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

Abstract

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