# Modeling a Wave Energy Harvester for Supplying Data Buoys
## 2 Materials and Methods
### Bang-Bang Mode
The limit value taken by \(L_{T/2}\) for \(\phi\longrightarrow 0\) is  
\[L_{T/2\_0}=\frac{2\,g\text{sin}\,\alpha_{M}}{k\,\omega^{2}}\,\frac{\pi}{2} \tag{15}\]  
The value of \(\phi\) for which \(L_{T/2}\) reaches its maximum value can be calculated by deriving (14) and equating the result to zero. The result is  
\[\phi_{M}=\text{atan}\frac{2}{\pi} \tag{16}\]  
and the maximum value of \(L_{T/2}\) is  
\[L_{T/2\_M}=\frac{2\,g\text{sin}\,\alpha_{M}}{k\,\omega^{2}}\bigg{[}\text{sin }\,\text{atan}\frac{2}{\pi}+\frac{\pi}{2}\text{cos}\,\text{atan}\frac{2}{\pi} \bigg{]} \tag{17}\]  
Let us now investigate the evolution of the cylinder position \(x(t)\) for \(t>t_{1}\). Firstly, consider the case that \(L_{T/2,0}\leq L\leq L_{T/2\_M}\), as assumed in Figure 4a, where the functions \(L_{inv}(\phi)\) and \(L_{T/2}(\phi)\) are represented together with the length of the track \(L\). Note that the function \(L_{T/2}(\phi)\) intersects the horizontal line \(L\) for \(\phi=\phi_{EQ}\), where  
\[\dot{x}=\frac{g\text{sin}\,\alpha_{M}}{k\,\omega}[\text{cos}\,(\omega\ t+\phi _{0})-\text{cos}\,\phi] \tag{10}\]0  
as obtained by equating Equation (14) to \(L\) and solving for \(\phi\). If the phase \(\phi\) is equal to the value \(\phi=\phi_{EQ}\) for which \(L=L_{T/2}\), then according to Figure 4a, the inversion length is greater than the track length. Therefore, the cylinder is able to reach the other end of the track without changing direction. Moreover, since \(L=\ L_{T/2}\), the cylinder takes a time exactly equal to \(T/2\) to reach the other end of the track. When the cylinder reaches the other end, its position is \(x=\ x_{2}\), its velocity is \(\dot{x}=0\) and the acceleration is  
\[\dot{x}=\frac{g\text{sin}\,\alpha_{M}}{k\,\omega}[\text{cos}\,(\omega\ t+\phi _{0})-\text{cos}\,\phi] \tag{10}\]1  
which is equal to the acceleration at time \(t=t_{1}\), but directed toward the track end 1. Thus, the instantaneous phase in \(x_{2}\) is equal to that in \(x_{1}\), but with opposite sign. Consequently, the motion back from track end 2 to track end 1 is equal to that from track end 1 to 2. When the cylinder comes back to \(x_{1}\), its phase returns exactly to \(\phi\) (wrapped to the interval \([0,\;2\pi]\)). Therefore, the phase \(\phi=\phi_{EQ}\) represents an equilibrium condition that allows the cylinder to move back and forth along the track periodically over a period equal to \(T\). If the phase \(\phi\) belongs to the interval \(0\leq\phi<\phi_{EQ}\), the curves in Figure 4a show that the track length \(L\) is greater than the half-period length \(L_{T/2}\) and less than the inversion length \(L_{inv}\). Thus, the cylinder is able to reach the other end of the track without making a change in direction, but it takes more time than \(T/2\), say \(T/2+dt\). When the cylinder reaches the other end of the track, its position is \(x=x_{2}\), its velocity is \(\dot{x}=0\) and the instantaneous phase of the acceleration is greater than \(\phi\), as it results  
\[\dot{x}=\frac{g\text{sin}\,\alpha_{M}}{k\,\omega}[\text{cos}\,(\omega\ t+\phi _{0})-\text{cos}\,\phi] \tag{10}\]2  
Thus, the instantaneous phase at \(x_{2}\) is \(|\phi+\omega\;dT|>\phi\) and has the opposite sign. When the cylinder comes back to \(x_{1}\), its instantaneous phase is further increased compared to \(\phi\). Therefore, cycle after cycle, the instantaneous phase at the beginning of each cycle will be larger and larger until the instantaneous phase reaches the value \(\phi_{EQ}\). If the instantaneous phase \(\phi\) belongs to the interval \(\phi_{EQ}<\phi<\phi_{L}\), the curves in Figure 4a show that the track length \(L\) is less than the half-period length \(L_{T/2}\) and less than the inversion length \(L_{inv}\). Thus, once again, there will be no reversal of direction, but the cylinder reaches the other end of the track in less time than \(T/2\) and returns to the initial position in less time than \(T\). The instantaneous phase at the next cycle becomes less than \(\phi\).