Linear models of asymptotic probability models describing the statistics of large wave heights in simple wind seas are reviewed, and potential effects of nonlinearities due to bound and free waves are explored. Nonlinearities due to second-order bound waves do not, on average, affect the statistics of large wave heights. This is confirmed by simulations. Wave heights representing approximately the largest 1/3 of waves are described quite accurately by Boccotti's asymptotic distribution (Boccotti, 1989). However, large waves observed in oceanic storm seas and relatively narrowband long-crested waves mechanically generated in wave flumes often display higher-order nonlinearities due to third-order bound and free waves, causing the wave-height statistics to deviate from Boccotti's asymptotic model. Here, Boccotti's model is generalized so as to include the potential effects of such nonlinearities. Analyses and comparisons of waveheight distributions extracted from four oceanic datasets representing measurements gathered during severe oceanic storms indicate that the generalized Boccotti model is able to describe the distribution of large wave heights fairly well and noticeably better than the original Boccotti model.
Since the pioneering works of Rice (1944) and Longuet-Higgins (1952), the Rayleigh probability law has been regarded as the standard model for predicting wave heights and related statistics. In theory, the Rayleigh distribution describes the statistics of linear wave envelopes exactly. A Gram-Charlier series expansion for second-order surface displacements indicates that it is also valid for nonlinear wave envelopes (Tayfun, 2008). However, the probability structure of zerocrossing wave heights depends on the two-time joint statistics of surface displacements or the associated wave envelope, approximately. If accuracy is not of primary concern and one is interested in an overall description of all wave heights, this approximation works reasonably well in most cases, notwithstanding the presence of surface stresses, dissipation and other complications.