## 4 Discussion
This sea state condition can be considered as the limit of a 3D smooth surface, resulting in the progressive increase in Te with the ship's speed and \(\alpha\) angle. This statement can be corroborated after representing the cases for \(\text{Tw}=10\) s (d), \(\text{Tw}=6\) s (e), and \(\text{Tw}=3\) s (f). From them, it is deduced that the maximum Te is not reached at the maximum ship speed and with \(\alpha\) angles near the beam. In fact, it is shown that negative values are produced, which means that the possibility of encountering the period has already been exceeded. However, these negative values are very close to the positive values of very high relative values, which means that any minimal changes in ship speed or \(\alpha\) angle will imply a very different situation, which has to be considered by shipmasters. It is also important to notice the very high peak value that occurs when \(\text{Tw}=6\) s, in spite of it being a smooth sea state condition. Figure 14: Rolling angles reached with different sea state conditions for \(\text{Td}=12\) s, (**a**) from 0 to 100 s and (**b**) from 200 s to 300 s.  
Therefore, although initially and from a mathematical point of view, it could be thought that the highest Te values would occur at low ship speeds and with \(\alpha\) angles close to the beam, it has been shown that this is not the case. Thus, in our opinion, the use of these original guides can be very useful for shipmasters to avoid dangerous situations on board, related to very large rolling angles before coupling occurs, and even the synchronism phenomenon, which causes the ship to capsize. However, although in some Figure 15d,e there are values out of the axes limit (tending to be infinite), it was represented in this way for the clarifying purposes of the remaining conditions. Figure 15: Surface graphs for different sea state conditions. (**a**) Tw = 21 s; (**b**) Tw = 18 s; (**c**) Tw = 14 s; (**d**) Tw = 10 s; (**e**) Tw = 6 s; and (**f**) Tw = 3 s.  
Knowing that the synchronism situation is reached when Te and Td are equal, and considering Equation (39) and all sea state conditions (21 types), it was studied which \(\alpha\) angles should be avoided for speeds from 2 to 12 knots, at intervals of 2 knots. A polar diagram with the results of combinations that produce synchronism for a Td = 9 s, where 0\({}^{\circ}\) and 360\({}^{\circ}\) correspond to the forward and 180\({}^{\circ}\) the stern, is shown in Figure 16. Each of the concentric circles represents the ship's speed. As it is shown, the only probability of synchronism occurs from 1 s \(<\) Tw \(<\) 8 s, i.e., for the lowest value of Tw, and within these ranges, a higher probability occurs for even the lowest values. Therefore, contrary to what might be supposed, synchronism does not occur only in the worst sea state conditions, a fact that shipmasters must take into account. Obviously, although the synchronism is only represented at the port side, considering symmetry, the same occurs at the starboard side. Another relevant result is that the possibilities of synchronism are concentrated for about 30\({}^{\circ}\)\(<\)\(\alpha\)\(<\) 110\({}^{\circ}\), i.e., the safest options to avoid it would be to receive the waves from quarter to aft, and more towards the forward. An important conclusion obtained from Figure 16 is that for a given ship's loading condition (Td) and waves (Tw and Lw), the limit condition of synchronism occurrence is constant and has a linear relationship determined by the ship's speed and \(\alpha\) angle. Therefore, if the shipmaster decides to alter the speed (Vs) and the heading (\(\alpha\)) at the same time in order to avoid synchronism, he must consider that the result of the product must be substantially different from the limiting value in order to avoid a new dangerous situation. From Equation (39) it is also possible to know what ship speed must be avoided when the shipmaster prefers not to alter the heading, i.e., when the \(\alpha\) is constant.