## 3 Results
### Influence of the Damping Factor on the Synchronism Phenomenon  
Using Equation (10) for the synchronism phenomenon, it can be deduced that \(\tan\beta=\infty\), so \(\beta=\frac{\pi}{2}\). The second term of Equation (11) reaches the maximum value \(\pm\theta_{\text{MW}}\) when the numerator is \(\sin\left[\frac{2\pi}{\text{TW}}\cdot\text{t}-\beta\right]=\pm 1\), which is achieved with a periodicity of \(\text{t}=\left(\frac{\text{n}+1}{2}\right)\cdot\text{Tw}\).  
This remarkable conclusion can be observed in the results shown in Figure 6, where the induced rolling motion generated by two different sea state conditions is simulated, considering 0.015 as an average value of the damping factor, according to the literature [34; 35]. In these simulations, this representative average value was used, although it is important to note that this parameter changes with the amplitude of different rolling angles. In the case of using another damping factor, the amplitude of the results shown in the figures will be different, but the appearance, and then the ship's behaviour, will be the same.  
Figure 5: Induced rolling angles for \(\text{Tw}=10\) s and \(\text{Td}=16\) s.  
Figure 6: Representation of induced rolling motion for \(\text{Td}=12\) s. (**a**) \(\text{Tw}=8\) s; (**b**) \(\text{Tw}=15\) s.  
This periodicity indicates that the maximum rolling angles are reached in the crests and the troughs of the waves. What is more, for the Td considered in the previous subsection (very small and large Td), the maximum rolling angles were reached in the middle of the wave slope, while during the process of approaching the synchronism phenomenon, these maximum rolling angles are shifted towards the crest and the trough.  
Taking into account the previously obtained equations, the relationship between the induced rolling motion and the wave slope maximum angle, \(\theta_{\text{MW}}\), is as follows:  
\[\frac{\theta}{\theta_{\text{MW}}}=\pm\frac{1}{\sqrt{\left(1-\frac{\text{Td}^{ 2}}{\text{Tw}^{2}}\right)^{2}+\frac{\lambda_{1}^{2}}{\pi^{2}}\cdot\frac{\text{ Td}^{2}}{\text{Tw}^{2}}}} \tag{20}\]  
\(\theta_{\text{MW}}\) does not usually exceed \(10^{\circ}\), and the wave slope maximum angle increases up to a finite value, which is achieved when there is equilibrium between the damping energy during a single rolling motion (port-starboard) and the energy induced by the wave in the passage of the trough or crest, i.e., half of wavelength.  
Therefore, once the new mathematical models of the rolling motion were independently analysed and represented, Figure 7 depicts the combination of both effects for a sea state condition of Tw = 15 s and a usual ship's loading condition of Td = 12 s.  
From Figure 7, it is deduced that, as time goes on, during the synchronism phenomenon, the ship's tendency is to roll without considering the periodicity of the waves. Furthermore, the amplitude of the rolling angles increases according to the interesting conclusions mentioned in the previous sub-sections. What is more, according to this theoretical approach, any ship in less than a minute reaches a rolling angle where flooding occurs as a consequence of a non-watertight element being submerged in water. For this reason, it is essential that officers on watch detect this dangerous situation in order to take action and avoid potential problems. According to the present results, this warning can be easily detected by carefully checking the time on the wave profile where the maximum rolling angles occur.