A high-order nonlinear envelope equation for gravity waves in finite-depth water

A third-order nonlinear envelope equation is derived for surface waves in finite-depth water by assuming small wave steepness, narrow-band spectrum, and small depth as compared to the modulation length. A generalized Dysthe equation is derived for waves in relatively deep water. In the shallow-water limit, one of the nonlinear dispersive terms vanishes. This limit case is compared with the envelope equation for waves described by the Korteweg-de Vries equation. The critical regime of vanishing nonlinearity in the classical nonlinear Schrödinger equation for water waves (when kh ≈ 1.363) is analyzed. It is shown that the modulational instability threshold shifts toward the shallow-water (long-wavelength) limit with increasing wave intensity.