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Lagrangian moments and mass transport in Stokes waves Part 2. Water of finite depth

Published online by Cambridge University Press:  21 April 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW and Institute of Oceanographic Sciences, Wormley, Surrey, GU8 5UB, UK
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Abstract

Some simple relations between the Lagrangian moments and cumulants in a steady finite-amplitude gravity wave on deep water are here generalized to water of finite depth. The first three Lagrangian moments are shown to be given in term of the mass-transport velocity U at the free surface, the potential and kinetic energy densities V and T, and the mean-square particle velocity $\overline{u^2_B}$ on the bottom.

A simple method of calculation is described, which exploits certain quadratic relations between the Fourier coefficients in Stokes's series. The ratio U/c and the associated Lagrangian skewness is calculated for periodic waves, as a function of the wave steepness and the mean water depth.

For limiting waves, i.e. those with sharp crests, it is found that the most symmetric orbits, in the Lagrangian sense, occur not in very deep or very shallow water, but at one intermediate value of the ratio of depth to wavelength. When the depth parameter kd equals 1.93 the vertical displacement of a marked particle at the free surface is closely sinusoidal in the time t.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 186 , January 1988 , pp. 321 - 336
Copyright
© 1988 Cambridge University Press

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