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The Quarterly Journal of Mechanics and Applied Mathematics 1999 52(4):525-542; doi:10.1093/qjmam/52.4.525
© 1999 by Oxford University Press
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Second harmonics generated by a bounded standing wave on a plane beach of arbitrary slope

GL Body and UT Ehrenmark

( Department of Computing, Information Systems and Mathematics, London Guildhall University, 100 Minories, London EC3N 1JY, UK )

The second-order Stokes wave expansion is obtained for flow over a beach of uniform slope with beach angle {alpha} = (2r - 1){pi}/(2m), m, r N. The first-order solution is written as an inverse Mellin transform and the Van Dyke principle of minimum singularity is invoked at the shoreline, whereby higher-order terms are no more singular than their predecessors. This condition establishes uniqueness at higher orders and in particular, for the first-order bounded standing wave, we obtain bounded second-order solutions with the presence of a quantifiable radiating second harmonic. This scattered wave is obtained as a finite sum of inverse Mellin transforms for r = 1 and is computed for this case. For r > 1 the Mellin transform of the velocity potential may be written as a (multiple) integral.


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