## 2 Definition of the 2D Water Entry  
Figure 1 shows the definition sketch of an impact model of a 2D wedge. According to the separation time of the jet root from the wedge surface, the impact process is divided into two stages: the stage before the flow separation and the stage after the flow separation, as shown in Figure 1a,b, respectively. As the wedge discussed in this paper is symmetrical, only half of the wedge is analyzed. As shown in Figure 1a,b, the global coordinate system \(o-xyz\) and the local coordinate system \(o^{\prime}-\zeta\eta\zeta\) are used. The global coordinate system is stationary and its origin is located at the intersection between the symmetry line and the initial water surface. The local coordinate system is fixed to the wedge and its origin is located at the keel of the wedge. The directions of \(\alpha x\), \(\varrho z\), \(\sigma_{\xi}^{\zeta}\) and \(\sigma_{\xi}\) are shown in Figure 1a,b. The directions of \(\varrho z\), \(o\eta\) are determined by the right-hand rule. \(V\) is the constant velocity of the impacting body. The initial state of the water is static. \(t\) is the time during the water entry, and \(t=0\) means the time instant when the lowest point of the wedge reaches the water surface. \(c(t)\) is the half width of the wetted surface. \(s(t)\) is the wetted length in the local coordinate system. \(\beta\) is the deadrise angle of the wedge. \(b\) is the half width of the wedge; \(L\) is the length of the wedge surface.  
In order to analyze the dynamic response after the flow separation, the semi-analytical hydrodynamic model reported by [PERSON] et al. [7] is adopted in the present study. This model is based on the concept of Fictitious Body Continuation (FBC), which assumes that the water surface rises along the fictitious section after the flow separation. The location of the fictitious section is defined by the angle \(\alpha\), as shown in Figure 1b. The detailed explanation of this model is listed in Appendix A. Based on this hydrodynamic model, the impact pressure distribution on the wetted surface can be described as:  
Figure 1: Impact model of a 2D wedge. (**a**) Before the flow separation; (**b**) after the flow separation.  
\[P_{r}(x,t)=\frac{1}{2}\rho V^{2}[\frac{2\dot{c}(t)}{V}\frac{c(t)}{\sqrt{c^{2}(t)- \left(\tau x\right)^{2}}}-\cos^{2}\beta\frac{c^{2}(t)}{c^{2}(t)-\left(\tau x \right)^{2}}-\sin^{2}\beta] \tag{1}\]  
\[\frac{dc}{dt}=\frac{c_{v}V}{\tan\beta}\text{ before the flow separation} \tag{2a}\] \[\frac{dc}{dt}=\frac{c_{v}V}{\tan\beta(1-\cos\theta_{s})+\tan\alpha\cos\theta_ {s}}\text{ after the flow separation} \tag{2b}\]  
where \(\tau\) is a parameter introduced to correct the unreasonable negative pressure; \(\theta_{s}\) is a parameter that related to the fictitious, and it satisfies the equation \(c(t)\cdot\sin\theta_{s}=L\cos\beta\); \(c_{v}\) is the pile-up coefficient of the fluid surface.  
In order to facilitate the calculation of the generalized excitation force, the impact pressure is expressed in the local coordinate system by the transformation \(c(t)=s(t)\cdot\cos\beta\) and \(x=\xi\cdot\cos\beta\), as shown in the following.  
\[P_{r}(\xi,t)=\rho V\cos\beta\frac{\dot{s}(t)s(t)}{\sqrt{s^{2}(t)-\left(\tau \xi\right)^{2}}}-\frac{1}{2}\rho V^{2}[\cos^{2}\beta\frac{s^{2}(t)}{s^{2}(t)- \left(\tau\xi\right)^{2}}-\sin^{2}\beta] \tag{3}\]  
\[\frac{ds}{dt}=\frac{c_{v}V}{\sin\beta}\text{ before the flow separation} \tag{4a}\] \[\frac{ds}{dt}=\frac{c_{v}V}{\sin\beta(1-\cos\theta_{s})+\cos\beta\tan\alpha \cos\theta_{s}}\text{ after the flow separation} \tag{4b}\]  
It should be noted that the pressure truncation in Equation (A14) is adopted in the present study.