## 3 Estimate of Thrust and Speed of a Wave Glider on the Basis of a Simplified Mathematical Model  
Proposed in the present paper is a method of estimation of thrust and speed of a WG based on a simplified mathematical model which employs the flapping wing theory [44; 45; 46]. Consider the vertical oscillations of a WG, with the upper module being represented with a float of a rectangular platform of length \(l\) and width \(b\), both measured at a waterline plane. Let this float have a draft \(d\) and a waterplane area \(S_{WL}^{0}=lb\) with axes \(x\) and \(y\) of the right Cartesian coordinate system \((x,y,z)\) lying in the plane of symmetry, whereas axes \(x\) and \(z\) lie in the plane of unperturbed water surface. Axis \(x\) is directed to the right, and axis \(y\) is directed upwards. In a general case of an inclined board of the float, the area of its cross-section by a plane parallel to \(x0z\) is described by a function \(S_{WL}(y)\), therewith \(S_{WL}(0)=S_{WL}^{0}\). A sketch of two-module WG in waves in shown in Figure 10.  
Let progressive waves run to the right with a phase speed \(U_{phase}=\omega(k_{\lambda})/k_{\lambda}\), where in the general case of water of finite depth \(H\), the time (circular) frequency \(\omega(k_{\lambda})\) is related to wave number \(k_{\lambda}=2\pi/\lambda\) (\(\lambda\): wave length) by the following relationship, ensuing from a combined linear condition at the free surface of water  
\[\omega^{2}(k_{\lambda})=gk_{\lambda}th(k_{\lambda}H) \tag{38}\]  
If the WG is moving left with speed \(U_{0}\), the waves run relative to the float with a speed  
\[U_{wave}=U_{phase}+U_{0}=\sqrt{\frac{g}{k_{\lambda}}th(k_{\lambda}H)}+U_{0}=U_ {0}\left(1+\frac{1}{Fr_{c}}\sqrt{\frac{\lambda}{2\pi\cdot c}th\frac{2\pi H}{ \lambda}}\right), \tag{39}\]  
where \(Fr_{c}=U_{0}/\sqrt{g\epsilon}\) is the Froude number based on the chord of the underwater module. Therewith, the apparent frequency equals  
\[\omega_{a}=k_{\lambda}U_{wave}. \tag{40}\]  
Figure 10: Schematized wave glider. Assuming that the upper and lower models are rigidly connected, one can write down the equation of forced oscillations of the WG in the direction of axis \(y\). Therewith, note that beside inertial forces, the WG is subject to a restoring time-dependent _Archimedes force_. \[R_{y}^{A}=\rho g(S_{WL}^{0}+\left.\frac{dS_{WL}}{dy}\right|_{y=0}y)y, \tag{41}\]  
(\(\rho\): water density, \(g\): gravitational acceleration), unsteady lift force on the wing \(R_{ywing}\), due to its oscillations and unsteady exciting, _Archimedes force_\(R_{ywave}\), due to the periodic variation of the wetted volume of the float. The final equation has been obtained in the form:  
\[(M+m)\ddot{y}+\rho g[S_{WL}^{0}+S^{\prime}w_{L}(0)y]y=R_{ywing}+R_{ywave}, \tag{42}\]  
where \(M\) is the mass of the WG, \(m\) is the added mass of the non-lifting part of the WG, \(S^{\prime}w_{L}(0)=dS_{WL}/dy\) for \(y=0\). Representing the lift force acting on the oscillating wing as an expansion with respect to kinematic parameters, and calculating the exciting force due to waves with the help of the _Krylov-Froude_ hypothesis as a corresponding variation of the _Archimedes force_,  
\[R_{ywing}=R_{ywing}^{\dot{y}}\dot{y}+R_{ywing}^{\ddot{y}}\ddot{y}, \tag{43}\]  
\[R_{ywave}=\rho g\dot{b}\int_{-l/2}^{l/2}y_{wave}dx=\rho gba_{w}\int_{-l/2}^{l/2 }\cos(k_{\lambda}x-\omega_{a}t)dx=\frac{2\rho gba_{w}}{k_{\lambda}}\sin(\frac{ \pi\cdot l}{\lambda})\cos(\omega_{a}t) \tag{44}\]  
where \(a_{w}\) is the wave amplitude, write the WG forced oscillations equation as follows:  
\[(M+m-R_{ywing}^{\ddot{y}})\ddot{y}-R_{ywing}^{\dot{y}}\dot{y}+\rho gS_{WL}^{0}y +\rho gS_{WL}^{\prime}(0)y^{2}=\frac{2\rho gba_{w}}{k_{\lambda}}\sin(\frac{ \pi\cdot l}{\lambda})\cos(\omega_{a}t) \tag{45}\]  
To reduce Equation (45) to a non-dimensional form, introduce the half-chord of the wing \(c/2\) as a typical length and speed \(U_{0}\) of the WG as a typical speed.