## 2 Materials and Methods
### Ship's Rolling Motion Affected by Successive Beam Waves in a Resistant Environment  
The hypothesis referred to in Section 2.1 can only be considered theoretical, because, in real conditions, ships are not only under the influence of a single wave (single force) but are subject to the influence of successive waves. For this reason, the effect of beam seas on an idle ship, together with the ship's behaviour as a consequence of its righting moment, is studied in the present sub-section. The rolling response of a ship under the influence of any beam wave is called its induced rolling motion (\(\theta\)f). This induced force is determined by the parameters of waves (maximum wave slope, \(\theta_{\text{MW}}\), and frequency, \(\omega_{\text{W}}\)), and by the ship parameters (the ship's natural rolling period, Td). This induced force can be expressed as follows [28]:  
\[\theta\text{f }=\ \frac{\theta_{\text{MW}}\cdot\ \cos\beta\cdot\ \sin[\omega_{\text{W}}\cdot\text{t }-\ \beta]}{1-\ \frac{\text{Td}^{2}}{\text{Tw}^{2}}}\ =\ \frac{\theta_{\text{MW}}\cdot\ \cos\beta\cdot\ \sin\left[\frac{2\pi}{\text{Tw}}\cdot\text{t }-\ \beta\right]}{1-\frac{\text{Td}^{2}}{\text{Tw}^{2}}}\ \, \tag{33}\]  
where \(\beta\) is equal to:  
\[\tan\ \beta\ =\ \frac{\lambda_{1}}{\pi}\cdot\frac{\frac{\text{Td}}{\text{Tw}} }{1\ -\ \frac{\text{Td}^{2}}{\text{Tw}^{2}}}, \tag{34}\]  
Therefore, taking into account Equation (28) and the induced rolling by successive waves, the equation of the rolling angle in zero-speed conditions and under the influence of beam waves is the following:  
Figure 1: (**a**) Wavelength as a function of period. Precision 100%. (**b**) Velocity in m\(\cdot\)s\({}^{-1}\) as a function of period. Precision 99.67%. (**c**) Velocity in knots as a function of period. Precision 99.93%. \[\theta=\ \theta_{\rm M}\cdot\rm e^{-\ \lambda\ \cdot\ t}\cdot\left[\cos(\omega_{1} \cdot\rm t)\ +\ \frac{\lambda}{\omega_{1}}\cdot\ \sin(\omega_{1}\cdot\rm t)\right]\ +\ \frac{\theta_{\rm MW}\cdot\cos\beta\cdot\sin(\omega_{\rm W}\cdot\rm t \ -\ \beta)}{1-\ \frac{Td^{2}}{Tw^{2}}}\ \, \tag{35}\]  
or  
\[\theta=\ \theta_{\rm M}\cdot\rm e^{-\ \lambda\ \cdot\ t}\cdot\left[\cos\sqrt{ \omega^{2}\ -\ \lambda^{2}}\cdot\rm t\ +\ \frac{\lambda}{\sqrt{\omega^{2}\ -\ \lambda^{2}}}\cdot\sin\sqrt{\omega^{2}\ -\ \lambda^{2}}\cdot\rm t\right]\ +\ \frac{\theta_{\rm MW}\cdot\ cos\beta\cdot\sin(\omega_{\rm W}\cdot\rm t \ -\ \beta)}{1-\ \frac{Td^{2}}{Tw^{2}}}\ \, \tag{36}\]  
For operative purposes, replacing the ship's natural frequency, \(\omega\), with its relationship with Td, we obtain the following:  
\[\theta=\ \theta_{\rm M}\cdot e^{-\ \lambda\ \cdot\ t}\cdot\left[\cos\sqrt{ \left(\frac{2\pi}{Td}\right)^{2}\ -\ \lambda^{2}}\cdot\rm t\ +\ \frac{\lambda}{\sqrt{\left(\frac{2\pi}{Td}\right)^{2}\ -\ \lambda^{2}}}\cdot\sin\sqrt{\left(\frac{2\pi}{Td}\right)^{2}\ -\ \lambda^{2}}\cdot\rm t \right]\ +\ \frac{\theta_{\rm MW}\cdot\ cos\beta\cdot\sin(\omega_{\rm W}\cdot\rm t \ -\ \beta)}{1-\ \frac{Td^{2}}{Tw^{2}}}\ . \tag{37}\]  
Ship's Rolling Motion at Non-Zero Speed and under the Influence of Waves Coming from Any Constant Direction  
As the hypotheses referred to previously are not universal, in the present sub-section the ship's rolling motion when the ship is sailing with zero speed is studied. In this real situation, the ship is receiving the impact of successive waves from a constant direction, \(\alpha\), defined as the angle formed between the ship's centre line (heading) and the waves' direction. Navigational parameters such as the ship's speed (Vs) and heading are under the control of the ship's operator (the decision criteria of the shipmaster), and both parameters are present as the only tools which can modify the ship's behaviour under certain weather conditions, given that they cannot be controlled by sailors. In this dynamic situation, the encounter velocity (Ve) is defined as follows [28]:  
\[\rm Ve\ =\ \rm Vw\ +\ \rm Vs\cdot\ cos\alpha \tag{38}\]  
where Vw is the wave's velocity.