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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: [email protected]
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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
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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References

1. Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009a Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis. J. Fluid Mech. 624, 191–224.CrossRefGoogle Scholar
2. Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009b Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation. J. Fluid Mech. 624, 225–253.CrossRefGoogle Scholar
3. Alam, M.-R., Liu, Y. & Yue, D. K. P. 2010 Oblique sub- and super-harmonic Bragg resonance of surface waves by bottom ripples. J. Fluid Mech. 643, 437–447.CrossRefGoogle Scholar
4. Alam, M.-R. & Mei, C. C. 2007 Attenuation of long interfacial waves over a randomly rough seabed. J. Fluid Mech. 587, 73–96.CrossRefGoogle Scholar
5. Bryant, P. J. J 1973 Periodic waves in shallow water. J. Fluid Mech. 59 (4) 925–644.CrossRefGoogle Scholar
6. Dalrymple, R. A. & Liu, P. L.-F. 1978 Waves over soft muds: a two-layer fluid model. J. Phys. Oceanogr. 8 (6), 1121–1131.2.0.CO;2>CrossRefGoogle Scholar
7. Dommermuth, D. G. & Yue, D. K. P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.CrossRefGoogle Scholar
8. Elgar, S. & Raubenheimer, B. 2008 Wave dissipation by muddy seafloors. Geophys. Res. Lett. 35 (7), 1–5.CrossRefGoogle Scholar
9. Goda, Y. 1967 Traveling secondary waves in channels. Report to Port and Harbor Research Institute, Minstry of Transport, Japan.Google Scholar
10. Hasselmann, K. 1962 On the nonlinear energy transfer in a gravity wave spectrum, part 1: general theory. J. Fluid Mech. 12, 481–501.CrossRefGoogle Scholar
11. Kaihatu, J., Sheremet, A. & Holland, T. 2007 A model for the propagation of nonlinear surface waves over viscous muds. Coast. Engng 54, 752–764.CrossRefGoogle Scholar
12. Kirby, J. T. 1986 A general wave equation for waves over rippled beds. J. Fluid Mech. 162, 171–186.CrossRefGoogle Scholar
13. Liu, Y. & Yue, D. K.-P. P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–326.CrossRefGoogle Scholar
14. Macpherson, H. 1980 The attenuation of water waves over a non-rigid bed. J. Fluid Mech. 97 (04), 721.CrossRefGoogle Scholar
15. Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.CrossRefGoogle Scholar
16. Mei, C. C., Krotov, M., Huang, Z. & Huhe, A. 2010 Short and long waves over a muddy seabed. J. Fluid Mech. 643, 33–58.CrossRefGoogle Scholar
17. Mei, C. C. & Unluata, U. 1971 Harmonic generation in shallow water waves. In Waves on Beaches and Resulting Sediment Transport: Proc. Advanced Seminar, Mathematics Research Center, the University of Wisconsin, and the Coastal Engineering Research Center, US Army, at Madison, October 11–13, 1971, p. 181. Academic.Google Scholar
18. Myrhaug, D. 1992 Bottom Friction beneath Random Waves. Coast. Engng 24, 259–273.CrossRefGoogle Scholar
19. Ng, C.-o. & Zhang, X. 2007 Mass transport in water waves over a thin layer of soft viscoelastic mud. J. Fluid Mech. 573, 105–130.CrossRefGoogle Scholar
20. Sheremet, A., Mehta, A., Liu, B. & Stone, G. 2005 Wave sediment interaction on a muddy inner shelf during Hurricane Claudette. Estuar. Coast. Shelf Sci. 63 (1–2), 225–233.CrossRefGoogle Scholar
21. Sheremet, A. & Stone, G. W. W. 2003 Observations of nearshore wave dissipation over muddy sea beds. J. Geophys. Res. 108 (C11), 1–11.CrossRefGoogle Scholar
22. Torres-Freyermuth, A. & Hsu, T.-J. 2010 On the dynamics of wave-mud interaction: a numerical study. J. Geophys. Res. 115 (C7), 1–18.CrossRefGoogle Scholar
23. Wu, G., Liu, Y. & Yue, D. K. P. 2006 A note on stabilizing the Benjamin Feir instability. J. Fluid Mech. 556, 45–54.CrossRefGoogle Scholar
24. Young, I. 1998 Observations of triad coupling of finite depth wind waves. Coast. Engng 33 (2–3), 137–154.CrossRefGoogle Scholar