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Experimental evidence of stable wave patterns on deep water*

Published online by Cambridge University Press:  16 August 2010

DIANE M. HENDERSON ,
HARVEY SEGUR  and
JOHN D. CARTER
DIANE M. HENDERSON*
Affiliation:
Department of Mathematics, Penn State University, University Park, PA 16803, USA
HARVEY SEGUR
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
JOHN D. CARTER
Affiliation:
Mathematics Department, Seattle University, Seattle, WA 98122, USA
*
†Email address for correspondence: [email protected]
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Abstract

Recent predictions from competing theoretical models have disagreed about the stability/instability of bi-periodic patterns of surface waves on deep water. We present laboratory experiments to address this controversy. Growth rates of modulational perturbations are compared to predictions from: (i) inviscid coupled nonlinear Schrödinger (NLS) equations, according to which the patterns are unstable and (ii) dissipative coupled NLS equations, according to which they are linearly stable. For bi-periodic wave patterns of small amplitude and nearly permanent form, we find that the dissipative model predicts the experimental observations more accurately. Hence, our experiments support the claim that these bi-periodic wave patterns are linearly stable in the presence of damping. For bi-periodic wave patterns of large enough amplitude or subject to large enough perturbations, both models fail to predict accurately the observed behaviour, which includes frequency downshifting.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 658 , 10 September 2010 , pp. 247 - 278
Copyright
Copyright © Cambridge University Press 2010

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Footnotes

*

With an appendix by Abhijit Chaudhuri and Erin Byrne

References

REFERENCES

Anderson, D. & Lisak, M. 1984 Modulational instability of coherent optical-fibre transmission signals. Opt. Lett. 9, 468–470.CrossRefGoogle Scholar
Baludin, S. I., Shrira, V. I., Kharif, C. & Ioualalen, M. 1995 On two approaches to the problem of instability of short-crested water waves. J. Fluid Mech. 303, 297–326.Google Scholar
Benjamin, B. & Feir, J. 1967 The disintegration of wavetrains in deep water. Part 1. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299, 59–75.Google Scholar
Benney, D. J. & Newell, A. C. 1967 The propagation of nonlinear wave envelopes. J. Math. Phys. (Stud. Appl. Math.) 46, 133–139.Google Scholar
Bridges, T. J. & Laine-Pearson, F. E. 2005 The long-wave instability of short-crested waves, via embedding in the oblique two-wave interaction. J. Fluid Mech. 543, 147–182.CrossRefGoogle Scholar
Chappelear, J. E. 1959 A class of three-dimensional shallow water waves. J. Geophys. Res. 64, 1883–1890.CrossRefGoogle Scholar
Craig, W., Henderson, D. M., Oscamou, M. & Segur, H. 2007 Stable three-dimensional waves of nearly permanent form on deep water. Math. Comput. Simul. 74, 135–144.CrossRefGoogle Scholar
Craig, W. & Nicholls, D. 2000 Travelling two and three dimensional capillary gravity water waves. SIAM: J. Math. Anal. 32, 323–359.Google Scholar
Craig, W. & Nicholls, D. 2002 Travelling gravity waves in two and three dimensions. Eur. J. Mech. B/Fluids 21, 615–641.CrossRefGoogle Scholar
Davies, J. T. & Vose, R. W. 1965 On the damping of capillary waves by surface films. Proc. R. Soc. Lond. A 260, 218–233.Google Scholar
Dhar, A. K. & Das, K. P. 1991 Fourth-order nonlinear evolution equations for two Stokes wave trains in deep water. Phys. Fluids A 3, 3021–3026.CrossRefGoogle Scholar
Dias, F., Dyachenko, A. I. & Zakharov, V. E. 2008 Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. Phys. Lett. A 372, 1297–1302.CrossRefGoogle Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary-gravity waves. Annu. Rev. Fluid Mech. 31, 301–346.CrossRefGoogle Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105–114.Google Scholar
Fuchs, R. A. 1952 On the theory of short crested oscillatory waves. In Gravity Waves. 521, pp. 187–200. United States National Bureau of Standards.Google Scholar
Fuhrman, D. R. & Madsen, P. A. 2006 Short-crested waves in deep water: a numerical investigation of recent laboratory experiments. J. Fluid Mech. 559, 391–411.CrossRefGoogle Scholar
Fuhrman, D. R., Madsen, P. A. & Bingham, H. B. 2006 Numerical simulation of lowest-order short-crested wave instabilities. J. Fluid Mech. 563, 415–441.CrossRefGoogle Scholar
Gordon, J. P. 1986 Theory of the soliton self-frequency shift. Opt. Lett. 11, 662–664.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Hammack, J. L., Scheffner, N. & Segur, H. 1989 Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, 567–589.CrossRefGoogle Scholar
Hasegawa, A. 1972 Theory and computer experiment on self-trapping instability of plasma cyclotron waves. Phys. Fluids 15, 870–881.CrossRefGoogle Scholar
Hasegawa, A. & Kodama, Y. 1995 Solitons in Optical Communications. Clarendon Press.CrossRefGoogle Scholar
Henderson, D. M. 1998 Effects of surfactants on Faraday-wave dynamics. J. Fluid Mech. 365, 89–107.CrossRefGoogle Scholar
Henderson, D., Patterson, M. S. & Segur, H. 2006 On the laboratory generation of two-dimensional, progressive, surface waves of nearly permanent form on deep water. J. Fluid Mech. 559, 413–427.CrossRefGoogle Scholar
Huhnerfuss, H., Lange, P. & Walter, W. 1985 Relaxation effects in monolayers and their contribution to water wave damping. Part II. The Marangoni phenomenon and gravity wave attenuation. J. Colloid Interface Sci. 108, 442–450.CrossRefGoogle Scholar
Iooss, G. & Plotnikov, P. 2009 Small divisor problem in the theory of three-dimensional water gravity waves. Mem. AMS 200 (940), 128 pp.Google Scholar
Ioualalen, M. & Kharif, C. 1994 On the subharmonic instabilities of steady three-dimensional deep water waves. J. Fluid Mech. 262, 265–291.CrossRefGoogle Scholar
Karlsson, M. 1995 Modulational instability in lossy optical fibers. J. Opt. Soc. Am. B 12, 2071–2077.CrossRefGoogle Scholar
Kimmoun, O., Branger, H. & Kharif, C. 1999 On short-crested waves: experimental and analytical investigations. Eur. J. Mech. B/Fluids 18, 889–930.CrossRefGoogle Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.CrossRefGoogle Scholar
Leblanc, S. 2009 Stability of bichromatic gravity waves on deep water. Eur. J. Mech. B/Fluids 28, 605–612.CrossRefGoogle Scholar
Lucassen, J. 1982 Effect of surface-active material on the damping of gravity waves: a reappraisal. J. Colloid Interface Sci. 85, 52–58.CrossRefGoogle Scholar
Luther, G. G. & McKinstrie, C. J. 1990 Transverse modulational instability of collinear waves. J. Opt. Soc. Am. B 7, 1125–1141.CrossRefGoogle Scholar
McKinstrie, C. J. & Bingham, R. 1989 The modulational instability of coupled waves. Phys. Fluids B 1, 230–237.CrossRefGoogle Scholar
Mei, C. C. & Hancock, M. J. 2003 Weakly nonlinear surface waves over a random seabed. J. Fluid Mech. 475, 247–268.CrossRefGoogle Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.CrossRefGoogle Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Mitschke, F. M. & Mollenauer, L. F. 1986 Discovery of the soliton self-frequency shift. Opt. Lett. 11, 659–661.CrossRefGoogle ScholarPubMed
Nemytskii, V. V. & Stepanov, V. V. 1960 Qualitative Theory of Differential Equations. Princeton University Press.Google Scholar
Onorato, M., Osborne, A. R. & Serio, M. 2006 Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys. Rev. Lett. 96, 014503.CrossRefGoogle Scholar
Ostrovsky, L. A. 1967 Propagation of wave packets and space-time self-focussing in a nonlinear medium. Sov. J. Exp. Theor. Phys. 24, 797–800.Google Scholar
Pierce, R. D. & Knobloch, E. 1994 On the modulational instability of travelling and standing water waves. Phys. Fluids 6, 1177–1190.CrossRefGoogle Scholar
Rayleigh, L. 1890 On the tension of water surfaces, clean and contaminated, investigated by the method of ripples. Phil. Mag. XXX, 386–400.CrossRefGoogle Scholar
Roskes, G. J. 1976 Nonlinear multiphase deep-water wavetrains. Phys. Fluids. 19, 1253–1254.CrossRefGoogle Scholar
Segur, H., Henderson, D., Carter, J., Hammack, J., Li, C.-M., Pheiff, D. & Socha, K. 2005 Stabilizing the Benjamin–Feir instability. J. Fluid Mech. 539, 229–271.CrossRefGoogle Scholar
Shukla, P. K., Kourakis, I., Eliasson, B., Marklund, M. & Stenflo, L. 2006 Instability and evolution of nonlinearly interacting water waves. Phys. Rev. Lett. 97, 094501.CrossRefGoogle ScholarPubMed
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441.Google Scholar
Stokes, G. G. 1966 Mathematical and Physical Papers, vol. 1. Johnson Reprint Corp.Google Scholar
Tai, K., Tomita, A. & Hasegawa, A. 1986 Observation of modulational instability in optical fibres. Phys. Rev. Lett. 56, 135–138.CrossRefGoogle Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.CrossRefGoogle Scholar
Zakharov, V. E. 1967 Instability of self-focusing of light. Sov. J. Exp. Theor. Phys. 24, 455–459.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar