On wave groups in a Gaussian sea

In the context of Gaussian waves, if two successive wave crests of amplitude h1h1 and h2h2, respectively, are recorded in time at a fixed point x0x0 then in the limit of h1→∞h1→∞ and h2→∞h2→∞, with probability approaching 1, a wave group has passed closed by the point x0x0 at the apex of its development stage, giving rise to an isolated extreme crest. The two large successive wave crests occur at x0x0 during the initial phase of decay of the wave group and they are lagged in time by T2*+O(h1-1,h2-1), T2* being the abscissa of the second absolute maximum of the time covariance function ψ(T)ψ(T) of the surface displacement.Thus, either an isolated extreme crest event or two consecutive extreme crest events are particular realizations of the space–time evolution of a wave group, in agreement with the theory of quasi determinism of Boccotti [2000. Wave Mechanics for Ocean Engineering. Elsevier, Oxford].This result is of relevant interest for offshore engineering. Firstly, the design of offshore structures resisting to a double wave impact can be based on the wave forces generated by the mechanics of a single wave group. On the other hand, in the context of nonlinear water waves, extreme events and their probability of occurrence can be investigated by studying the nonlinear evolution of a wave group.