Journal of Fluid Mechanics

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

Peter A. E. M.  Janssen a1
a1 Applied Mathematics, California Institute of Technology, Pasadena, CA 91125 1

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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).

(Published Online April 20 2006)
(Received March 4 1982)


1 Permanent address: K.N.M.I, De Bilt, The Netherlands.