Theoretical distributions for describing the crest-to-trough heights of linear waves are reviewed briefly. To explore the effects of nonlinearities, these and two approximations that follow from the Tayfun [28] model are generalized to nonlinear waves via the second-order quasi-deterministic model of Fedele and Arena [7]. The potential utility of Gram-Charlier type approximations [17, 18, 29, 31] in representing the statistics of nonlinear wave heights is also explored. All models and a fifth-order Stokes-Rayleigh type model recently proposed by Dawson [5] are compared with linear and nonlinear waves simulated from the JONSWAP spectrum representative of long-crested extreme seas, and also with observational data gathered during two severe storms in the North Sea. The results indicate first that nonlinearities do not appear to affect the crest-to-trough wave heights significantly. Most models and their nonlinear extensions yield similar and reasonable predictions of the data trends observed. The present comparisons do not confirm the efficacy of Gram-Charlier type approximations in modeling the statistics of unusually large wave heights.

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