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Evolution of a short surface wave on a very long surface wave of finite amplitude

Published online by Cambridge University Press:  26 April 2006

Mamoun Naciri  and
Chiang C. Mei
Mamoun Naciri
Affiliation:
Ralph M. Parsons Laboratory, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Chiang C. Mei
Affiliation:
Ralph M. Parsons Laboratory, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

To facilitate the theoretical prediction of the evolution of a short gravity wave on a long wave of finite amplitude, we consider a model where the long wave is represented by Gerstner's exact but rotational solution in Lagrangian coordinates. Analytical formulae for the modulation of an infinitesimal irrotational short wave are shown to be qualitatively accurate in comparison with the numerical results by Longuet-Higgins (1987) and with the analytical results by Henyey et al. (1988) for irrotational long waves. Discrepancies are generally of second order in the long-wave steepness, consistent with the vorticity in Gerstner's solution. Weakly nonlinear short waves are shown to be parametrically excited by the long wave over a long time. In particular, multiple bands of modulational instability appear in the parameter space. Numerical calculations of the nonlinear evolution equation show the onset of chaos for sufficiently large parameter $\alpha = \epsilon (k\overline{A})^2/2\Omega/\sigma $, where $\epsilon k\overline{A} $ is the short-wave steepness and (εΩ, σ) the frequency of the (long, short) wave. Furthermore, if the short-wave amplitude A is approximated by a two-mode truncated Fourier series, the evolution equation reduces to a non-autonomous Hamiltonian system. The numerical solutions confirm that the onset of chaos is an inherent feature of the parametrically excited nonlinear system.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 235 , February 1992 , pp. 415 - 452
Copyright
© 1992 Cambridge University Press

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References

Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. S. 1962 Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1818–1939.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430.Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1969 Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A 302, 529–544.Google Scholar
Bryant, P. J. 1973 Periodic waves in shallow water. J. Fluid Mech. 59, 625–644.Google Scholar
Gerstner, F. J. von 1802 Theorie der wellen. Abh. k. boUhm. Ges. Wiss.Google Scholar
Henyey, F. S., Creamer, D. B., Dysthe, K., Schult, R. L. & Wright, J. A. 1988 The energy and action of small waves riding on large waves. J. Fluid Mech. 189, 443–462.Google Scholar
Infeld, E. 1981 Quantitative theory of the Fermi-Pasta-Ulam recurrence in the nonlinear SchroUdinger equation. Phy. Rev. Lett. 47, 717–718.Google Scholar
Iooss, G. & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.
Kharif, C. 1990 Some aspects of the kinematics of short waves over longer gravity waves on deep water. In Water Wave Kinematics (ed. A. ToSrum), pp. 265–279. Klumer.
Kochin, N. E., Kibel, I. A. & Roze, N. V. 1965 Theoretical Hydrodynamics, 2nd edn. Interscience.
Lo, E. & Mei, C. C. 1985 A numerical study of water-wave modulation based on a higher-order nonlinear SchroUinger equation. J. Fluid Mech. 150, 395–416.Google Scholar
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293–306.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 566–583.Google Scholar
Mei, C. C. & uUnluUata, uU. 1973 Harmonic generation in shallow water waves. In Waves on Beaches (ed. R. E. Meyer), pp. 181–202. Academic.
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley-Interscience.
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Stiassnie, M. & Kroszynski, U. I. 1982 Long-time evolution of an unstable water-wave train. J. Fluid Mech. 116, 207–225.Google Scholar
Zhang, J. & Melville, W. K. 1990 Evolution of weakly nonlinear short waves riding on long gravity waves. J. Fluid Mech. 214, 321–346.Google Scholar