Introduction

Since the discovery of the Benjamin-Feir instability in the Stokes wave train, much attention has been paid to the behavior of nonlinear deep-water waves. Last two decades, the instability of the gravity waves have been studied by many researchers using the nonlinear Schrödinger type equations (Caponi et al., 1982; Dysthe, 1979; Yuen and Lake, 1982), mode-coupling equations (Stiassnie and Shemer, 1987), pseudo-spectral methods (Yasuda and Mori, 1997b) and experiments (Su, 1982). However, most of them concentrated on amplitude modulations of the Stokes wave for the purpose of scientific interests. The rest of the studies were related energy transfer of random waves for the purpose of prediction of ocean wave spectra (Masuda, 1980; Hasselmann, 1962). Little is known about the high-order resonant interaction effects on random wave statistics in deep and shallow-water.

Alber (1978) mathematically demonstrated that randomness of waves makes wave trains stabilized and Yuen and Ferguson (1978) stated that the instability is confined within an initially unstable range and become weak if the spectral bandwidth becomes broad. However, it is not clear that how stability and instability of the gravity wave are connected between the Stokes and random waves having broad band spectra. It is found that relatively broad banded spectrum waves can transfer the Fourier mode energy in deep-water (Yasuda et al., 1992; Yasuda and Mori, 1994).

On the other hand, a freak wave becomes an important topic recently and is sometimes featured by a single and steep crest causing severe damage to offshore structures and ships. There is no doubt on the occurrence of a freak wave from many reports (e.g. Yasuda and Mori, 1997a) and the mechanisms and detailed statistical properties of the freak wave are getting clear (Mori et al., 2000; Yasuda et al., 1997). The state of the art on the freak wave was summarized at NATO Advanced Research Workshop the last decade. It was concluded that both of nonlinearity and directionality effects are primary possible causes of the freak wave (Dean, 1990). Experimental studies demonstrate that the freak wave like wave can be generated in two-dimensional wave flume without current, refraction and diffraction (Stansberg, 1990). Numerical studies also indicates that the freak wave having a single and steep crest can be generated by the third-order nonlinear interaction in deep-water (Yasuda et al., 1992). It is, however, not clear what influences statistical properties, occurrence probabilities and effects of spectrum shape and water depth have on the instability generated freak wave.

The purpose of this study is to investigate influence of spectrum band width and water depth on the stability of random waves solving highly nonlinear equations of a potential flow by the pseudo spectrum method. On the basis of the numerical results, importance of the high-order nonlinearities is evaluated in comparison with the second-order solution. Finally it is demonstrated the generation mechanisms of the freak wave in unidirectional wave train.