# Design and Study of a New Wave Actuator for a Boat
## 3 Computer Simulation of the Ship Thrusting Performance of the Wave Actuator
### Simulation Results and Discussions
The calculated results for \(P_{c}\) (Equation (31)), \(P\) (Equation (30)), and \(Mean\ V_{drag}\) (Equation (29)) are presented corresponding to each cycle. At the bottom of the table, the \(Cruise\ speed=1/M\sum\limits_{i=1}^{M}\left(Mean\ V_{drag}\right)_{i}\), which represents the average speed achieved over 30 s of simulation. Here, \(M\) denotes the total number of wave cycles considered in the simulation, and \(\left(Mean\ V_{drag}\right)_{i}\) is the average velocity during the i-th wave cycle. The table also presents the corresponding efficiency values \(\eta\) as defined in Equation (32). A time-domain simulation is conducted to evaluate the wave actuator's performance under regular waves, while considering velocity and displacement constraints. The MATLAB code is employed to examine the interaction between the simulated boat motion and wave motion, as well as to analyze the wave force exerted on the boat over time. In Case 1, the boat had a damping coefficient of 20, a spring coefficient of 20, an initial speed of 2.06 m/s, and a wave angle of 180\({}^{\circ}\). The simulation included displacement and velocity limits, suggesting that increasing wave amplitudes or draft size could improve wave energy conversion efficiency as depicted in Figure 11 and Table 3. A higher draft makes the boat more resistant to undulation, and changes to the boat's initial velocity limit had little impact. In Case 2, from Figure 12 and Table 4, it can be observed that the frequency for calculating the total force on the boat is adjusted, and similar to Case 3 (Table 5), reducing wave frequency decreased wave energy conversion. The simulation results for Case 4 did not fully meet expectations, as shown in Table 6. The simulation results for Case 4 did not fully meet expectations, as shown in Table 6. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Wave Parameters** & **Unit** & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline Frequency, \(\omega\) & Hz & 0.4 & 0.3 & 0.2 & 0.2 \\ Wave height, \(wh\) & m & 1.25 & 1.25 & 1.25 & 1.25 \\ Wave angle & deg & 180 & 180 & 180 & 180 \\ Damping coefficient of tank, \(bt\) & Ns/m & 20 & 20 & 20 & 25 \\ Spring coefficient of tank, \(ct\) & N/m & 20 & 20 & 20 & 40 \\ Initial velocity of boat, \(V\) & m/s & 2.06 & 2.06 & 2.06 & 4.12 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of waves used in the simulation. All waves have different speeds according to their wavelength. The wavelength (\(wl\)) is calculated using: \(wl=gT^{2}/2\pi\), where \(g=9.8\) m/s\({}^{2}\) and \(T\) is the wave period in seconds. Wavelength is inversely proportional to the frequency of the wave: waves with higher frequencies have shorter wavelengths, while waves with lower frequencies have longer wavelengths. Compared to Case 1, the wavelength in Cases 3 and 4 is much longer due to the reduced wave frequency. This is clearly illustrated in Figures 13 and 14. The variation in boat motion and wave motion is not large, unlike Case 1. In all cases, the oscillations of the waves and the boat both stabilized after about 5 s. As shown in Table 7 for comparison, Case 1 had a _Cruise speed_ of 4.312 m/s and \(\eta\) of 63.3% with wave angle of 180\({}^{\circ}\). Case 2 had the highest _Cruise speed_ of 6.098 m/s and \(\eta\) of 67.9% with a frequency of 0.3 Hz, indicating good wave energy exploitation. Longer wave periods are expected to produce better results. However, in Case 3, _Cruise speed_ and \(\eta\) decreased significantly (4.768 m/s and 38.8%). Case 4 had a slight increase in _Cruise speed_ and \(\eta\) (5.233 m/s and 49.5%) due to changes in initial velocity and tank coefficients, but Case 2 was more efficient. Wave analysis and boat behavior are influenced by wave frequency and phase. Figure 12: Amplitude graph for Case 2.