Modulational development of nonlinear gravity-wave groups

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Abstract
We present observations of the development of nonlinear surface gravity-wave groups. We calculate the amplitude and phase modulations using Hilbert-transform techniques. With increasing propagation distance and wave steepness, the phase modulation develops local phase reversals whose locations correspond to amplitude minima or nodes. The concomitant frequency modulation develops jumps or discontinuities. The observations are compared with recent similar results for wavetrains. The observations are modelled numerically using the cubic nonlinear Schrödinger equation. The motivation is twofold: to examine quantitatively the evolution of phase as well as amplitude modulation and to test the inviscid predictions for the asymptotic behaviour of groups versus long-time observations. Although dissipation rules out recurrence, there is a long-time coherence of the groups. The phase modulation is found to distinguish between dispersive and soliton behaviour.

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