## 3 Results
### Ship's Rolling Motion Influenced by Successive Beam Waves in a Resistance Environment  
This sub-section provides the results of a realistic ship's behaviour under the influence of beam seas, which affect the ship's rolling motion successively, just as it happens in a regular swell. As it was mentioned in Section 2.2, the real sea state conditions are referred to by 21 different types. However, the focus of this sub-section is not to analyse the influence of all the waves in the spectrum on a certain ship's loading condition, it is to obtain conclusions about the ship's performance under the influence of certain types of waves, which can be used for the rest. With this approach, as the weather conditions cannot be modified, the ship operator has an idea about the ship's final behaviour after a relatively long period of elapsed time (t). With the objective of obtaining a similar approach to Section 2.1, Figure 4 represents Equation (37) for a Tw = 18 s, which is considered a very bad sea state condition, corresponding to a wave amplitude (Aw) of 8.40 m, i.e., a wave height of 16.80 m. As seen in Figure 4, it is remarkable to note that between 150 s and 200 s onwards, a coupling of the rolling frequencies with the Tw occurs for all of the ship loading conditions (Td) simulated, periodically repeated, and with the same rolling angle amplitudes, which depend on each ship's loading condition. What is the same, once the mentioned period of time has elapsed, the ship begins to roll according to the wave's frequency and with the same amplitude, regardless of its loading condition. Figure 3: Representation (red colour) of the exponential limit curve of a ship’s rolling motion under the influence of a single and external force. In order to validate these remarkable results, it was necessary to obtain the corresponding angular velocities. In this case, the angular velocity of Equation (35) in a dynamic condition of waves, \(\theta^{\prime}_{W}\), can be expressed as follows:  
\[\theta^{\prime}_{W}=\theta_{M}\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot \left[\lambda\cdot\cos(\omega_{1}\cdot\mathrm{t})-\omega_{1}\cdot\sin(\omega_{ 1}\cdot\mathrm{t})\right]\ -\ \theta_{M}\cdot\lambda\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot\left[ \cos\left(\omega_{1}\cdot\mathrm{t}\right)+\frac{\lambda}{\omega_{1}}\cdot \sin(\omega_{1}\cdot\mathrm{t})\right]-\frac{\theta_{MW}\cdot\omega_{W}\cdot \cos(\beta\ -\ \omega_{W}\cdot\mathrm{t})\cdot\cos\beta}{\frac{\pi\theta^{2}}{\mathrm{Tw}^{2}}-1}\, \tag{43}\]  
In Figure 5, the angular velocity for the same sea state condition (Tw = 18 s) and for the same studied cases, i.e., Td = 8 s, 10 s, 12 s, and 14 s, are plotted. These results indicate again that between 150 s and 200 s, a perfect coupling of the ship's own angular velocities is produced with the motion of the angular velocities produced by the waves. In order to show the coupling process produced between both frequencies (the ship and waves) after the first 150-200 s under the influence of the waves, regardless of the ship's and wave parameters, the rolling angle produced by induced forces of waves with a Tw = 12 s and where the ship has a Td = 9 s (yellow colour) is shown in Figure 6. Over this graph, the exponential limit curve (purple colour) is also plotted where it is deducted that, from 150 to 200 s onwards, a perfect coupling of the ship to the waves' frequencies is registered, keeping the amplitude of the rolling angles constant over time. Figure 4: Rolling angles reached by different ship loading conditions for Tw = 18 s.  
Figure 5: Rolling angular velocities reached by different ship loading conditions for Tw = 18 s.  
Furthermore, and for comparative purposes, in the same Figure 6, the rolling angles originating from the intact stability (Td = 9 s) as a result of a single external force (blue colour) and the corresponding exponential limit curve (red colour) are depicted, where it is observed that, after the first 300 s, the roll angles are practically null.