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A high-order cnoidal wave theory

Published online by Cambridge University Press:  19 April 2006

J. D. Fenton
J. D. Fenton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, Australia 2033
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Abstract

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.

Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 94 , Issue 1 , 11 September 1979 , pp. 129 - 161
Copyright
© 1979 Cambridge University Press

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