This paper concerns the short-term crest-height statistics arising in deep water sea states characterised by realistic (JONSWAP) frequency spectra. The relevant data has been generated using direct numerical simulations based upon the Zakharov equation. This provides a Hamiltonian formulation of the surface water wave evolution for which it is possible to isolate the contributions arising at increasing orders of nonlinearity. Specifically, it allows the quantitative comparison between the two-wave bound interactions and the three-wave resonant interactions; the latter being important because they are not commonly adopted in design practice. Both long-crested and short-crested seas have been considered and the role of directional spreading specifically addressed. Further comparisons with fully nonlinear boundary element (BEM) calculations and available laboratory observations provide important validation and allow the following conclusions to be drawn:
In the steepest sea states, the third-order resonant interactions can be responsible for local and rapid spectral change and are crucial in defining the relevant crest height statistics.
The wave nonlinearities arising above third-order have little effect.
The limiting effects of wave breaking are important, particularly in the tail of the distribution.
The bandwidth of the energy spectrum affects the probability of occurrence of extreme waves.
The paper addresses each of these points and emphasises their practical significance for engineering design.
Crest-height statistics are of practical importance to offshore engineers as they constitute a vital input to design calculations of offshore structures. For example, they are used to determine extreme crest heights and thus the deck elevation of fixed structures. Since wave loading on slender structures is proportional to the crest elevation squared, small errors in crest-height are magnified in load calculations. As such, the structural reliability of offshore structures depends on the correct definition of crest-height statistics and their associated errors.
Presently, the most commonly used crest-height probability distributions fall short of providing a full description of the observed data. Probability distributions from a linear description of the sea surface are nonconservative, whereas probability distributions at second-order in steepness offer a significant improvement. Matching analytical theories to experimental measurements of extreme waves will help explain the origin of departures from second-order distributions.
