# Research on Influence of Tank Sloshing on Ship Motion Response under Different Wavelengths
## 5 Simulation of Rolling Motion of Liquid Carrier in Transverse Waves Condition
### Influence of Wavelength of Incident Wave on Rolling Motion of Liquid-Carrying Ship  
Cases 1-4 in Table 4 calculate the rolling motion of the ship with a liquid loading rate of 0.8 under different incident wave wavelengths (3 m, 6 m, 11 m, 15 m). The influence of the incident wave wavelength on the rolling motion of the ship under this liquid loading rate is explored, and the calculation results are shown in Figure 18.  
It can be seen from Figure 18a that when the liquid loading rate is 0.8, the order of the roll angles in the stable stage corresponding to different incident wave wavelengths is: \(\theta_{11}>\theta_{15}>\theta_{6}>\theta_{3}\). This is mainly because the ratio of the incident wave frequency to the natural frequency of the ship's roll is different. It can be seen from Table 5 that the natural frequency of the ship's roll is 2.417 rad/s under this liquid loading rate. The ratio of the wave frequency corresponding to different incident wave wavelengths to the natural frequency of ship roll is shown in Table 6.  
It can be seen from Table 6 that the frequency of the wave corresponding to the incident wave wavelength of 11 m is closest to the natural frequency of the ship's roll, and the frequency of the wave corresponding to the incident wave wavelength of 3 m is the farthest from the natural frequency. Therefore, the roll amplitude of the ship is the largest when the incident wave wavelength is 11 m, and the roll amplitude is the smallest when the incident wave wavelength is 3 m.  
Time dimensionless (the ratio of time to incident wave period) is shown in Figure 18b. It can be seen from the figure that under this liquid loading rate, when the wavelength of the incident wave is 6 m, 11 m, and 15 m, the rolling period of the ship is basically the same \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Wavelength (m)**} & **Incident Wave Frequency** & **Ratio of Incident Wave Frequency** \\  & **(rad/s)** & **to Natural Frequency** \\ \hline
3 & 4.532 & 1.875 \\
6 & 3.205 & 1.326 \\
11 & 2.367 & 0.979 \\
15 & 2.027 & 0.838 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The ratio of the incident wave frequency to the roll natural frequency when the loading rate is 0.8. Figure 18: Time history curves of ship rolling under different wavelengths when the liquid loading rate is 0.8. (**a**) Time history curve of stable phase; (**b**) the result after time dimensionless. as that of the incident wave. However, under the high frequency incident wave (when the wavelength is 3 m), the ship roll period is smaller than the incident wave period, that is, under the high frequency incident wave, the ship roll amplitude is smaller, but the speed is faster. In Cases 5-8 and 9-12 in Table 4, the rolling motion law of the ship with the liquid loading rate of 0.4 and 0 under different incident wave wavelengths is studied, respectively. The calculation results of rolling under different loading rates are compared in Figure 19.  
It can be seen from Figure 19 that when the loading rate is 0.4, the order of the roll angles in the stable stages corresponding to different incident wave wavelengths is: \(\theta_{11}>\theta_{6}>\theta_{15}>\theta_{3}\); when the loading rate is 0, the stable stages corresponding to different incident wave wavelengths are in the following order: \(\theta_{6}>\theta_{3}>\theta_{11}>\theta_{15}\). It can be seen that the order of roll angles corresponding to ships with different liquid loading rates is different, which is mainly due to the difference in the natural frequency of rolling of ships due to different liquid loading rates. Tables 7 and 8 are the ratios of the incident wave frequency to the natural frequency of the ship's roll when the loading rate is 0.4 and 0, respectively.