Results and Discussions

The theory of maximum wave height distribution is generally difficult to verify by observed data, quantitatively. This is because it depends on the number of waves. Moreover, Eq.(12) also depends on $\mu _4$ . This means that the observed data have to be classified both the number of waves and the value of kurtosis. This section attempts to verify the theory by observed data, although the available data is insufficient.

Wave data to be analyzed here were obtained at a location 3 km off the Yura fishery harbor facing the Sea of Japan. The observations were made during the period from September 1986 to July 1990 by the National Maritime Research Institute (formerly the Ship Research Institute), the Ministry of Land, Infrastructure and Transport, Japan. Temporal sea surface elevations were measured with ultrasonic-type wave gages installed at three points at a water depth of 43 m. The measurements were continuous recordings over 20 hours when sea conditions became stormy in winter. To obtain sufficient data having similar number of waves and the value of kurtosis, a basic stationarity test (e.g. Bendat and Piersol, 1971) was used to examine the parameters $\eta _{rms}$ and $H_{1/3}$ in order to select a data set that passed the stationarity test from the over 20 hours of continuous recordings. Then, the continuous data set was divided into 30 min recorded units, from which wave statistics were calculated at intervals of 30 min. Freak waves were observed when the maximum wave height $H_{max}$ exceeded 10 m during winter storm conditions where the mean wind speeds exceeded 13 m/s and were wind waves with a well-established equilibrium range on the high frequency side of the spectrum, $f^{-4}$ . General wave statistics of the observed waves were presented by Yoshimoto and Kato (1992) and wind field statistics including the weather conditions were reported by Mori et al. (2002,2000).

**Table 1:** Comparison of theories of freak wave occurrence probability in the Sea of Japan
Date	period(hr))	$\bar{N}$		$P_{obs}$ (%)	$P_{ER}$ (%)	$P_{R}$ (%)
1987/12/24	15:00	221.4	3	10.0	13.51	7.01
1988/01/09	11:30	249.3	4	19.0	15.04	7.83
1988/02/02	13:00	200.2	3	11.5	12.37	6.39
1988/12/14	14:30	212.0	2	6.7	13.05	6.39
1990/01/25	11:30	230.9	1	4.3	14.08	7.31

Table 1 shows extreme wave statistics of the data set. The first column in the table indicates the observation date, the second column is the total length of observation, $\bar{N}$ is mean number of waves in each divided data set, and the maximum wave heights were 10-15 m.

in the table indicates the number of freak waves, and $P_{obs}$ , $P_{ER}$ and $P_{R}$ are the freak wave occurrence probabilities by the observed, Eq.(12) and the Rayleigh theory, respectively. The value of $\mu _4$ is used the mean value of each data set. The observed waves had weak nonlinearity, as evidenced from the values of skewness $\mu _3$ =0.25-0.4 and kurtosis $\mu _4$ =3.1-3.4. The occurrence probabilities of freak waves from the top three in Table 1 show better agreement with Eq.(12) than the Rayleigh theory on balance. However, the rest of two are close to the Rayleigh based theory. The occurrence probability of freak wave predicted by Eq.(12) gives the maximum occurrence, although the number of runs is insufficient to compare the theory, quantitatively. However, there is no doubt that on the occurrence of a freak wave and that the freak wave sometimes appears more frequently than expected by the Rayleigh theory. The further verifications of the nonlinear dependence of freak wave appearance will be required.