On wave groups in a Gaussian sea

Abstract

In the context of Gaussian waves, if two successive wave crests of amplitude h 1 and h 2 , respectively, are recorded in time at a fixed point x 0 then in the limit of h 1 → ∞ and h 2 → ∞ , with probability approaching 1, a wave group has passed closed by the point x 0 at the apex of its development stage, giving rise to an isolated extreme crest. The two large successive wave crests occur at x 0 during the initial phase of decay of the wave group and they are lagged in time by T 2 * + O ( h 1 - 1 , h 2 - 1 ) , T 2 * being the abscissa of the second absolute maximum of the time covariance function 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.