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Attenuation of short surface waves by the sea floor via nonlinear sub-harmonic interaction

Published online by Cambridge University Press:  08 November 2011

Mohammad-Reza Alam ,
Yuming Liu  and
Dick K. P. Yue
Mohammad-Reza Alam
Affiliation:
Department of Mechanical Engineering, Center for Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Mechanical Engineering, University of California, Berkeley, CA, 94720, USA
Yuming Liu
Affiliation:
Department of Mechanical Engineering, Center for Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Dick K. P. Yue*
Affiliation:
Department of Mechanical Engineering, Center for Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: yue@mit.edu
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Abstract

We consider the indirect mechanism for dissipation of short surface waves through their near-resonant interactions with long sub-harmonic waves that are dissipated by the bottom. Using direct perturbation analysis and an energy argument, we obtain analytic predictions of the evolution of the amplitudes of two short primary waves and the long sub-harmonic wave which form a near-resonant triad, elucidating the energy transfer, from the short waves to the long wave, which may be significant over time. We obtain expressions for the rate of total energy loss of the system and show that this rate has an extremum corresponding to a specific value of the (bottom) damping coefficient (for a given pair of short wavelengths relative to water depth). These analytic results agree very well with direct numerical simulations developed for the general nonlinear wave–wave and wave–bottom interaction problem.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Coastal engineering

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 529 - 540
Copyright
Copyright © Cambridge University Press 2011

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