# Design and Study of a New Wave Actuator for a Boat
## 2 Mathematical Modeling of Wave Actuator
### Modeling of the Wave Actuator  
After exploring the action of vertical wave forces on the bottom of the boat, the transformation of the vertical wave forces into power to propel the boat forward using the designed new wave actuator is analyzed. The wave actuator, as proposed in this paper, consists of a flexible water tank, inlet and outlet nozzles, and a revolving spring. It is engineered to be mounted on the underside of the boat's hull, as illustrated in Figure 3. During an upward wave motion, the flexible water tank expels compressed water through the outlet nozzle, generating thrust to drive the boat. In the downward phase of the wave, the springs within the actuator rotate the tank to draw in water, producing damping effects  
Figure 2: Frequency of encounter. due to the flexible material and rapid water flow. The simulation in the paper assumes that the output nozzle should be in water. Figure 3 is only an illustration of the system. With Equation (12), we can calculate the total wave force applied on the boat. In this regard, the dynamic model for the structure of a boat with a wave actuator can be expressed as a second-order differential equation with springs and a damper. \[F_{w}=b_{t}\dot{z}_{t}+c_{t}z_{t}+F_{r} \tag{13}\]  
where  
\(\dot{z}_{t},\ z_{t}\) is the velocity and position of bottom plate of water tank (m/s, m);  
\(b_{t}\dot{z}_{t}\) is the damping force (N);  
\(c_{t}z_{t}\) is the spring force (N);  
\(F_{r}\) is the propulsion force through the nozzle (N). The Bernoulli theorem can also be applied to the measurement of flow rate. The passage of an incompressible fluid through the constriction is assumed. The increase in kinetic energy as the velocity increases from \(V_{1}\) to \(V_{2}\) is derived from the pressure energy of the fluid; the pressure drops from \(P_{1}\) to \(P_{2}\). Since there is no change in height, and equation can be rearranged to give the following:  
\[\frac{{V_{2}}^{2}}{2}-\frac{{V_{1}}^{2}}{2}=\frac{{P_{1}-P_{2}}}{\rho} \tag{14}\]  
The volumetric flow rate \(Q=V_{1}A_{1}=V_{2}A_{2}\). Therefore, by rearrangement,  
\[V_{1}=\frac{{V_{2}A_{2}}}{{A_{1}}} \tag{15}\]  
Substituting for \(V_{1}\) gives  
\[\frac{{V_{2}}^{2}}{2}-\frac{{V_{2}}^{2}{(A_{2}/A_{1})}^{2}}{2}=\frac{{P_{1}-P _{2}}}{\rho} \tag{16}\]  
Therefore,  
\[V_{2}=\sqrt{\frac{{2(P_{1}-P_{2})}}{{\rho\Big{(}1-{(A_{2}/A_{1})}^{2}\Big{)}}} }\Longrightarrow Q=V_{2}A_{2}=A_{2}\sqrt{\frac{{2(P_{1}-P_{2})}}{{\rho\Big{(} 1-{(A_{2}/A_{1})}^{2}\Big{)}}}} \tag{17}\]  
This derivation neglects the correction of kinetic energy loss due to the non-uniformity of flow in both cross sections and the frictional degradation of energy during passage  
Figure 3: Structure of the wave actuator attached to the boat. through the constriction. This is expressed by the introduction of a numerical coefficient, \(c_{\text{discharge}}\), known as the coefficient of discharge (Figure 4) as the following:  
\[c_{\text{discharge}}=\frac{Q}{A_{2}\sqrt{\frac{2\Delta P}{\rho\left(1-\left( \frac{A_{2}}{A_{1}}\right)^{2}\right)}}} \tag{18}\]  
where  
\(c_{\text{discharge}}\) is the discharge coefficient through the constriction (dimensionless). \(Q\) is the volumetric flow rate of fluid through constriction (volume per time). \(\Delta P\) is the pressure drop across constriction (force per area). The value of \(c_{\text{discharge}}\) depends on the conditions of flow and the shape of the constriction. For a well-shaped constriction (notably circular cross-section), it would vary between 0.95 and 0.99 for turbulent flow. The value is much lower in laminar flow because the kinetic energy correction is larger. Values of the coefficient of discharge are given in the preceding paragraph. The diverging section or diffuser is designed to induce a gradual return to the original velocity.