Nonlinear Wave Groups in Deep Water

Nonlinear wave groups in deep water consist of wave modes for which nonlinear interactions and dispersion are in balance. The evolution equations for the wave modes are derived, and properties of nonlinear wave groups are found from these equations. It is shown that the nonlinear wave groups are linearly unstable to sideband modulations in the sense that the linearized perturbation theory, in providing a good fit over the initial time interval, predicts that the growth of the modulations is exponential. Instead the perturbed wave group is shown to return cyclically to a state close to its initial state. The cyclic recurrence is demonstrated analytically for the simpler wave groups and numerically otherwise. The interactions between nonlinear wave groups of the same and of nearly the same central wavenumbers are calculated.