## 3 Estimate of Thrust and Speed of a Wave Glider on the Basis of a Simplified Mathematical Model
Noting that for steady motion this drag is a sum of the wave resistance of the float and viscous drag of the whole vehicle, one can write  
\[2\beta=\frac{-\frac{c}{4d}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)C_{y}^{\tilde{ \eta}}wing}{1+\overline{m}-C_{y}^{\tilde{\eta}}\frac{\tilde{\eta}}{d\tilde{d}} \left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{50}\]6  
In this paper, for the purpose of an approximate evaluation, the wave resistance of the float is determined with the help of the formulae given in [47] for a shallow hull with wedge-like extremities, and viscous drag is found as a sum of the friction drag of the modules, calculated by the method of equivalent plate with account of the regime of the flow with addition of the drag of the umbilical cord. For the calculation of the wave drag, take the following formula, relating a complex function of wave amplitudes \(A(\theta)\) with wave resistance [48]:  
\[2\beta=\frac{-\frac{c}{4d}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)C_{y}^{\tilde{ \eta}}wing}{1+\overline{m}-C_{y}^{\tilde{\eta}}\frac{\tilde{\eta}}{d\tilde{d}} \left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{50}\]7For the case of a straight-board hull of overall length \(l\) with wedge-like extremities and insert of constant width \(b\) and length \(l_{p}\) at constant draft \(d\), [47] gives the following expression for the calculation of the wave amplitude complex function \(A(\theta)\):  
\[2\beta=\frac{-\frac{c}{4d}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)C_{y}^{\tilde{ \eta}}wing}{1+\overline{m}-C_{y}^{\tilde{\eta}}\frac{\tilde{\eta}}{d\tilde{d}} \left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{50}\]8  
where \(i=\sqrt{-1},v_{w}=g/U_{0}^{2},C_{p}=b/(l-l_{p})\). Viscous drag has been estimated using the formula  
\[2\beta=\frac{-\frac{c}{4d}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)C_{y}^{\tilde{ \eta}}wing}{1+\overline{m}-C_{y}^{\tilde{\eta}}\frac{\tilde{\eta}}{d\tilde{d}} \left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{50}\]9  
where \(\Omega_{raft}\) m \(\Omega_{wing}\) are wetted surfaces of the float and wing, correspondingly, \(C_{fraft}\) and \(C_{fwing}\) are friction coefficients, determined through formulae of Prandtl-Schlichting for the turbulent flow and Blasius formulae for the laminar flow  
\[\kappa^{2}=\frac{\frac{c}{4F_{T}^{2}d}}{1+\overline{m}-C_{y}^{\tilde{\eta}} \frac{\tilde{c}}{d\tilde{d}}\left(\frac{S_{ving}}{S_{WL}^{0}}\right)}, \tag{51}\]0  
In the Equation (68), the Reynolds numbers are composed using the length of the float and the chord of the wing, correspondingly. \(R_{xumb}\) represents the drag of the umbilical cord connecting the upper and lower modules. To validate the approach, a number of calculations have been carried out for a concrete variant of WG with the following parameters: \(l=2\) m, \(b=0.6\) m, \(d=0.2\) m, \(l_{w}=1.5\) m, \(c=0.3\) m, \(m=0.3\) M, \(h=5\) m (\(h\): submergence of the wing), \(\rho=1000\) kg/m\({}^{3}\), \(\
u=1.3\times 10^{-6}\) m\({}^{2}\)/s, \(d_{umb}=h=5\) m, \(C_{xumb}=1\) (assuming a laminar separation), and the wave length varied in the limits \(3\) m \(\leq\lambda\leq 50\) m. Based on statistics of sea waves, the wave amplitude was assumed as \(a_{w}=\lambda/40\). In the calculation of the wave resistance, the length of the insert of constant width was adopted as \(l_{p}=0.97l\). Typical results of the calculation of curves of the drag and available thrust are presented in Figures 11 and 12, separately, for the wavelengths in the range \(\lambda=10\) m-\(50\) m and in the range \(\lambda=3\) m-\(4\) m, in order to demonstrate possible thrust resonant modes which for the input vehicle (float) length of \(2\) m are found to occur at wavelength \(\lambda=3\)--see the thrust maximum around the UG's speed \(U_{0}\approx 0.34\) m/s--and at \(\lambda\approx 4\) m--see the thrust maximum around the UG's speed \(U_{0}\approx 0.72\) m/s. Naturally, for other lengths of the float, the resonant wavelength would be different.