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  • Journal of Fluid Mechanics, Volume 126
  • January 1983, pp. 1-11

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

  • Peter A. E. M. Janssen (a1)
  • DOI: http://dx.doi.org/10.1017/S0022112083000014
  • Published online: 01 April 2006
Abstract

The ordinary nonlinear Schrödinger equation for deep-water waves (found by a perturbation analysis to O3) 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 O4). 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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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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