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Observation of resonant interactions among surface gravity waves

Published online by Cambridge University Press:  20 September 2016

F. Bonnefoy ,
F. Haudin ,
G. Michel ,
B. Semin ,
T. Humbert ,
S. Aumaître ,
M. Berhanu  and
E. Falcon
F. Bonnefoy*
Affiliation:
École Centrale de Nantes, LHEEA, UMR 6598 CNRS, F-44 321 Nantes, France
F. Haudin
Affiliation:
Univ. Paris Diderot, Sorbonne Paris Cité, MSC, UMR 7057 CNRS, F-75 013 Paris, France
G. Michel
Affiliation:
École Normale Supérieure, LPS, UMR 8550 CNRS, F-75 005 Paris, France
B. Semin
Affiliation:
École Normale Supérieure, LPS, UMR 8550 CNRS, F-75 005 Paris, France
T. Humbert
Affiliation:
Univ. Paris-Scalay, CEA-Saclay, SPEC, DRF, UMR 3680 CNRS, F-91 191 Gif-sur-Yvette, France
S. Aumaître
Affiliation:
Univ. Paris-Scalay, CEA-Saclay, SPEC, DRF, UMR 3680 CNRS, F-91 191 Gif-sur-Yvette, France
M. Berhanu
Affiliation:
Univ. Paris Diderot, Sorbonne Paris Cité, MSC, UMR 7057 CNRS, F-75 013 Paris, France
E. Falcon
Affiliation:
Univ. Paris Diderot, Sorbonne Paris Cité, MSC, UMR 7057 CNRS, F-75 013 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

We experimentally study resonant interactions of oblique surface gravity waves in a large basin. Our results strongly extend previous experimental results performed mainly for perpendicular or collinear wave trains. We generate two oblique waves crossing at an acute angle, while we control their frequency ratio, steepnesses and directions. These mother waves mutually interact and give birth to a resonant wave whose properties (growth rate, resonant response curve and phase locking) are fully characterized. All our experimental results are found in good quantitative agreement with four-wave interaction theory with no fitting parameter. Off-resonance experiments are also reported and the relevant theoretical analysis is conducted and validated.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 805 , 25 October 2016 , R3
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Copyright
© 2016 Cambridge University Press 

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References

Aubourg, Q. & Mordant, N. 2015 Nonlocal resonances in weak turbulence of gravity–capillary waves. Phys. Rev. Lett. 114, 144501.Google Scholar
Bonnefoy, F., Haudin, F., Michel, G., Semin, B., Humbert, T., Aumaître, S., Berhanu, M. & Falcon, E.2015 Other supplementary material, observation of resonant interactions among gravity surface waves.Google Scholar
Bordes, G., Moisy, F., Dauxois, T. & Cortet, P.-P. 2012 Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid. Phys. Fluids 24, 014105.Google Scholar
Dalrymple, R. A. 1989 Directional wavemaker theory with sidewall reflection. J. Hydraul. Res. 27 (1), 23–24.Google Scholar
Hammack, J. L., Henderson, D. M. & Segur, H. 2005 Progressive waves with persistent two-dimensional surface patterns in deep water. J. Fluid Mech. 532, 1–52.Google Scholar
Haudin, F., Cazaubiel, A., Deike, L., Jamin, T., Falcon, E. & Berhanu, M. 2016 Experimental study of three-wave interactions among capillary–gravity surface waves. Phys. Rev. E 93, 043110.Google Scholar
Henderson, D. M. & Hammack, J. L. 1987 Experiments on ripple instabilities. Part 1. Resonant triads. J. Fluid Mech. 184, 15–41.Google Scholar
Janssen, P. A. E. M. 2009 On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J. Fluid Mech. 637, 1–44.Google Scholar
Joubaud, S., Munroe, J., Odier, P. & Dauxois, T. 2012 Experimental parametric subharmonic instability in stratified fluids. Phys. Fluids 24, 041703.Google Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.Google Scholar
Lake, B. & Yuen, H. 1977 A note on some nonlinear water–wave experiments and the comparison of data with theory. J. Fluid Mech. 83, 75–81.Google Scholar
Leblanc, S. 2009 Stability of bichromatic gravity waves on deep water. Eur. J. Mech. (B/Fluids) 28 (5), 605–612.Google Scholar
Liu, Z., Xu, D. L., Li, J., Peng, T., Alsaedi, A. & Liao, S. J. 2015 On the existence of steady-state resonant waves in experiments. J. Fluid Mech. 763, 1–23.Google Scholar
Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12, 321–332. We have noticed a misprint in (6.4) in Longuet-Higgins (1962): the term $-(6+\unicode[STIX]{x1D709}^{2})^{1/2}$ should be replaced by $-\text{sgn}(\unicode[STIX]{x1D709})(6+\unicode[STIX]{x1D709}^{2})^{1/2}$ where $\unicode[STIX]{x1D709}=(1-r)/r$ .Google Scholar
Longuet-Higgins, M. S. & Smith, N. D. 1966 An experiment on third-order resonant wave interactions. J. Fluid Mech. 25, 417–435.Google Scholar
Martin, B. S., Simmons, W. & Wunsch, C. 1972 The excitation of resonant triads by single internal waves. J. Fluid Mech. 53, 17–44.Google Scholar
Mcgoldrick, L. F. 1970 An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251–271.CrossRefGoogle Scholar
Mcgoldrick, L. F., Phillips, O. M., Huang, N. E. & Hodgson, T. H. 1966 Measurements of third-order resonant wave interactions. J. Fluid Mech. 25, 437–456.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part I. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Shemer, L. & Chamesse, M. 1999 Experiments on nonlinear gravity–capillary waves. J. Fluid Mech. 380, 205–232.Google Scholar
Stiassnie, M. & Shemer, L. 2005 On the interaction of four water–waves. Wave Motion 41 (4), 307–328.Google Scholar
Su, M.-Y., Bergin, M., Marler, P. & Myrick, R. 1982 Experiments on nonlinear instabilities and evolution of steep gravity–wave trains. J. Fluid Mech. 124, 45–72.Google Scholar
Tomita, H. 1989 Theoretical and experimental investigations of interaction among deep-water gravity waves. Rep. Ship Res. Inst. 26 (5), 251–350.Google Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.CrossRefGoogle Scholar
Waseda, T., Kinoshita, T., Cavaleri, L. & Toffoli, A. 2015 Third-order resonant wave interactions under the influence of background current fields. J. Fluid Mech. 784, 51–73.Google Scholar
Zakharov, V. 1968 Stability of periodic waves of finite amplitude on a surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–198.Google Scholar

Bonnefoy et al. supplementary movie

Generation of a daughter wave by a resonant interaction between two oblique crossing waves in a large basin (50 m long): 0-4 s: basin at rest; 4-10 s: generation of the mother wave 1 only; 10-19 s: generation of the mother wave 3 only; 19-42 s: simultaneous generation of the two mother waves 1 and 3. In the latter, note the growth of waves in the expected direction of the daughter wave. Additional dashed lines are aligned with crests and separated by a wavelength. Arrows indicate the wave direction (Perspective view, resonance conditions, mother-wave steepnesses = 0.05)

Download Bonnefoy et al. supplementary movie(Video)
Video 37.1 MB

Bonnefoy et al. supplementary movie

Generation of a daughter wave by a resonant interaction between two oblique crossing waves in a large basin (50 m long): 0-4 s: basin at rest; 4-10 s: generation of the mother wave 1 only; 10-19 s: generation of the mother wave 3 only; 19-42 s: simultaneous generation of the two mother waves 1 and 3. In the latter, note the growth of waves in the expected direction of the daughter wave. Additional dashed lines are aligned with crests and separated by a wavelength. Arrows indicate the wave direction (Perspective view, resonance conditions, mother-wave steepnesses = 0.05)

Download Bonnefoy et al. supplementary movie(Video)
Video 114 MB
Supplementary material: PDF

Bonnefoy et al. supplementary material

Supplementary material

Download Bonnefoy et al. supplementary material(PDF)
PDF 108.3 KB