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Published January 1983 | Published
Journal Article Open

On a fourth-order envelope equation for deep-water waves

Abstract

The ordinary nonlinear Schrödinger equation for deep-water waves (found by a perturbation analysis to O(ε^3) in the wave steepness ε) compares unfavourably with the exact calculations of Longuet-Higgins (1978) for ε > 0·10. Dysthe (1979) showed that a significant improvement is found by taking the perturbation analysis one step further to O(ε^4). One of the dominant new effects is the wave-induced mean flow. We elaborate the Dysthe approach by investigating the effect of the wave-induced flow on the long-time behaviour of the Benjamin–Feir instability. The occurrence of a wave-induced flow may give rise to a Doppler shift in the frequency of the carrier wave and therefore could explain the observed down-shift in experiment (Lake et al. 1977). However, we present arguments why this is not a proper explanation. Finally, we apply the Dysthe equations to a homogeneous random field of gravity waves and obtain the nonlinear energy-transfer function recently found by Dungey & Hui (1979).

Additional Information

© 1983 Cambridge University Press. Received 4 March 1982. Published online: 20 April 2006. The author is pleased to acknowledge useful discussions with M. J. McGuinness, P. G. Saffman and G. B. Whitham.

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