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

Janssen, Peter A. E. M. (1983) On a fourth-order envelope equation for deep-water waves. Journal of Fluid Mechanics, 126 . pp. 1-11. ISSN 0022-1120. doi:10.1017/S0022112083000014.

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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).

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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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Official Citation:Peter A. E. M. Janssen (1983). On a fourth-order envelope equation for deep-water waves. Journal of Fluid Mechanics, 126 , pp 1-11 doi:10.1017/S0022112083000014
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:32391
Deposited On:13 Jul 2012 16:57
Last Modified:09 Nov 2021 21:27

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