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The almost-highest wave: a simple approximation

Published online by Cambridge University Press:  19 April 2006

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

The crest of a steep, symmetric gravity wave is shown to be closely approximated by the expression \[ x+iy = \frac{\alpha +\gamma i\chi}{(\beta + i\chi)^{\frac{1}{3}}}, \] where x, y are co-ordinates in the vertical plane, χ is the complex velocity potential and α, β, γ are certain constants. This expression is asymptotically correct both for small and for large values of |χ|; and the free surface agrees with the exact profile calculated by Longuet-Higgins & Fox (1977) everywhere to within 1·5 per cent. The pressure at the surface is constant to within 5 per cent.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 94 , Issue 2 , 25 September 1979 , pp. 269 - 273
Copyright
© 1979 Cambridge University Press

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References

Longuet-higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721–741.Google Scholar
Longuet-higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786.Google Scholar