Introduction

Wave height statistics ( wave height distribution, run length, etc.) of random waves play important roles 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 a few decades, a considerable number of studies have been made on the validity of the Rayleigh distribution. It is commonly known that large wave heights in field do not necessarily obey the Rayleigh distribution. For example, Haring et al. (1976) shows that large wave heights observed in storms are on the order of 10 percent less than those predicted by the Rayleigh distribution. After that, Forristall (1984), and Myrhaug and Kjeldsen (1987) also reported that occurrence probabilities of large wave heights in field are smaller than the predicted value of the Rayleigh distribution.

On the contrary, Yasuda et al. (1992, 1994, 2002) numerically investigated that the third order nonlinear interactions have significant effects on the statistical properties of random wave train. That is, the third order nonlinear solution in deep water increases the occurrence probabilities of large wave heights more than the linear and second order one do. Stansberg (1993) also found the same results in his experimental work. However, there is no theoretical distribution which agrees with the data, although many studies have attempted to establish the wave height distribution without a linear or narrow banded spectrum assumption.

The Rayleigh distribution is put to practical use under the assumption that water surface elevations are regarded as independent stochastic processes, since the nonlinear wave-wave interactions are weak in deep water. Thus, the probability density function of the surface elevation had been assumed to be the Gaussian on the basis of the central-limit theorem. For the statistical point of view, the fourth order moment of the surface elevation is directly related to the third order nonlinear interaction (Longuet-Higgins, 1963). It is therefore necessary to include the effects of the fourth order moment of the surface elevation for the wave height distribution to consider the influences of the third order nonlinear interaction.

The aim of this study is , It is to formulated that a weakly nonlinear wave height distribution for narrow banded random wave train and check the validity of the distribution with experimental and field data.