## 4 Numerical Results and Analysis
### Effect of Wavelengths on the Foil Performance  
In this section, the effects of wavelengths on the propulsive performance are discussed. Assuming the motion frequency of the foil is equal to its encounter frequency with waves, it should be illustrated that different wavelengths mean different motion periods for the foil in infinite flow fields and under calm water. The thrust force coefficient \(C_{T}\) and the wave power utilization coefficient \(C_{pr}\) with the non-dimensional wavelength \(\lambda/c\) can then be calculated, as shown in Figure 10.  
The thrust force descends rapidly with increasing wavelength \(\lambda/c\), because longer wavelengths mean longer motion periods for the foil, so the velocity of the foil, relative to the water particles, will descend in this situation with a given heating amplitude. As a result, the thrust force of the foil will become smaller. However, as wave motion benefits the thrust force generation, the descending rate of the foil's thrust force in waves is slower than in infinite flow fields and under calm water. Meanwhile, the wave power utilization coefficient \(C_{pr}\) of the rigid and flexible foils first ascends and then descends with increasing wavelength \(\lambda/c\). According to its expression, \(C_{pr}\) reflects the difference in the thrust force of the foil between waves and calm water. Therefore, as seen in Figure 10, with larger or smaller wavelengths, the wave effect on improving foil thrust force is limited. In addition, when \(\lambda/c\approx 54\), \(C_{pr}\) of the flexible and rigid foil can reach peak values of 1.78 and 1.44, respectively.  
Figure 9: The percentage of wave energy recovery \(\eta_{r}\) and propulsive efficiency \(\eta\) for the foil versus non-dimensional wave amplitudes.  
The propulsive efficiency \(\eta\) ascends with increasing wavelengths, as shown in Figure 11. For the rigid foil, \(\eta\) in the infinite flow fields is always larger than under the free surface, and the difference gets larger with increasing wavelengths. However, for the flexible foil, the difference of \(\eta\) between the foil in waves and in infinite flow fields is not significant, and the former will be larger than the latter when \(\lambda/c>130\). At an arbitrary wavelength, \(\eta\) of the flexible foil is always larger than that of the rigid foil, and is larger by 48.2% and 39.7% when \(\lambda/c=20\) and 140, respectively. Differing from the trend of \(\eta\) monotonically increasing with the rigid foil, \(\eta\) of the flexible foil can reach a peak value when \(\lambda/c=100\). Combined with Figure 10, it can then be seen that the flexible deformation contributes to a higher thrust force and propulsive efficiency in waves than in infinite flow fields or under calm water.  
Similarly, the percentage of wave energy recovery \(\eta_{r}\) of the rigid foil descends with increasing \(\lambda/c\), but for the flexible foil, the trend is to first ascend and then descend, and its maximum value appears at \(\lambda/c=60\). Meanwhile, \(\eta_{r}\) of the rigid foil is slightly larger than that of the flexible foil at smaller wavelengths, by 16.5% at \(\lambda/c=20\). However, when \(\lambda/c=35\), they are approximately equal. Nevertheless, when \(\lambda/c>35\), \(\eta_{r}\) of the flexible foil will be larger than that of the rigid foil, by 31.4% at \(\lambda/c=140\). Therefore, we can conclude that the effect of the flexible deformation on improving the percentage of wave energy recovery works with certain wavelengths.  
Figure 11: The percentage of wave energy recovery \(\eta_{r}\) and propulsive efficiency \(\eta\) for the foil versus non-dimensional wavelengths.  
Figure 10: The average thrust and wave power utilization coefficients for the foil versus non-dimensional wavelengths.