Quantitative prediction of the freak wave occurrence probability in co-propagating mixed waves

The bimodal sea state, which occupies around 15%–25% of the unidirectional sea states suggested in the critical operational environment, is one of the harshest wave conditions in the field of ocean engineering. Quantitative prediction of the freak wave occurrence probability in such co-propagating mixed wave environments is crucial for marine structure designs. Based on our previous study of statistics of long-crested extreme waves, here the quantitative relation between the occurrence probability of freak waves and sea-state parameters is further investigated under bimodal spectral sea-states with equivalent energy but different energy distribution in the frequency band. Unlike the unimodal sea states, the nonlinear interaction process of the co-propagating mixed waves is challenging to theoretically obtain the maximum kurtosis versus their characteristics. In this study, we develop the relation of the maximum kurtosis between single-peak and bimodal wave trains by introducing energy distribution and frequency separation with an empirical formula. A quantitative relation between the occurrence probability of freak waves and kurtosis in co-propagating mixed waves is obtained. It enables the fast and accurate prediction of the occurrence probability of freak waves in the operational area for a given bimodal sea state. This work provides a conservative reference threshold for offshore engineering and ship navigation, as well as ideas and methods for seeking a universal result applicable for practical application.