Note on a modification to the nonlinear Schrödinger equation for application to deep water waves

Abstract

Abstract The ordinary nonlinear Schrödinger equation for deep water waves, found by perturbation analysis to O(∊3) in the wave-steepness ∊ ═ ka, is shown to compare rather unfavourably with the exact calculations of Longuet-Higgins (1978b) for ∊ > 0.15, say. We show that a significant improvement can be achieved by taking the perturbation analysis one step further O(∊4). The dominant new effect introduced to order ∊4 is the mean flow response to non-uniformities in the radiation stress caused by modulation of a finite amplitude wave.