Statistical properties of wave groups in nonlinear random waves of finite bandwidth

The theoretical treatment of the nonlinear effect on the formation of a wave group in a random wave field of finite bandwidth is an overdue task hampered by the complicated form of nonlinear random waves. In this study, the mean number of waves in a group, the mean number of waves in a high run and the phase distribution in nonlinear random waves of finite bandwidth are derived. The study employs the complex envelope and total phase function, random variable transformation technique and perturbation method. It turns out that the phase distribution is modified significantly by nonlinearities, showing a systematic excess of values near the mean phase and the corresponding symmetrical deficiency on both sides away from the mean. In the case of the threshold crossing rate, it turns out that it reaches its maximum just below the mean water level rather than at zero. A considerable amount of the threshold crossing rate is shifted toward the larger values of water-surface elevation as nonlinearity increases markedly. Further, the mean waves in a high run associated with highly nonlinear waves are shown to have larger values than their weakly nonlinear counterparts when the reference level is relatively small. A similar trend can also be found in the average number of waves in a group.