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Large amplitude progressive interfacial waves

Published online by Cambridge University Press:  18 April 2017

Judith Y. Holyer
Judith Y. Holyer*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Extract

This paper contains a study of large amplitude, progressive interfacial waves moving between two infinite fluids of different densities. The highest wave has been calculated using the criterion that it has zero horizontal fluid velocity at the interface in a frame moving at the phase speed of the waves. For free surface waves this criterion is identical to the criterion due to Stokes, namely that there is a stagnation point at the crest of each wave. I t is found that as the density of the upper fluid increases relative to the density of the lower fluid the maximum height of the wave, for fixed wavelength, increases. The maximum height of a Boussinesq wave, which has the density almost the same above and below the interface, is 2·5 times the maximum height of a surface wave of the same wavelength. A wave with air over the top of it can be about 2% higher than the highest free surface wave. The point at which the limiting criterion is first satisfied moves from the crest for free surface waves to the point half-way between the crest and the trough for Boussinesq waves. The phase speed, momentum, energy and other wave properties are calculated for waves up to the highest using Padé approximants. For free surface waves and waves with air above the interface the maximum value of these properties occurs for waves which are lower than the highest. For Boussinesq waves and waves with the density of the upper fluid onetenth of the density of the lower fluid these properties each increase monotonically with the wave height.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 93 , Issue 3 , August 1979 , pp. 433 - 448
Copyright
Copyright © Cambridge University Press 1979

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References

Baker, G. A. 1965 The Theory And Application Of The Pade Approximant Method. In Advances In Theoretical Physics (Ed. Breuckner, K.), Vol. 1, P. 1. Academic Press.Google Scholar
Benjamin, T. B. 1966 Internal Waves Of Finite Amplitude And Permanent Form. J. Fluid Mech. 25, 241.CrossRefGoogle Scholar
Cokblet, E. D. 1977 Steep Gravity Waves In Water Of Arbitrary Uniform Depth. Phil. Trans. Roy. Soc. A 286, 183.Google Scholar
Graves-Morris, P. R. 1973 Pade Approximants And Their Applications. London: Academic Press.Google Scholar
Hunt, J. N. 1961 Interfacial Waves Of Finite Amplitude. La Houille Blanche 16, 515.Google Scholar
Levi-Civita, T. 1925 Determination Rigoureuse Des Ondes Permanentes D'Ampleur Finie. Math. Ann. 93, 264.Google Scholar
Lonouet-Higgins, M. S. 1975 Integral Properties Of Periodic Gravity Waves Of Finite Amplitude. Proc. Boy. Soc. Lond. A 342, 157.Google Scholar
Lonouet-Higgins, M. S. & Fenton, J. D. 1974 On The Mass, Momentum, Energy And Circulation Of A Solitary Wave. II . Proc. Roy. Soc. A 350, 1.Google Scholar
Orlanski, I. & Bryan, K. 1969 Formation Of The Thermocline Step Structure By Large Amplitude Internal Gravity Waves. J. Oeophys. Res. 74, 6975.Google Scholar
Schwartz, L. W. 1974 Computer Extension And Analytic Continuation Of Stokes Expansion For Gravity Waves. J. Fluid Mech. 62, 553.Google Scholar
Stokes, G. G. 1880 Considerations Relative To The Greatest Height Of Oscillatory Irrotational Waves. In Mathematical And Physical Papers, Vol. 1, P. 225. Cambridge University Press.Google Scholar
Thorpe, S. A. 1968 A On Standing Internal Gravity Waves Of Finite Amplitude. J. Fluid Mech. 32, 489.Google Scholar
Thorpe, S. A. 1968 B On The Shape Of Progressive Internal Waves. Phil. Trans. Roy. Soc. A 263, 563.Google Scholar
Thorpe, S. A. 1978 On The Shape And Breaking Of Finite Amplitude Internal Gravity Waves In A Shear Flow. J. Fluid Mech. 85, 7.Google Scholar
Tsuji, Y. & Nagata, Y. 1973 Stokes Expansion Of Internal Deep Water Waves To The Fifth Order. J. Ocean. Soc. Japan 29, 61.CrossRefGoogle Scholar
Woods, J. D. 1968 Wave Induced Shear Instability In The Summer Thermocline. J. Fluid Mech. 32, 791.Google Scholar
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