## 3 Estimate of Thrust and Speed of a Wave Glider on the Basis of a Simplified Mathematical Model
Then, the nondimensional time \(\tau\) and the nondimensional vertical displacement of the float \(\eta\) can be written as follows:  
\[\omega^{2}(k_{\lambda})=gk_{\lambda}th(k_{\lambda}H) \tag{38}\]0  
Introducing the instantaneous magnitude \(C_{ywing}\) of the lift coefficient of oscillating underwater foil and its derivatives with respect to nondimensional kinematic parameters \(\ddot{\eta}\) and \(\ddot{\eta}\),  
\[\omega^{2}(k_{\lambda})=gk_{\lambda}th(k_{\lambda}H) \tag{38}\]1  
(where \(S_{wing}\) is the reference area of the wing), and also noting that on calm water the mass of the WG is counterbalanced by buoyancy force, that is \(M=\rho S_{WL}^{0}d\), where \(d\) is the draft of WG, one can rewrite the equation of the oscillations of WG in waves as follows:  
\[ \begin{array}{l}\left[1+\overline{m}-C_{ywing}^{\ddot{y}}\frac{c}{4d}\left( \frac{S_{wing}}{S_{WL}^{0}}\right)\right]\ddot{\eta}-\frac{c}{4}\left(\frac{S_ {wing}}{S_{WL}^{0}}\right)C_{ywing}^{\ddot{\eta}}\dot{\eta}+\left(\frac{gc}{U_ {0}^{2}}\right)\frac{c}{4d}\eta\\ +\left(\frac{gc}{U_{0}^{2}}\right)S_{WL}^{\prime}\frac{c}{4d}\eta^{2}=\left( \frac{gc}{U_{0}^{2}}\right)\frac{\lambda}{2\pi}\frac{ba_{w}}{S_{WL}^{0}d} \sin\left(\frac{\pi\lambda}{\lambda}\right)\cos\left(\frac{\omega_{c}c}{2U_{0 }}\tau\right)\end{array}  \tag{48}\]  
Dividing both parts of the equation by the coefficient of \(\ddot{\eta}\), and again introducing the Froude number based on the wing chord \(Fr_{\epsilon}=U_{0}/\sqrt{gc}\), one obtains the desired equation in the following nondimensional form:  
\[\ddot{\eta}+2\beta\dot{\eta}+\kappa^{2}\eta+\epsilon\eta^{2}=\delta_{w}\cos(k _{a}\tau) \tag{49}\]where  
\[2\beta=\frac{-\frac{c}{4d}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)C_{y}^{\tilde{ \eta}}wing}{1+\overline{m}-C_{y}^{\tilde{\eta}}\frac{\tilde{\eta}}{d\tilde{d}} \left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{50}\]  
\[\kappa^{2}=\frac{\frac{c}{4F_{T}^{2}d}}{1+\overline{m}-C_{y}^{\tilde{\eta}} \frac{\tilde{c}}{d\tilde{d}}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{51}\]  
\[\varepsilon=\frac{\frac{c^{2}S_{WL}^{0}}{4F_{T}^{2}S_{WL}d}}{1+\overline{m}-C _{y}^{\tilde{\eta}}\frac{\tilde{c}}{d\tilde{d}}\left(\frac{S_{ving}}{S_{WL}^{0 }}\right)}, \tag{52}\]  
\[\delta_{w}=\frac{\frac{\lambda d\mu_{w}b}{2\pi F_{T}^{2}d}\sin\left(\frac{ \pi l}{\lambda}\right)}{1+\overline{m}-C_{y}^{\tilde{\eta}}\frac{\tilde{c}}{ d\tilde{d}}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{53}\]  
\[k_{a}=\frac{\omega_{a}c}{2U_{0}}, \tag{54}\]  
\(k_{a}\) can be considered as a _Strouhal number_ based on the apparent circular frequency \(\omega_{a}\). To simplify, consider first the case of a straight-walled board float, for which  
\[S_{WL}(y)\equiv S_{WL}^{0},S_{WL}^{\prime}(0)=0,\varepsilon=0.\]  
It is easy to see that in this case the Equation (49) represents a standard equation of the theory of oscillations for which the solution can be obtained in analytical form. The homogeneous solution \(\eta_{h}(\tau)\) of Equation (49) (\(\delta_{w}=0\)) for the initial conditions \(\eta(0)=\eta_{0}\) and \(\dot{\eta}(0)=\dot{\eta}_{0}\) has the form  
\[\eta_{h}(\tau)=\exp(-\beta\tau)[\eta_{0}\cos(\sqrt{\kappa^{2}-\beta^{2}}\tau )+\frac{\dot{\eta}_{0}+\beta\eta_{0}}{\sqrt{\kappa^{2}-\beta^{2}}}\sin(\sqrt{ \kappa^{2}-\beta^{2}}\tau)]. \tag{55}\]  
A particular (nonhomogeneous) solution \(\eta_{nh}(\tau)\) of Equation (49), satisfying the right-hand side (\(\delta_{w}\
eq 0\)), has the form  
\[\eta_{nh}(\tau)=\frac{a_{w}[(\kappa^{2}-k_{a}^{2})\cos(k_{a}\tau)+2\beta k_{a }\sin(k_{a}\tau)]}{(\kappa^{2}-k_{a}^{2})^{2}+4\beta^{2}k_{a}^{2}} \tag{56}\]  
As the homogeneous solution of (49), corresponding to free oscillations with damping decays exponentially, to calculate the thrust due to force oscillations of WG it is sufficient to consider nonhomogeneous solution (49).