Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-26T19:22:02.266Z Has data issue: false hasContentIssue false
  • English
  • Français

Oblique sub- and super-harmonic Bragg resonance of surface waves by bottom ripples

Published online by Cambridge University Press:  15 January 2010

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
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
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of higher order (quartet) Bragg resonance involving two incident wave components and a bottom ripple component (so called class III Bragg resonance). In this case, unlike class I/II Bragg resonance involving a single incident wave and one/two bottom ripple components, the frequency of the resonant wave, which can be reflected or transmitted, is a sum or difference of the incident wave frequencies. In addition to transferring energy across the spectrum leading to potentially significant spectral transformation, such resonances may generate long (infragravity) waves of special importance to coastal processes and engineering applications. Of particular interest here is the case where the incident waves are oblique to the bottom undulations (or to each other) which leads to new and unexpected wave configurations. We elucidate the general conditions for such resonances, offering a simple geometric construction for obtaining these. Perturbation analysis results are obtained for these resonances predicting the evolutions of the resonant and incident wave amplitudes. We investigate special cases using numerical simulations (applying a high-order spectral method) and compare the results to perturbation theory: infragravity wave generation by co- and counter-propagating incident waves normal to bottom undulations; longshore long waves generated by (bottom) oblique incident waves; and propagating–standing resonant waves due to (bottom) parallel incident waves. Finally, we consider a case of multiple resonance due to oblique incident waves on bottom ripples which leads to complex wave creation and transformations not easily tractable with perturbation theory. These new wave resonance mechanisms can be of potential importance on continental shelves and in littoral zones, contributing to wave spectral evolution and bottom processes such as sandbar formation.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 643 , 25 January 2010 , pp. 437 - 447
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ardhuin, F. & Herbers, T. H. C. 2002 Bragg scattering of random surface gravity waves by irregular seabed topography. J. Fluid Mech. 451, 1–33.CrossRefGoogle Scholar
Ardhuin, F. & Magne, R. 2007 Scattering of surface gravity waves by bottom topography with a current. J. Fluid Mech. 576, 235–264.CrossRefGoogle Scholar
Babcock, J. M., Kirkendall, B. A. & Orcutt, J. A. 1994 Relationships between ocean bottom noise and the environment. Bull. Seismol. Soc. Am. 84 (6), 1991–2007.CrossRefGoogle Scholar
Ball, F. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19, 465–478.Google Scholar
Chen, Y. & Guza, R. T. 1999 Resonant scattering of edge waves by longshore periodic topography: finite beach slope. J. Fluid Mech. 387, 255–269.CrossRefGoogle Scholar
Davies, A. G. 1982 The reflection of wave energy by undulation on the seabed. Dyn. Atmos. Oceans 6, 207–232.CrossRefGoogle Scholar
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419–443.CrossRefGoogle Scholar
Dommermuth, D. G. & Yue, D. K. P. 1987 A higher-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.CrossRefGoogle Scholar
Elgar, S., Raubenheimer, B. & Herbers, T. H. C. 2003 Bragg reflection of ocean waves from sandbars. Geophys. Res. Lett. 30 (1), 1016.CrossRefGoogle Scholar
Guazzelli, E., Rey, V. & Belzons, M. 1991 Higher-order Bragg reflection of gravity surface waves by periodic beds. J. Fluid Mech. 245, 301–317.CrossRefGoogle Scholar
Hancock, M. J., Landry, B. J. & Mei, C. C. 2008 Sandbar formation under surface waves: theory and experiments. J. Geophys. Res. 113, C07022.Google Scholar
Hara, T. & Mei, C. C. 1987 Bragg scattering of surface waves by periodic bars: theory and experiment. J. Fluid Mech. 178, 221–241.Google Scholar
Heathershaw, A. D. 1982 Seabed-wave resonance and sand bar growth. Nature 296 (5855), 343–345, 10.1038/296343a0.Google Scholar
Heathershaw, A. D. & Davies, A. G. 1985 Resonant wave reflection by transverse bedforms and its relation to beaches and offshore bars. Mar. Geol. 62, 321–338.CrossRefGoogle Scholar
Henderson, S. M., Guza, R. T., Elgar, S., Herbers, T. H. C. & Bowen, A. J. 2006 Nonlinear generation and loss of infragravity wave energy. J. Geophys. Res. 111.Google Scholar
Herbers, T. H. C., Elgar, S. & Guza, R. T. 1995 Generation and propagation of infragravity waves. J. Geophys. Res. 100 (C12), 24,863–24,872.CrossRefGoogle Scholar
Huntley, D. A. 1976 Long-period waves on a natural beach. J. Geophys. Res. 81 (36), 6441–6449.CrossRefGoogle Scholar
Kirby, J. T. 1986 A general wave equation for waves over rippled beds. J. Fluid Mech. 162, 171–186.CrossRefGoogle Scholar
Landry, B. J., Hancock, M. J. & Mei, C. C. 2007 Note on sediment sorting in a sandy bed under standing water waves. Coast. Engng 54 (9), 694–699, doi: 10.1016/j.coastaleng.2007.02.003.CrossRefGoogle Scholar
Lippmann, T. C., Holman, R. A. & Bowen, A. J. 1997 Generation of edge waves in shallow water. J. Geophys. Res. 102 (C4), 8663–8679.CrossRefGoogle Scholar
Liu, Y. & Yue, D. K. P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–326.CrossRefGoogle Scholar
Madsen, P. A. & Fuhrman, D. R. 2006 Third-order theory for bichromatic bi-directional water waves. J. Fluid Mech. 557, 369–397.Google Scholar
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.Google Scholar
Mei, C. C., Hara, T. & Naciri, M. 1988 Note on Bragg scattering of water waves by parallel bars on the seabed. J. Fluid Mech. 186, 147–162.CrossRefGoogle Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves. art 1, 2, Advanced Series on Ocean Engineering, vol. 23. World Scientific.Google Scholar
Munk, W. H. 1949 Surf beats. EOS Trans. AGU 30, 849–854.Google Scholar
Naciri, M. & Mei, C. C. 1988 Bragg scattering of water waves by a doubly periodic seabed. J. Fluid Mech. 192, 51–74.CrossRefGoogle Scholar
Porter, R. & Porter, D. 2001 Interaction of water waves with three-dimensional periodic topography. J. Fluid Mech. 434, 301–335.Google Scholar
Renaud, M., Rezende, F., Waals, O., Chen, X. B. & van Dijk, R. 2008 Second-order wave loads on a lng carrier in multi-directional waves. In Proceeding of OMAE 2008, 26th international Conference on Offshore Mechanics and Arctic Engineering, June 2008 Estoril, Portugal.CrossRefGoogle Scholar
Yu, J. & Mei, C. C. 2000 a Do longshore bars shelter the shore? J. Fluid Mech. 404, 251–268.CrossRefGoogle Scholar
Yu, J. & Mei, C. C. 2000 b Formation of sand bars under surface waves. J. Fluid Mech. 416, 315–348.Google Scholar