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

The ordinary nonlinear Schrodinger equation for deep-water waves (found by a perturbation analysis to 0(€3 ) in the wave steepness €) compares unfavourably with the exact calculations of Longuet-Higgins (1978) for € > 0·10. Dysthe (1979) showed that a significant improvement is found by taking the perturbation analysis one step further to 0(€ ). 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).