## 3 Results
### Ship's Rolling Motion Influenced by Successive Beam Waves in a Resistance Environment
It is worth noting that in order to define the time (t) in which the final coupling is reached, it is necessary to define the minimum admissible rolling angle (\(\theta_{\text{MIN}}\)). For this reason, in dynamic conditions at zero speed, i.e., receiving the impact of successive waves, it was chosen as 0.23 rad. Then, for a given Td and Tw, and establishing the minimum admissible rolling angle (\(\theta_{\text{MIN}}\)), it is possible to determine the time (t) of the coupling from the following equation:  
\[\theta_{\text{MIN}}=\ \theta_{\text{M}}\cdot\text{e}^{-\ \lambda\cdot t}\ +\ \frac{\theta_{\text{MW}}}{1-\ \frac{\text{Td}^{2}}{\text{Tw}^{2}}}\, \tag{44}\]  
As it was mentioned, in our simulations, certain sea state conditions valid for the full spectrum of waves were selected. However, in the particular cases when a Tw < Td was selected, relevant results are also obtained, which are different from the previously commented. For instance, in the studied cases of Td = 8 s, 10 s, 12 s, and 14 s, if waves of Tw = 11 s are selected, the corresponding rolling angles plotted in Figure 7 are reached. Figure 6: Representation of the maximum rolling angles in static (due to a single and external force) and in dynamic conditions (under the influence of successive waves) and their corresponding exponential limit curves. Figure 7: Rolling angles reached by different ship loading conditions for Tw = 11 s.  
From Figure 7, it can be concluded that the results regarding the coupling are being produced in the same periods, i.e., after the first 150-200 s. Nevertheless, it is remarkable that for Td = 8 and 10 s, the rolling angles occur on the opposite side when Td = 12 and 14 s. Considering that the simulated waves were Tw = 11 s, it can be deduced that if Td < Tw, the rolling angles will be produced on the opposite side as that of Td > Tw. What is more, a constant lag equivalent to half of Td is registered between both situations, regardless of the Td value. Furthermore, in order to have a more detailed idea of the ship's behaviour at the first moment after being influenced by successive waves, the case where Tw = 5 s, which corresponds to a very smooth condition, was simulated. Figure 8 shows the rolling angles in the first 100 s, where the ship's erratic behaviour between the different loading conditions is observed, unlike what is observed in Figure 9, in which with a Tw = 18, i.e., very rough sea conditions, a more harmonious and smooth ship behaviour can be seen. Despite this, as it has been concluded, in any case, a full coupling will take place, regardless of the ship's loading conditions and always at the same range of time. Once it was shown that a coupling between the sea state conditions and the ship's behaviour is produced, the approximate instants when the coupling with the Tw is produced was represented in Figure 10. These moments were established when the rolling angle amplitude was almost constant. It should be noted that at the lowest values of Tw, the coupling time would tend to infinity, due to the force generated by very smooth waves, which has hardly any influence on the ship's behaviour. For this reason, the tendency of the ship will be to roll according to its restoring moment. Figure 8: Simulation of the rolling angle for Tw = 5 s in the first moments. \[\theta^{\prime}_{W}=\theta_{M}\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot \left[\lambda\cdot\cos(\omega_{1}\cdot\mathrm{t})-\omega_{1}\cdot\sin(\omega_{ 1}\cdot\mathrm{t})\right]\ -\ \theta_{M}\cdot\lambda\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot\left[ \cos\left(\omega_{1}\cdot\mathrm{t}\right)+\frac{\lambda}{\omega_{1}}\cdot \sin(\omega_{1}\cdot\mathrm{t})\right]-\frac{\theta_{MW}\cdot\omega_{W}\cdot \cos(\beta\ -\ \omega_{W}\cdot\mathrm{t})\cdot\cos\beta}{\frac{\pi\theta^{2}}{\mathrm{Tw}^{2}}-1}\, \tag{43}\]0 Simulation of the rolling angle for Tw = 18 s in the first moments.