Introduction

Wave height statistics (e.g. wave height distribution, run length and so on) of random waves are important factors in designing coastal and ocean structures. The Rayleigh distribution is regarded as the distribution of wave heights in stochastic processes with a linear and narrow banded spectrum. Over the last decades, a considerable number of studies have been conducted on the validity of the Rayleigh distribution and have shown that large wave heights in the field do not necessarily obey the Rayleigh distribution. For example, Haring et al. (1976) shows that large wave heights observed in storms are in the order of 10% less than those predicted by the Rayleigh distribution. In addition, Forristall (1984), and Myrhaug and Kjeldsen (1987) reported that the occurrence probabilities of large wave heights in the field are always smaller than the predicted value of the Rayleigh distribution. However, Mori (2002b,2002a) found that the fourth order moment of the surface elevation, kurtosis, enhances the occurrence probability of large amplitude waves in deep-water statistically and numerically. Janssen (2002) investigated the similar results theoretically.

Freak waves are sometimes characterized by a single, steep crest and giving severe damage to offshore structures and ships. The occurrence of freak waves, their mechanism and detailed dynamic properties are becoming clearer (e.g. Mori et al., 2002; Haver, 2001). The state of the art on freak waves was recently summarized at the Rogue Wave Conference, held in December 2000 (Olagnon and Athanassoulis, 2000). It was concluded that nonlinear wave-wave interaction enhances freak wave appearance and is the primary possible cause of freak wave in a general wave field. Numerical and experimental studies have demonstrated that freak waves like waves can be generated frequently in a two-dimensional wave flume without current, refraction or diffraction (Stansberg, 1990; Yasuda et al., 1992). Moreover, a numerical study indicates that a freak wave having a single, steep crest can be generated by third-order nonlinear interactions in deep-water (Yasuda et al., 1992). However, the occurrence probabilities of freak waves in the ocean remains unclear and few studies have investigated this phenomena (e.g. Yasuda and Mori, 1997).

This study formulates a maximum wave height distribution using a non-Gaussian theory for a unidirectional wave train, expanding on previous our work (Mori and Yasuda, 2002b). Using the definition of a freak wave as $H_{max}/H_{1/3}\ge2$, the probability of freak wave occurrence is discussed, compared with the Rayleigh (Gaussian) wave theory, and then the theory is applied to field data.