## 4 Numerical Results and Analysis
### Effect of Wave Amplitudes on Foil Performance  
According to Equation (14), a given wave amplitude \(a\) denotes a certain velocity of water particles in waves. This motion disturbs the flow field of the nearby foil, thus impacting its hydrodynamic performance. The thrust force coefficient \(C_{T}\) and the wave power utilization coefficient \(C_{pr}\) of the foil are shown in Figure 8, relative to the non-dimensional wave amplitude \(a/c\), where \"-infinite\" means the foil is located in the infinite flow field, and \"-calm water\" means the foil is located under the calm water surface with the same submergence as in waves. As no wave motion exists in the \"infinite\" and \"calm water\" flow fields, corresponding \(C_{T}\) and \(C_{pr}\) are constant.  
As seen in Figure 8, \(C_{T}\) and \(C_{pr}\) ascend with the increasing wave amplitude \(a/c\), and the thrust force of the foil in waves is clearly larger than that in infinite flow fields and under calm water. Compared to calm water, the thrust force of the flexible and rigid foils in waves can be increased by 33.5% and 44.0% at \(a/c=0.1\), respectively, and by 142.3% and 168.0% at \(a/c=0.4\), respectively. Meanwhile the force of flexible foil is 13.5% and 16.4% higher than that of rigid foil at \(a/c=0.1\) and 0.4, respectively. Therefore, the wave motions and the flexible deformation are beneficial in obtaining a higher thrust force. The trend of \(C_{T}\) curves in Figure 8 is similar to that of the heaving amplitude of the foil in reference [38]. This phenomenon can be explained as maintaining a constant motion period, and increasing wave amplitudes will increase the velocity of water particles near the foil. Therefore, larger wave amplitudes or foil heaving amplitude can increase the moving velocity of the foil related to its surrounding water particles, and the thrust force can be improved correspondingly. In addition, flexible deformation can also cause this phenomenon, and thus result in higher thrust force than that of rigid foil.  
The wave energy utilization power coefficient \(C_{pr}\) of the flexible foil is also higher than that of the rigid foil, but its increment gets smaller with increasing wave amplitudes, specifically, 38.0% and 24.2% higher than the rigid foil at \(a/c=0.1\) and 0.4, respectively. This means the contribution of the flexible deformation to improving \(C_{pr}\) reduces with decreasing wave amplitudes. After obtaining the  
Figure 8: Average thrust and wave power utilization coefficients for the foil versus non-dimensional wave amplitudes.  
thrust force and the wave power utilization coefficients, the propulsive efficiency and the percentage of wave energy recovery can be calculated, as shown in Figure 9.  
Changes in wave amplitudes have less influence on foil propulsive efficiency \(\eta\), as shown in Figure 9. From Figure 8 and Equation (9), in waves, increasing \(a/c\) not only increases the thrust force coefficient, but also the lateral force and moment coefficients, which means the input and output power of the foil ascend and then change less in \(\eta\). However, these values in waves are still larger than under calm water. Although the propulsive efficiency of the foil under a free surface is smaller than that in an infinite flow field, the flexible deformation can reduce this decreasing trend, and the flexible foil \(\eta\) in waves is higher than the rigid foil \(\eta\) in the infinite flow fields. However, increasing wave amplitudes reduces the percentage of wave energy recovery \(\eta_{r}\). To summarize, a higher sea state is beneficial to improving the thrust force of the foil, but not wave energy utilization ability. The flexible deformation can produce a larger thrust force, and have fewer adverse effects of the free surface on reducing the propulsive efficiency of the foil.