## 2 Mathematical Formulation  
Figure 3 shows the parameters of the impact with three degrees of freedom, due to the relative motion of the wedge with the water. The new components appearing for vertical and tangential velocity are calculated as:  
\[\mathrm{w_{1}\;=\;w+\frac{1}{2}v\sin(2\beta_{1})-v_{r}\sin(\alpha)\cos(\beta_ {1})} \tag{20}\]  
\[\mathrm{w_{2}\;=\;w-\frac{1}{2}v\sin(2\beta_{2})+v_{r}\sin(\alpha  
\[\hat{\mathtt{z}}\ =\ \left[ \begin{array}{c}V_{\mathrm{cg}}\cos(\phi)-\mathrm{ ytan}(\beta_{2}),y<0\\ V_{\mathrm{cg}}\cos(\phi)-\mathrm{ytan}(\beta_{1}),y>0\end{array} \right] \tag{27}\]  
\[\gamma\ =\ \cos^{-1}\left(\frac{2}{|\mathrm{r}|}\right) \tag{28}\]  
\[\alpha\ =\ \left[ \begin{array}{c}\gamma-\beta_{2},y<0\\ \gamma-\beta_{1},y>0\end{array} \right] \tag{29}\]  
where \(\hat{y}\) and \(\hat{\mathtt{z}}\) are the horizontal distance and vertical distance in a point on the hull to the CG (Gravity centre), \(|\mathrm{r}|\) is the total distance between the centre of rotation and a point in the wedge, and \(\gamma\) is the angle between the vector \(\mathrm{v}_{\mathrm{r}}\) to the horizontal axis \(\mathrm{y}\). The potential function that represents the flow during the impact with three degrees of freedom is:  
\[\varphi\ =\ \left\{ \begin{array}{c}-\mathrm{A}_{\beta 1}\mathrm{w}_{2}\sqrt{ \mathrm{c}^{2}-\left(-\mu+\mathrm{y}\right)^{2}},\ \mathrm{y}<0\\ -\mathrm{A}_{\beta 2}\mathrm{w}_{1}\sqrt{\mathrm{c}^{2}-\left(-\mu+\mathrm{y} \right)^{2}},\ \mathrm{y}>0\end{array} \right\}. \tag{30}\]  
The models by [PERSON] [2] and [PERSON] [3] over-predict the pressure, and two correction factors are proposed: a correction factor of the deadrise angle \(\mathrm{A}_{\beta}\), and a correction factor for the jet energy \(\mathrm{c}_{\mathrm{jet}}(\mathrm{y})\), \(\mathrm{A}_{\beta}\) is a correction factor that depends on the deadrise angle of the wedge. By substitution in Equation (1) the pressure distribution can be computed as:  
Figure 4: Asymmetric impact with rotational motion. Figure 3: Impact with three degrees of freedom. \[\frac{\mathrm{p}}{\mathrm{p}}\ =\ \left[ \begin{array}{c}\mathrm{c}_{\mathrm{jet}}(y) \left[\mathrm{A}_{\mathrm{B1}}\left(\dot{w}-\frac{1}{2}\dot{v}\sin(2\beta_{2})+ \mathrm{|r}\dot{w}\sin(\alpha)\cos(\beta_{2})+v\mathrm{v}\cos(2\beta_{1})+v_{ \mathrm{r}}\omega\sin(\alpha)\sin(\beta_{2})\right)\\ \cdot\sqrt{c^{2}-\left(-\mu+y\right)^{2}}+\mathrm{A}_{\mathrm{B2}}w_{2}\frac{ \mathrm{c}\dot{c}+\left(-\mu+y\right)\dot{w}}{\sqrt{c^{2}-\left(-\mu+y\right) ^{2}}}-\frac{1}{2}\left(\frac{\mathrm{A}_{\mathrm{B2}}w_{2}\left(-\mu+y\right) }{\sqrt{c^{2}-\left(-\mu+y\right)^{2}}}+v_{\mathrm{r2}}\right)^{2}\right]y<0\\ \mathrm{c}_{\mathrm{jet}}(y)\left[\mathrm{A}_{\mathrm{B1}}\left(\dot{w}+\frac{ 1}{2}\dot{v}\sin(2\beta_{1})+\mathrm{|r}\dot{w}\sin(\alpha)\cos(\beta_{1})+v \mathrm{v}\cos(2\beta_{1})+v_{\mathrm{r}}\omega\sin(\alpha)\sin(\beta_{1}) \right)\right]\\ \cdot\sqrt{c^{2}-\left(-\mu+y\right)^{2}}+\mathrm{A}_{\mathrm{B1}}w_{1}\frac{ \mathrm{c}\dot{c}+\left(-\mu+y\right)\dot{w}}{\sqrt{c^{2}-\left(-\mu+y\right) ^{2}}}-\frac{1}{2}\left(\frac{\mathrm{A}_{\mathrm{B1}}w_{1}\left(-\mu+y\right) }{\sqrt{c^{2}-\left(-\mu+y\right)^{2}}}+v_{\mathrm{t1}}\right)^{2}\right]y>0 \end{array} \right] \tag{31}\]  
where \(\mathrm{c}_{\mathrm{jet}}(y)\) is a correction factor due to jet energy, which is evaluated as:  
\[\mathrm{c}_{\mathrm{jet}}(y)\ =\ \tan\left(12\left|1-\frac{\left(-\mu+y\right)}{ \mathrm{c}}\right|\right) \tag{32}\]  
In the two dry chines-phase, the values \(\mathrm{c}\), \(\dot{\mathrm{c}}\), \(\mu\), and \(\dot{\mu}\) are calculated with Equations (14)-(17). In the one wet chines-phase, the boundary condition is:  
\[