## 2 Mathematical Models Used for Route Development
### Environmental Forces Originating from Sea Waves  
The mathematical model of sea waves acting on the ship's hull depends on the wave spectrum used. In the literature, two spectra are most commonly employed: JONSWAP [34] and the Pierson-Moskowitz spectrum [34]. The classification society Det Norske Veritas (DNV) [35], in its studies on position-keeping capabilities of vessels, uses the Moskowitz spectrum. Below is a mathematical model approved (found in official document--_\"Assessment of station keeping capability of dynamic positioning vessels\"_, _DNV-ST-0111_) by DNV for determining the individual components of wave forces acting on a ship's hull:  
\[X_{waves}=q\cdot B\cdot h\left(Dir,bow_{a},C_{WLaf}\right)\cdot f\left(T_{s}^{ \prime}\right) \tag{13}\]  
\[h\left(Dir,bow_{a},C_{WLaf}\right)=0.09\cdot h_{1}\left(Dir,bow_{a},C_{WLaf} \right)\cdot h_{2}\left(Dir\right) \tag{14}\]  
Figure 2: Hull shape coefficients depending on the wind attack angle.  
\[h_{1A}(bowu_{a})=0.8\cdot bow_{a}^{0.45} \tag{15}\]  
\[h_{1B}\Big{(}C_{WLafl}\Big{)}=0.7\cdot C_{WLafl}^{2},C_{WLafl}\in[0.85,1.15] \tag{16}\]  
\[dir(Dir)=\left\{\begin{matrix}Dir,0\leq Dir\leq\pi\\ 2\pi-Dir,\pi\leq Dir\leq 2\pi\end{matrix}\right. \tag{17}\]  
\[h_{1}\Big{(}Dir,bowu_{a},C_{WLafl}\Big{)}=h_{1A}(bowu_{a})+\frac{dir(Dir)}{ \pi}\Big{(}h_{1B}\Big{(}C_{WLafl}\Big{)}-h_{1A}(bowu_{a})\Big{)} \tag{18}\]  
\[h_{2}(Dir)=0.05+0.95\cdot\tan^{-1}(1.45\cdot(dir(Dir)-1.75)) \tag{19}\]  
\[f\big{(}t^{\prime}\big{)}=\left\{\begin{matrix}1,ift^{\prime}<1\\ t^{\prime-3}\cdot\mathrm{e}^{1-t^{\prime-3}},ift^{\prime}\geq 1\end{matrix}\right. \tag{20}\]  
\[\mathrm{Y}_{waves}=q\cdot L_{\mathrm{OS}}\cdot(0.09\cdot\sin(Dir))\cdot f \big{(}T_{sw}^{\prime}\big{)} \tag{21}\]  
\[\mathrm{M}_{waves}=\mathrm{Y}_{waves}\cdot\left(X_{\mathrm{Los}}+\left(0.05-0.1 4\cdot\frac{dir(Dir)}{\pi}\right)\cdot L_{\mathrm{OS}}\right. \tag{22}\]  
\[T_{s}^{\prime}=\frac{T_{2}}{0.9\cdot L_{PP}^{0.33}} \tag{23}\]  
\[T_{sw}^{\prime}=\frac{T_{2}}{0.75\cdot B^{0.5}} \tag{24}\]  
\[H_{s}=0.3125\cdot V_{w}-0.62 \tag{25}\]  
\[T_{z}=0.741\cdot V_{w}+0.536 \tag{26}\]  
where \(X_{waves}\)--sea waves force acting on the \(X\)-axis of the ship [N]; \(q\)--water pressure on the ship's hull [kg/m\(\cdot\)s\({}^{2}\)]; \(B\)--maximum breadth at water line [m]; _Dir_--waves coming from direction [\({}^{\circ}\)]; \(bow_{a}\)--angle between the vessel \(x\)-axis and a line drawn from the forecast point in the water line to the point at y = B/4 on the water line [\({}^{\circ}\)]; \(C_{WLafl}\)--water plane area coefficient of the water plane area behind the midship [-]; \(\mathrm{Y}_{waves}\)--sea wave force acting on the \(Y\)-axis of the ship [N]; \(L_{\infty}\)--longitudinal distance between the forecast and affmost point under water [m]; \(\mathrm{M}_{waves}\)--torque generated by sea current force [N\(\cdot\)m]; \(X_{Los}\)--longitudinal position of \(\mathrm{L}_{\infty}/2\) [m]; \(L_{PP}\)--length between perpendiculars [m]; \(H_{s}\)--significant wave height [m]; \(A_{WLafl}\)--water plane area for x < 0 [m\({}^{2}\)]; \(T_{s}^{\prime}\)--sea wave period for surge [s]; \(T_{sw}^{\prime}\)--sea wave period for sway [s]; \(T_{Z}\)--zero crossing periods [s]; \(V_{W}\)--actual wind speed [m/s].