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The instability of periodic surface gravity waves

Published online by Cambridge University Press:  17 March 2011

Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA
Mathematics Department, Seattle University, Seattle, WA 98122-1090, USA
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Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this paper, we discuss the stability of periodic travelling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem, modified from that of Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, p. 313), restricted to a one-dimensional surface. Transforming the non-local formulation to a travelling coordinate frame, we obtain a new formulation for the stationary solutions in the travelling reference frame as a single equation for the surface in physical coordinates. We demonstrate that this equation can be used to numerically determine non-trivial travelling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically examine the spectral stability of the periodic travelling wave solutions by extending Fourier–Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin–Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation and are difficult to find numerically. To address this problem, we propose a strategy to estimate a priori the location in the complex plane of the eigenvalues associated with the instability.

Copyright © Cambridge University Press 2011

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Ablowitz, M. & Segur, H. 1981 Solitons and the Inverse Scattering Transform. Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313343.CrossRefGoogle Scholar
Ablowitz, M. J. & Haut, T. S. 2008 Spectral formulation of the two fluid Euler equations with a free interface and long wave reduction. Anal. Appl. 6, 323348.CrossRefGoogle Scholar
Benjamin, T. B. 1967 Instability of periodic wave trains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299, 5979.CrossRefGoogle Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417430.CrossRefGoogle Scholar
Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48, 377385.CrossRefGoogle Scholar
Bohr, H. 1947 Almost Periodic Functions. Chelsea Publishing Company.Google Scholar
Bottman, N. & Deconinck, B. 2009 KdV cnoidal waves are linearly stable. DCDS A 25, 11631180.CrossRefGoogle Scholar
Bridges, T. H. & Mielke, A. 1995 A proof of the Benjamin–Feir instability. Arch. Rat. Mech. Anal. 133, 145198.CrossRefGoogle Scholar
Chandler, G. A. & Graham, I. G. 1993 The computation of water waves modelled by Nekrasov's equation. SIAM J. Numer. Anal. 30 (4), 10411065.CrossRefGoogle Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183230.CrossRefGoogle Scholar
Craig, W. & Nicholls, D. P. 2002 Traveling gravity water waves in two and three dimensions. Eur. J. Mech. B/Fluids 21, 615641.CrossRefGoogle Scholar
Craig, W. & Sternberg, P. 1988 Symmetry of solitary waves. Commun. Part. Diff. Equ. 13, 603633.CrossRefGoogle Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 7383.CrossRefGoogle Scholar
Curtis, C. W. & Deconinck, B. 2010 On the convergence of Hill's method. Maths Comput. 79, 169187.CrossRefGoogle Scholar
Deconinck, B. & Kutz, J. N. 2006 Computing spectra of linear operators using the Floquet–Fourier–Hill method. J. Comput. Phys. 219, 296321.CrossRefGoogle Scholar
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 7379.CrossRefGoogle Scholar
Francius, M. & Kharif, C. 2006 Three-dimensional instabilities of periodic gravity waves in shallow water. J. Fluid Mech. 561, 417437.CrossRefGoogle Scholar
Grimshaw, R. 2005 Nonlinear Waves in Fluids: Recent Advances and Modern Applications. Springer.CrossRefGoogle Scholar
Haragus, M. 2008 Stability of periodic waves for the generalized BBM equation. Rev. Roumaine Maths. Pures Appl. 53, 445463.Google Scholar
Kharif, C. & Ramamonjiarisoa, A. 1990 On the stability of gravity waves on deep water. J. Fluid Mech. 218, 163170.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1978 a The instabilities of gravity waves of finite amplitude in deep water. Part I. Superharmonics. Proc. R. Soc. Lond. A 360, 471488.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1978 b The instabilities of gravity waves of finite amplitude in deep water. Part II. Subharmonics. Proc. R. Soc. Lond. A 360, 489505.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1988 Lagrangian moments and mass transport in Stokes waves. Part 2. Water of finite depth. J. Fluid Mech. 186, 321336.CrossRefGoogle Scholar
MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115125.CrossRefGoogle Scholar
McLean, J. W. 1982 a Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331341.CrossRefGoogle Scholar
McLean, J. W. 1982 b Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315330.CrossRefGoogle Scholar
McLean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1981 Three dimensional instability of finite-amplitude water waves. Phys. Rev. Lett. 46, 817821.CrossRefGoogle Scholar
Nicholls, D. P. 1998 Traveling gravity water waves in two and three dimensions. PhD thesis, Brown University.Google Scholar
Nicholls, D. P. 2009 Spectral data for travelling water waves: singularities and stability. J. Fluid Mech. 625, 339360.CrossRefGoogle Scholar
Nivala, M. & Deconinck, B. 2010 Periodic finite-genus solutions of the KdV equation are orbitally stable. Physica D 239, 11471158.CrossRefGoogle Scholar
Okamoto, H. & Shoji, M. 2001 The Mathematical Theory of Permanent Progressive Water-Waves. World Scientific Publishing.CrossRefGoogle Scholar
Plotnikov, P. I. & Toland, J. F. 2002 The Fourier coefficients of Stokes' waves. In Nonlinear Problems in Mathematical Physics and Related Topics, I, International Mathematical Series (N.Y.), vol. 1, pp. 303315. Kluwer/Plenum.CrossRefGoogle Scholar
Rienecker, M. M. & Fenton, J. D. 1981 A Fourier approximation method for steady water waves. J. Fluid Mech. 104, 119137.CrossRefGoogle Scholar
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 30473055.CrossRefGoogle Scholar
Vanden-Broeck, J. M. 1983 Some new gravity waves in water of finite depth. Phys. Fluids 26, 23852387.CrossRefGoogle Scholar
Whitham, G. B. 1967 Non-linear dispersion of water waves. J. Fluid Mech. 27, 399412.CrossRefGoogle Scholar
Wiegel, R. L. 1982 A presentation of cnoidal wave theory for practical application. J. Fluid Mech. 7, 273286.CrossRefGoogle Scholar
Yuen, H. C. & Lake, B. M. 1980 Instabilities of waves in deep water. Annu. Rev. Fluid Mech. 12, 303334.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190194.CrossRefGoogle Scholar