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Wave action and set-down for waves on a shear current

Published online by Cambridge University Press:  12 April 2006

I. G. Jonsson ,
O. Brink-Kjaer  and
G. P. Thomas
I. G. Jonsson
Affiliation:
Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, DK-2800 Lyngby
O. Brink-Kjaer
Affiliation:
Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, DK-2800 Lyngby Present address: Computational Hydraulics Centre, Danish Hydraulic Institute, DK-2970 Hørsholm.
G. P. Thomas
Affiliation:
School of Mathematics, University of Bristol, England Present address: Department of Civil Engineering, University of Bristol, England.
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Abstract

This paper considers steady, slowly varying water waves propagating over a gently sloping bed on a steady current. The current varies linearly with depth, and so has constant vorticity ω. The analysis is two-dimensional and dissipation is neglected. Definitions, and expressions correct to second order in the amplitude, are given for the radiation stress, wave energy density E and total energy flux. An average La-grangian [Lscr ], obtained by heuristic arguments from Clebsch potentials, leads to the result that for this particular problem E equals the wave action [Lscr ]ω times the angular frequency ωrm relative to a frame of reference moving with the average-over-depth current velocity Um. This determines the variation of the amplitude with distance explicitly. An analytical expression for the height of the mean water surface is found by a heuristic argument which compares the conservation equations for total energy and wave action. All the results have been checked directly by substitution back into the basic equations. Graphs illustrate the effect of the vorticity ω on the wavelength, amplitude and set-down.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 87 , Issue 3 , 15 August 1978 , pp. 401 - 416
Copyright
© 1978 Cambridge University Press

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