We have developed an approximate solution to Zakharov’s equations for the evolution of ocean surface waves. We have applied it with a spectral response surface method to estimate the probability of exceedance of crest elevation in random seas. The method avoids time-marching and allows probabilities to be estimated on practical time scales. The results indicate that quartet resonance can produce very large amplification of extreme crests in uni-directional seas. However, the effect is very much smaller in spread seas. Although the solution is approximate it provides a great deal of physical insight into the mechanisms by which a freak wave might develop and explains the difference in results between uni-directional and spread seas.

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