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Long-time evolution of an unstable water-wave train

Published online by Cambridge University Press:  20 April 2006

Michael Stiassnie  and
Uri I. Kroszynski
Michael Stiassnie
Affiliation:
Department of Civil Engineering and Coastal & Marine Engineering Research Institute, Technion, I.I.T., Technion City, Haifa 32000, Israel
Uri I. Kroszynski
Affiliation:
Department of Civil Engineering and Coastal & Marine Engineering Research Institute, Technion, I.I.T., Technion City, Haifa 32000, Israel
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Abstract

The long-time evolution of an unstable wave train, consisting of a carrier wave and two 'side-band’ components, is investigated analytically. Mathematical expressions, involving Jacobian elliptic functions, for the wave envelope characteristics are derived. The solution yields the dependence of the long-time evolution on the initial disturbance. Of special interest is the simple formula for the modulation-demodulation recurrence period. The latter is shown to yield results in good agreement with those obtained from numerical solutions of the nonlinear Schrödinger equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 116 , March 1982 , pp. 207 - 225
Copyright
© 1982 Cambridge University Press

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References

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
Bryant, P. J. 1979 Nonlinear wave groups in deep water. Stud. Appl. Math. 61, 1–30.Google Scholar
Byrd, P. F. & Friedman, M. D. 1971 Handbook of Elliptic Integrals for Engineers and Scientists. Springer.
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Djordjević, V. D. & Redekopp, L. G. 1978 On the development of packets of surface gravity waves moving over an uneven bottom. Z. angew. Math. Phys. 29, 950–962.Google Scholar
Dysthe, K. B. 1980 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
Gajewski, H. 1978 Über Näherungsverfahren zur Lösung der nichtlinearen Schrödinger-Gleichung. Math. Nachr. 85, 283–302.Google Scholar
Hasimoto, H. & Ono, H. 1972 Nonlinear modulation of gravity waves. J. Phys. Soc. Japan 33, 805–811.Google Scholar
Johnson, R. S. 1977 On the modulation of water waves in the neighbourhood of kh 1363. Proc. R. Soc. Lond. A 357, 131–141.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. 1980 Modulation of the amplitude of steep wind waves. J. Fluid Mech. 99, 705–713.Google Scholar
Lonquet-Higgins, M. S. & Stewart, R. W. 1962 Radiation stress and mass transport in gravity waves with application to ‘surf beats’. J. Fluid Mech. 13, 481–504.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves; a physical discussion, with applications. Deep-Sea Res. 11, 529–562.Google Scholar
Martin, D. U. & Yuen, H. C. 1980 Spreading of energy in solutions of the nonlinear Schrödinger equation. Phys. Fluids 23, 1269–1271.Google Scholar
Stokes, G. G. 1849 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Yuen, H. C. & Ferguson, W. E. 1978 Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids 21, 1275–1278.Google Scholar