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Resonance theory of water waves in the long-wave limit

Published online by Cambridge University Press:  28 March 2013

Takeshi Kataoka
Takeshi Kataoka*
Affiliation:
Department of Mechanical Engineering, Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
*
†Email address for correspondence: kataoka@mech.kobe-u.ac.jp
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Abstract

The instability due to resonant interactions of finite-amplitude water waves is examined in the long-wave limit. In contrast to the well-known case of a small-amplitude limit in which the resonance is considered for a flat surface, here we consider a periodic approximate of the finite-amplitude solitary wave which is the long-wave limit of the periodic wave. The resonance conditions for the corresponding perturbations yield a new family of resonance curves that are totally different from those of the small-amplitude limit obtained by Phillips and Mclean. Under these resonance conditions, we conduct a systematic asymptotic analysis for small wavenumbers to obtain the growth rates of the perturbations explicitly and clarify whether each resonance curve is associated with instability. These results are verified numerically by showing that the instability bands for finite-amplitude periodic waves in shallow water are located along these unstable resonance curves.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 461 - 495
Copyright
©2013 Cambridge University Press

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References

Amick, C. J. & Toland, J. F. 1981a On solitary water-waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 9–95.Google Scholar
Amick, C. J. & Toland, J. F. 1981b On periodic water-waves and their convergence to solitary waves in the long-wave limit. Phil. Trans. R. Soc. Lond. A 303, 633–669.Google Scholar
Benjamin, T. B. 1967 Instability of periodic wave trains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299, 59–75.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Bridges, T. J. 2001 Transverse instability of solitary-wave states of the water-wave problem. J. Fluid Mech. 439, 255–278.Google Scholar
Bridges, T. J. & Dias, F. 2007 Enhancement of the Benjamin–Feir instability with dissipation. Phys. Fluids 19, 104104.Google Scholar
Byatt-Smith, J. G. & Longuet-Higgins, M. S. 1976 On the speed and profile of steep solitary waves. Proc. R. Soc. Lond. A 350, 175–189.Google Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.Google Scholar
Crawford, D., Lake, B. M., Saffman, P. G. & Yuen, H. C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177–191.Google Scholar
Francius, M. & Kharif, C. 2006 Three-dimensional instabilities of periodic gravity waves in shallow water. J. Fluid Mech. 561, 417–437.Google Scholar
Fructus, D., Kharif, C., Francius, M., Kristiansen, O., Clamond, D. & Grue, J. 2005 Dynamics of crescent water wave patterns. J. Fluid Mech. 537, 155–186.Google Scholar
Garabedian, P. 1965 Surface waves of finite depth. J. d’Anal. Math. 14, 161–169.Google Scholar
Hunter, J. K. & Vanden-Broeck, J. M. 1983 Accurate computations for steep solitary waves. J. Fluid Mech. 136, 63–71.Google Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in a weakly dispersing medium. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kataoka, T. 2006 On the superharmonic instability of surface gravity waves on fluid of finite depth. J. Fluid Mech. 547, 175–184.Google Scholar
Kataoka, T. 2008 Transverse instability of interfacial solitary waves. J. Fluid Mech. 611, 255–282.Google Scholar
Kataoka, T. 2010 Transverse instability of surface solitary waves. Part 2. Numerical linear stability analysis. J. Fluid Mech. 657, 126–170.Google Scholar
Kataoka, T. & Tsutahara, M. 2004 Transverse instability of surface solitary waves. J. Fluid Mech. 512, 211–221.Google Scholar
Keady, G. & Norbury, J. 1978 On the existence theory for irrotational water waves. Math. Proc. Camb. Phil. Soc. 83, 137–157.Google Scholar
Kharif, C. & Ramamonjiarisoa, A. 1988 Deep-water gravity wave instabilities at large steepness. Phys. Fluids 31, 1286–1288.Google Scholar
Kharif, C. & Ramamonjiarisoa, A. 1990 On the stability of gravity waves on deep water. J. Fluid Mech. 218, 163–170.CrossRefGoogle Scholar
Krasovskii, Yu. P. 1961 On the theory of steady state waves of large amplitude. U.S.S.R. Comput. Maths. and Math. Phys. 1, 996–1018.Google Scholar
Kuznetsov, E. A., Spektor, M. D. & Fal’kovich, G. E. 1984 On the stability of nonlinear waves in integrable models. Physica D 10, 379–386.Google 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.Google Scholar
Longuet-Higgins, M. S. 1974 On mass, momentum, energy, and circulation of a solitary wave. Proc. R. Soc. A 337, 1–13.Google Scholar
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. & Tanaka, M. 1997 On the crest instabilities of steep surface waves. J. Fluid Mech. 336, 51–68.Google Scholar
Mackay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
McCowan, J. 1891 On the solitary wave. Phil. Mag. (Ser. 5) 32, 45–58.Google Scholar
Mclean, J. W. 1982a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
Mclean, J. W. 1982b Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.Google Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Phillips, O. M. 1967 Theoretical and experimental studies of gravity wave interactions. Proc. R. Soc. Lond. A 299, 104–119.Google Scholar
Saffman, P. G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169–174.Google Scholar
Spektor, M. D. 1988 Stability of conoidal waves in media with positive and negative dispersion. Sov. Phys. JETP 67, 104–112.Google Scholar
Starr, V. T. 1947 Momentum and energy integrals for gravity waves of finite height. J. Mar. Res. 16, 175–193.Google Scholar
Su, M.-Y. 1982 Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric wave patterns. J. Fluid Mech. 124, 73–108.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
Su, M. -Y & Green, A. W. 1984 Coupled two- and three-dimensional instabilities of surface gravity waves. Phys. Fluids 27, 2595–2597.Google Scholar
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 3047–3055.CrossRefGoogle Scholar
Tanaka, M. 1985 The stability of steep gravity waves. Part 2. J. Fluid Mech. 156, 281–289.Google Scholar
Tanaka, M. 1986 The stability of solitary waves. Phys. Fluids 29, 650–655.CrossRefGoogle Scholar
Tanaka, M., Dold, J. W., Lewy, M. & Peregrine, D. H. 1987 Instability and breaking of a solitary wave. J. Fluid Mech. 185, 235–248.Google Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.Google Scholar
Wilkinson, J. H. 1965 The Algebraic Eigenvalue Problem. Clarendon.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