## 4 Generalized Impact Forces of 2D Sections
### General Hydrodynamic Coefficients  
Incompressible flow and the potential flow theory are adopted in the present study. According to [PERSON] [27] and the simplified method in [PERSON] [26], the velocity potential before the flow separation caused by the elastic vibration of the wedge is:  
\[\varphi(\xi,t)=\overline{\widetilde{w}}(s(t))\cdot\sqrt{s^{2}(t)-\xi^{2}} \tag{12}\]  
where \(\overline{\widetilde{w}}\) is the mean deflection velocity in the normal direction of the wedge surface. Based on the linear Bernoulli equation, the impact pressure over the wetted surface representing the effect of the added mass can be achieved:  
\[ \begin{array}{l}P_{\varepsilon}(\xi,t)=-\rho\frac{\partial\varphi}{ \partial t}\\ P(\xi,t)=-\rho\widetilde{w}(s(t))\cdot\sqrt{s^{2}(t)-\xi^{2}}-\rho\widetilde{ \widetilde{w}}(s(t))\cdot\frac{s(t)}{\sqrt{s^{2}(t)-\xi^{2}}}\cdot\frac{d\xi} {dt}\end{array}  \tag{13}\]  
For the \(k\)-th 2D section of the 3D wedge \(s^{k}(t)\), the generalized hydrodynamic force \(f_{ela,i}(s^{k}(t))\) representing the effect of the added mass can be written as:  
\[f_{ela,i}(s^{k}(t))=\int\limits_{0}^{s^{k}(t)}P_{\varepsilon}(\xi,t)\cdot\psi_ {i}^{k}(\xi)d\xi \tag{14}\]  
Substituting Equation (13) into Equation (14), then  
\[f_{ela,i}(s^{k}(t))=\int\limits_{0}^{s^{k}(t)}\left[-\rho\widetilde{\widetilde{ w}}\cdot\sqrt{\left(s^{k}(t)\right)^{2}-\xi^{2}}-\rho\widetilde{\widetilde{w}} \cdot\frac{s^{k}(t)}{\sqrt{\left(s^{k}(t)\right)^{2}-(\tau_{\xi}^{2})^{2}}} \cdot\dot{s}^{k}(t)\right]\cdot\psi_{i}^{k}(\xi)d\xi \tag{15}\]  
In Equation (15), it can be seen that \(f_{ela,i}(s^{k}(t))\) is related to the mean deflection velocity \(\overline{\widetilde{w}}\) and acceleration \(\overline{\widetilde{w}}\) in the normal direction of the wedge surface. By defining \(p_{j}(t)\) as the \(j\)-th order principal coordinate of the 3D elastic wedge, the mean displacement, velocity and acceleration can be obtained as:  
\[\overline{w}(s^{k}(t))=\sum\limits_{j=1}^{m}\overline{\psi}_{j}(s^{k}(t)) \cdot p_{j}(t) \tag{16a}\] \[\overline{\widetilde{w}}(s^{k}(t))=\sum\limits_{j=1}^{m}\overline{\psi}_{j}(s^ {k}(t))\cdot\dot{p}_{j}(t)\] (16b) \[\overline{\widetilde{w}}(s^{k}(t))=\sum\limits_{j=1}^{m}\overline{\psi}_{j}(s^ {k}(t))\cdot\ddot{p}_{j}(t) \tag{16c}\]  
where \(m\) is the number of modes used in the calculation; \(\dot{p}_{j}(t)\) is the first order derivative of \(p_{j}(t)\); \(\ddot{p}_{j}(t)\) is the second order derivative of \(p_{j}(t)\); \(\overline{\psi}_{j}(s^{k}(t))\) is the mean deflection of the wetted area. \[\overline{\psi}_{j}(s^{k}(t))=\int_{0}^{s^{k}(t)}\frac{\psi_{j}(\xi)}{s^{k}(t) }d\xi \tag{17}\]By substituting Equations (16b) and (16c) into Equation (15), \(f_{ela,i}(s^{k}(t))\) can be expressed as follows:  
\[f_{ela,i}(s^{k}(t))=-\sum\limits_{j=1}^{m}A_{ij}^{k}(s^{k}(t))\cdot\tilde{p}_{j}( t)-\sum\limits_{j=1}^{m}B_{ij}^{k}(s^{k}(t))\cdot\dot{p}_{j}(t) \tag{18a}\]  
\[A_{ij}^{k}(s^{k}(t))=\int\limits_{0}^{s^{k}(t)}\rho\overline{\dot{\psi}}_{j}(s ^{k}(t))\cdot\sqrt{\left(s^{k}(t)\right)^{2}-\xi^{2}}\cdot\psi_{i}^{k}(\xi)d\xi \tag{18b}\]  
\[B_{ij}^{k}(s^{k}(t))=\int\limits_{0}^{s^{k}(t)} \begin{array}{l}[\rho \overline{\dot{\psi}}_{j}(s^{k}(t))\cdot\frac{s^{k}(t)}{\sqrt{\left(s^{k}(t) \right)^{2}-\left(\tau_{\xi}^{2}\right)^{2}}}\cdot\dot{s}^{k}(t)\\ -\rho V\cos\beta\overline{\dot{\psi}}_{j}(s^{k}(t))\cdot\left(\frac{\left(s^{ k}(t)\right)^{2}}{\left(s^{k}(t)\right)^{2}-\left(\tau_{\xi}\right)^{2}}-\tan^{2} \beta\right)]\cdot\psi_{i}^{k}(\xi)d\xi\end{array}  \tag{18c}\]  
where \(i\), \(j\) are order numbers of modes;\(A_{ij}^{k}(s^{k}(t))\), \(B_{ij}^{k}(s^{k}(t))\) are the generalized hydrodynamic coefficients of the \(k\)-th section. For the stage after the flow separation, the above derivation is assumed to be valid and this assumption will be verified in Section 6.3.