wind Waves. Englewood Cliffs, N.J.: Prentice-Hall.
Lamb, H. 1932 Hydrodynamics (6th ed.) Cambridge University Press.
Levi-Civita, T. 1925 Determination rigoureuse des ondes permanentes d'ampleur finie. Math. Ann. 93, 264.Google Scholar
Michell, J. H. 1893 The highest wave in water. Phil. Mag. 36, 430.Google Scholar
Nayfeh, A. H. 1965 A perturbation method for treating nonlinear oscillation problems. J. Math. & Phys. 44, 368.Google Scholar
Nayfeh, A. H. 1968 Forced oscillations of the van der Pol oscillator with delayed amplitude limiting. IEEE Trans. on Circuit Theory, 15, 192.Google Scholar
Nayfeh, A. H. 1969 Triple- and quintuple-dimpled wave profiles in deep water. Phys. Fluids. (To be published.)Google Scholar
Pierson, W. J. & Fife, P. 1961 Some nonlinear properties of long-crested periodic waves with lengths near 2·44 centimetres. J. Geophys. Res. 66, 163.Google Scholar
Sekerzh-Zenkovich, Y. I. 1956 On the theory of stationary capillary waves of finite amplitude on the surface of a heavy fluid. Doklady Akad. Nauk. SSSR 109, 913.Google Scholar
Schooley, A. H. 1958 Profiles of wind-created water waves in the capillary-gravity transition region. J. Marine Res. Sears Foundation, 16, 100.Google Scholar
Schooley, A. H. 1960 Double, triple, and higher-order dimples in the profiles of wind-generated water waves in the capillary-gravity transition region. J. Geophys. Res. 65, 4075.Google Scholar
Stoker, J. J. 1957 Water Waves. Interscience Publishers.
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 1 (also in Mathematical and Physical Papers, vol. 1, Cambridge University Press).Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. New York: Academic Press.
Wilton, J. R. 1914 On deep water waves. Phil. Mag. 27, 385.Google Scholar
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688.Google Scholar