## 2 Mathematical Model
### Boundary Integral Equation for the \(TGF\_N\_V\)  
According to (A7) in Appendix A, one can obtain the boundary integral equation  
\[ \begin{array}{ll}2\pi\psi(t,\mathbf{p})+\int_{\text{S}_{\text{B}}(t)}\Big{(} \overline{G}\frac{\partial\psi(t,\mathbf{q})}{\partial n_{q}}-\psi(t,\mathbf{q})\frac {\partial\overline{G}}{\partial n_{q}}\Big{)}\text{d}s_{q}=\int_{0}^{t} \mathrm{d}\tau\int_{\text{S}_{\text{B}}(t)}\Big{(}\widetilde{G}\frac{\partial \psi(\tau,\mathbf{q})}{\partial n_{q}}-\psi(\tau,\mathbf{q})\frac{\partial\widetilde{G} }{\partial n_{q}}\Big{)}\text{d}s_{q}\\ +\frac{1}{8}\int_{0}^{t}\Big{(}\psi(\tau,\mathbf{q})\frac{\partial\widetilde{G}}{ \partial\tau}-\widetilde{G}\frac{\partial\psi(\tau,\mathbf{q})}{\partial\tau}-2 \epsilon\omega_{0}\psi(\tau,\mathbf{q})\widetilde{G}\Big{)}\dot{\xi}(\tau)\Bigg{|} \begin{array}{ll}B\\ A\end{array}\ \mathrm{d}\tau\end{array}  \tag{10}\]  
where \(A\) and \(B\) are water surface--wetted surface of the hull intersections (see Figure 1). The boundary integral Equation (10) is different from that in [PERSON] et al. [21]. There is an addition term at the right end of Equation (10) that reveals the body-nonlinear effect, while there is no similar term in boundary integral equation from [PERSON] et al. [21]. The boundary integral Equation (10) is a mixed source and dipole distribution model. One might prefer the pure source distribution model in numerical practice. Let \(\psi_{\text{i}}\) be the velocity potential of the flow in the interior of the floating body. Through the same procedure, one can obtain the boundary integral equation with respect to \(\psi_{\text{i}}\) as follows:  
\[ \begin{array}{ll}\int_{\text{S}_{\text{B}}(t)}\Big{(}\overline{G}\frac{ \partial\psi(t,\mathbf{q})}{\partial n_{q}}-\psi_{\text{i}}(t,\mathbf{q})\frac{ \partial\overline{G}}{\partial n_{q}}\Big{)}\text{d}s_{q}=\int_{0}^{t} \mathrm{d}\tau\int_{\text{S}_{\text{B}}(t)}\Big{(}\widetilde{G}\frac{\partial \psi(\tau,\mathbf{q})}{\partial n_{q}}-\psi_{\text{i}}(\tau,\mathbf{q})\frac{\partial \widetilde{G}}{\partial n_{q}}\Big{)}\text{d}s_{q}\\ +\frac{1}{8}\int_{0}^{t}\Big{(}\psi_{\text{i}}(\tau,\mathbf{q})\frac{\partial \widetilde{G}}{\partial\tau}-\widetilde{G}\frac{\partial\psi(\tau,\mathbf{q})}{ \partial\tau}-2\epsilon\omega_{0}\psi_{\text{i}}(\tau,\mathbf{q})\widetilde{G} \Big{)}\dot{\xi}(\tau)\Bigg{|}\begin{array}{ll}B\\ A\end{array}\ \mathrm{d}\tau\end{array}  \tag{11}\]  
It is assumed that the following conditions are satisfied on the wetted surface of the hull:  
\[\left\{ \begin{array}{ll}\psi(\tau,\mathbf{q})=\psi_{\text{i}}(\tau,\mathbf{q})\\ \sigma(\tau,\mathbf{q})=\frac{\partial\psi(\tau,\mathbf{q})}{\partial n_{q}}-\frac{ \partial\psi(\tau,\mathbf{q})}{\partial n_{q}}\end{array} \,\ \ \ \text{on}\ S_{\text{B}}(t)\right. \tag{12}\]  
where \(\sigma\) is the source density. Subtracting Equation (11) from Equation (10) and taking Equation (12) into consideration yields  
\[ \begin{split} 2\pi\psi(t,\mathbf{p})+\int_{S_{\mathrm{B}}(t)} \sigma(t,\mathbf{q})\overline{\mathcal{G}}\mathrm{d}s_{q}\\ =\int_{0}^{t}\mathrm{d}\tau\int_{S_{\mathrm{B}}(t)}\sigma(\tau, \mathbf{q})\widetilde{\mathcal{G}}\mathrm{d}s_{q}-\frac{1}{8}\int_{0}^{t}\left( \frac{\partial\psi(\tau,\mathbf{q})}{\partial\tau}-\frac{\partial\phi_{\mathrm{i}} (\tau,\mathbf{q})}{\partial\tau}\right)\widetilde{G}\dot{\xi}(\tau)\left|\begin{array} []{c}B\\ A\end{array}\mathrm{d}\tau\end{split}  \tag{13}\]  
According to [PERSON] (2015),  
\[\frac{\partial\psi(\tau,\mathbf{q})}{\partial\tau}-\frac{\partial\psi_{\mathrm{i}} (\tau,\mathbf{q})}{\partial\tau}=-\sigma(\tau,\mathbf{q})n_{x}\dot{\xi}(\tau) \tag{14}\]  
where \(n_{x}\) is the horizontal component of the normal vector on the intersection of the wetted surface. Substituting Equations (14) into (13), one gets a boundary integral equation only with source points distributing on the instantaneous wetted body surface.