Conclusion

For a narrow-band, random wave train we have shown that the kurtosis of the surface elevation is mainly determined by resonant and non-resonant wave-wave interactions, while bound waves only give a small contribution. Thus, the kurtosis and related high-order cumulants can be evaluated on the basis of ()'s work. Second, we have shown that for a narrow-band wave train the wave height and the maximum wave height probability distribution depends to a good approximation on the wave variance and the kurtosis. As a consequence it is possible to formulate the freak wave occurrence probability in terms of the kurtosis and the number of waves in a time series. From the comparison with laboratory and field data, we conclude the following:

• The second order cross-cumulant $kappa_{22}$ is 1/3 of the fourth cumulant, $kappa_{40}$, of the surface elevation.
• The weakly non-Gaussian theory shows the dependence of the expected maximum wave height on kurtosis, which is supported by the observed data.
• The occurrence probability of freak waves is significantly enhanced by the kurtosis increase caused by four-wave interactions.

In order to check the validity of the approach developed here, in particular the dependence of freak wave occurrence on the kurtosis and the number of waves, systematic and continuous field measurements of freak waves including wave spectra and nonlinear statistics will be critically required.