Short wave groups in deep water

Waves on the water surface propagate typically in wave groups, each group consisting of a small number of waves. The envelopes of the wave groups change, in general, as the groups propagate. Particular envelope shapes remain constant for certain ranges of the group height to wavelength ratio and the group length to wavelength ratio k₀. Envelopes for groups containing a large number of waves (k₀ »1) of small amplitude ( ≪ 1) are modelled by the cubic Schrödinger equation. Short periodic groups of permanent envelope exist only for larger values of . A numerical method is described for obtaining solutions of the nonlinear water wave equations representing periodic wave groups of permanent envelope without small or large k₀ assumptions. The method, which is based on the fast Fourier transform technique, has applications elsewhere in nonlinear wave problems. Examples of short wave groups of permanent envelope are presented