## 3 Calculations of Wave Power  
### Wave Power and Its Absorption by the Buoy  
For the calculation of wave profiles and velocities, the 1 st-order Stokes wave theory was employed. In unidirectional waves, the wave power per unit width of the wave front, or wave energy flux, can be estimated as [32]:  
\[P_{wf}=C_{g}E_{d} \tag{4}\]  
Figure 3: Schematic representation of the cylindrical buoy’s WEC implementation process. This equation is also known as the power density, since \(E_{d}\) is the wave energy density (J/m\({}^{2}\)) and \(C_{g}\) is the group velocity (m/s). According to the dispersion relationship, water waves can be classified by their relative depth as shallow water waves (\(h/\lambda<1/20\)), intermediate depth waves (\(1/20<h/\lambda<1/2\)), and deep-water waves (\(h/\lambda>1/2\)), where \(\lambda\) is the wavelength and \(h\) is the water depth. In the case of the numerical tests described in this manuscript, where \(h=3\) m and wave periods are of 4 and 8 s, we considered intermediate depth waves, for which the power per wave front unit is defined as:  
\[P_{wf}=\frac{\rho g^{2}}{16\omega}H^{2}D(\mathrm{kh}) \tag{5}\]  
where \(H\) is the local wave height, \(\omega\) is the angular frequency (\(2\pi/T\)), \(k\) is the wavenumber (\(2\pi/\lambda\)), \(h\) is the water depth, and:  
\[D(kh)=\left(1+\frac{2 kh}{\sinh(2 kh)}\right)\tanh(\mathrm{kh}) \tag{6}\]  
The available power density can be related to the power absorbed by the WEC device through a parameter called the capture or absorption width [33], defined as the ratio of the power absorbed by the buoy (\(P_{ab}\)) to the available wave energy flux per unit crest length, and it is given by:  
\[C_{w}=\frac{P_{ab}}{P_{wf}} \tag{7}\]  
where \(P_{ab}=\tau\cdot\omega_{r}\), where the torque \(\tau\) and angular velocity \(\omega_{r}\) are obtained by the analysis of the buoy's movement. Since \(P_{ab}\) is in W and \(P_{wf}\) in W/m, \(C_{w}\) has units in meters. [PERSON] [34] established that, for axisymmetric bodies that are subjected to regular waves, the maximum theoretical capture width ratio is defined as:  
\[C_{wMax}=\epsilon_{df}\frac{\lambda}{2\pi} \tag{8}\]  
where \(\lambda\) is the wavelength and the dimensionless value \(\epsilon_{df}\) is linked to the degrees of freedom. In WECs with a vertical axis of symmetry oscillating in heave mode \(C_{wMax}=k^{-1}\) and for surge or pitch modes, as in our case, \(\epsilon_{df}=2\)[35]. In this sense, the ratio of the maximum extracted wave power to the available wave power flux from Equation (8) is equal to Equation (7), i.e., \(C_{wMax}=P_{abMax}/P_{wf}\). This result implies that the buoy's geometry is not related to the maximum \(C_{w}\); rather, it is linked to the degrees of freedom. In conclusion, the shape of the floating object is independent of its maximum-capture width, which refers to the well-known _antenna effect_, where a small float can absorb waves over a width greater than its own dimension [36]. Regarding the energy conversion from the wave power available to the buoy, or the design of the device, different types of efficiency can be highlighted. One simple means to assess the capture efficiency, or hydrodynamic efficiency, is to divide the capture width (Equation (7)) by a characteristic length. \[\epsilon_{cw}=\frac{P_{ab}}{P_{wf}B}=\frac{C_{w}}{B} \tag{9}\]  
In this case, \(B\) is such a length that represents the physical WEC length, or in our case, this would be the buoy width (see Figure 2). A more detailed description of the annual average capture width ratio of different WEC technologies can be consulted in [37].