## 3 Results
### Ship's Rolling Motion Influenced by Successive Beam Waves in a Resistance Environment
Ship's Rolling Motion at Non-Zero Speed and under the Influence of Waves Coming from Any Constant Direction  
This sub-section shows the results which can be considered as a practical guide for shipmasters to take into account when sailing under given weather conditions. Considering the real sea state conditions previously studied, for practical purposes, two waves of the full spectrum were selected. The first one was \(\mathrm{Tw}=6\) s, considered as smooth conditions, and the second one was \(\mathrm{Tw}=18\) s, which represents a rough sea state. Regarding the ship's speed, a range from 2 to 20 knots was analysed, with intervals of 2 knots, knowing that in many ships of a merchant fleet, the minimum speed is over 5 knots. However, speeds lower than 5 knots were considered in order to be useful to small and medium fishing vessels too. With regards to the heading angle, \(\alpha\), changes of 10\({}^{\circ}\) were considered, from 000\({}^{\circ}\) to 180\({}^{\circ}\), as 10\({}^{\circ}\) is understood to be the minimum alteration to steering the ship and improve its behaviour under bad weather conditions. Considering Equation (40), Figure 11 represents the needed Te of a ship sailing at a certain speed and for all heading alterations considered for a \(\mathrm{Tw}=18\) s. As it can be observed, for heading alterations \(\alpha<90^{\circ}\), i.e., receiving the seas between the forward and the beam, the Te decreases with increasing ship speed. For \(\alpha=90^{\circ}\), the Te is constant, regardless of the ship's speed, and it matches exactly with \(\mathrm{Tw}\), due to the intervention of \(\cos\alpha\). This particular case, and at zero-speed conditions, would correspond with those studied in the previous sub-sections, i.e., under the influence of beam seas. Furthermore, it can be concluded that for \(\alpha>90^{\circ}\), Te increases as the ship's speed increases. The highest time of Te occurs for \(\alpha=180^{\circ}\), i.e., sailing following seas. \[\theta^{\prime}_{W}=\theta_{M}\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot \left[\lambda\cdot\cos(\omega_{1}\cdot\mathrm{t})-\omega_{1}\cdot\sin(\omega_{ 1}\cdot\mathrm{t})\right]\ -\ \theta_{M}\cdot\lambda\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot\left[ \cos\left(\omega_{1}\cdot\mathrm{t}\right)+\frac{\lambda}{\omega_{1}}\cdot \sin(\omega_{1}\cdot\mathrm{t})\right]-\frac{\theta_{MW}\cdot\omega_{W}\cdot \cos(\beta\ -\ \omega_{W}\cdot\mathrm{t})\cdot\cos\beta}{\frac{\pi\theta^{2}}{\mathrm{Tw}^{2}}-1}\, \tag{43}\]1 Te for different combinations of ship speeds and \(\alpha\) angles, for a \(\mathrm{Tw}=18\) s.  
\[\theta^{\prime}_{W}=\theta_{M}\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot \left[\lambda\cdot\cos(\omega_{1}\cdot\mathrm{t})-\omega_{1}\cdot\sin(\omega_{ 1}\cdot\mathrm{t})\right]\ -\ \theta_{M}\cdot\lambda\cdot\mathrm{e}^{-\lambda\cdot\mathrm{t}}\cdot\left[ \cos\left(\omega_{1}\cdot\mathrm{t}\right)+\frac{\lambda}{\omega_{1}}\cdot \sin(\omega_{1}\cdot\mathrm{t})\right]-\frac{\theta_{MW}\cdot\omega_{W}\cdot \cos(\beta\ -\ \omega_{W}\cdot\mathrm{t})\cdot\cos\beta}{\frac{\pi\theta^{2}}{\mathrm{Tw}^{2}}-1}\, \tag{43}\]2 Time to coupling depending on the \(\mathrm{Tw}\) state. In Figure 12, the influence of \(\mathrm{Tw}=6\) s in the relationship between the navigational parameters of ship speed and heading (or \(\alpha\) angle) is depicted. For clarifying purposes, the results were plotted on two graphs, i.e., an \(\alpha\) angle between \(000^{\circ}\) and \(90^{\circ}\) (Figure 12a), and \(\alpha\) angle from \(90^{\circ}\) to \(180^{\circ}\) (Figure 12b). In Figure 12a, the same tendency as in the previous case is observed, i.e., a decrease in \(\mathrm{Te}\) as the ship's speed increases. However, erratic behaviour in the \(\mathrm{Te}\) tendency is clearly observed in Figure 12b, except for \(\alpha=100^{\circ}\) and \(110^{\circ}\) where a smooth progressive increase is deducted.