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New integral relations for gravity waves of finite amplitude

Published online by Cambridge University Press:  20 April 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, England, and Institute of Oceanographic Sciences, Wormley, Godalming, Surrey
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Abstract

Some new exact relations are derived between integral properties of a progressive irrotational gravity wave of finite amplitude in deep water. In particular it is shown that the Eulerian-mean angular momentum $\overline{A}_{\rm E}$ is directly proportional to the Lagrangian T−V, through the relation \[ \overline{A}_{\rm E} = 2c(T-V)/g, \] where c is the phase speed and g denotes the acceleration due to gravity. Moreover, for waves of constant length, the differential relation \[ {\rm d}\overline{A}_{\rm E} = 2(3T-V)\,{\rm d}c/g \] also holds.

In a wave of limiting steepness it was shown previously that the level of action ya is very nearly equal to the crest level ymax. This is further discussed, and is shown to be probably a numerical coincidence.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 149 , December 1984 , pp. 205 - 215
Copyright
© 1984 Cambridge University Press

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References

Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157–174.Google Scholar
Longuet-Higgins, M. S. 1978 Some new relations between Stokes's coefficients in the theory of gravity waves. J. Inst. Maths Applics 22, 261–273.Google Scholar
Longuet-Higgins, M. S. 1980 Spin and angular momentum in gravity waves. J. Fluid Mech. 97, 1–25.Google Scholar
Williams, J. M. 1981 Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302, 139–188.Google Scholar