## 4 Wave-Ship Interaction and Green-Water Loads
### Generation of the Freak Wave  
The target freak wave was similar to the one recorded at Yura fishery in the Japanese sea area by [PERSON] et al. [48]. The measured freak wave had a height of 11.2 m, crest height of 7.4 m, and significant wave height of 4.6 m, occurring in water with a depth of 43 m. Considering that the container ship model used in present study possesses dimensions of 398 m in length, 57 m in width, and 34 m in height, the freak wave was scaled with a scaling factor of 1:3.3. A wave surface elevation monitoring point was set 150 m away from the wave-making boundary to record the wave surface elevation at that location within the numerical wave tank. Subsequently, the free surface elevation of the freak wave was transformed into the displacement of the Piston. As mentioned by [PERSON] et al. [49], the displacement \(S(t)\) of the Piston wave-maker for freak waves can be expressed as:  
\[S(t)=\frac{\sum\limits_{i=1}^{N_{1}}a_{1i}\cos(o_{1i}x-w_{1i}t+\theta_{1i})}{W _{1i}}+\frac{\sum\limits_{i=1}^{N_{2}}a_{2i}\cos[o_{2i}(x-x_{c})-w_{2i}(t-t_{c} )]}{W_{2i}} \tag{14}\]  
\[W_{i}=\frac{4\sinh^{2}(o_{i}d)}{\sinh^{2}(o_{i}d)+2o_{i}d} \tag{15}\]  
where \(W_{i}\) is the transfer function associated with the \(i\)th component of the propagating wave, \(d\) is the water depth, while the other variables have the same definitions as in Equation (1). Additionally, the theoretical value of the surface elevation at this location was calculated using Equation (1) as a reference for comparison with the numerical simulation results.  
Figure 5: Layout of the numerical wave tank: (**a**) Front view, (**b**) top view. The plot is not to scale.  
Figure 6: A sketch of the container ship model.  
Figure 7a displays the free surface elevations of freak waves generated using three different grid groups: coarse, medium, and fine. The coarse grid had a size of 1 m \(\times\) 1 m \(\times\) 1 m, the medium grid had a size of 0.5 m \(\times\) 0.5 m \(\times\) 0.5 m, and the fine grid had a size of 0.25 m \(\times\) 0.25 m \(\times\) 0.25 m. Figure 7b gives a comparison between the theoretical and medium-grid results. As can be observed, the numerical simulation results are in excellent agreement with the theoretical values. Despite some small differences in the random waves used as background waves, the heights of the freak-wave crest and peak are highly consistent.  
[PERSON] and [PERSON] [50] put forward \(\alpha\), \(\beta_{1}\), \(\beta_{2}\) and \(\mu\) to describe the features of freak waves. These characteristic parameters are defined as follows: (1) \(\alpha=H_{f}/H_{a}\); (2) \(\beta_{1}=H_{f}/H_{f-1}\); (3) \(\beta_{2}=H_{f}/H_{f+1}\); (4) \(\mu=\eta_{f}/H_{f}\). Here, \(H_{f}\) is the freak wave's height, \(H_{f+1}\) and \(H_{f-1}\) are the wave heights of adjacent waves before and after the freak wave, \(H_{a}\) is the significant wave height of the wave train, and \(\eta_{f}\) is the peak height corresponding to \(H_{f}\). As shown in Table 2, it can be observed that the characteristic parameters of the generated freak wave largely agree with those of the measured freak wave, indicating the effectiveness of the numerical wave tank and freak-wave model in generating such waves.