## 2 Mathematical Model
### [PERSON]'s Function for the \(TGF\_N\_V\)  
Analogous to the TGF_V [21], the [PERSON]'s function method is employed to solve the Equation (2). Let \(G(\mathbf{p},t;\mathbf{q},\tau;\epsilon)\) be the [PERSON]'s function, which can be expressed as the combination of instantaneous term \(\overline{G}(\mathbf{p},\mathbf{q};\epsilon)\) and free-surface memory term \(\widetilde{G}(\mathbf{p},t;\mathbf{q},\tau;\epsilon)\). \[G(\mathbf{p},t;\mathbf{q},\tau;\epsilon)=\delta(t-\tau)\overline{G}(\mathbf{p},\mathbf{q}; \epsilon)-H(t-\tau)\widetilde{G}(\mathbf{p},t;\mathbf{q},\tau;\epsilon) \tag{3}\]  
where \(\delta(\cdot)\) is the Dirac function, \(H(\cdot)\) is the Heaviside function, \(\mathbf{p}(x,y)\)**is** the field point, \(\mathbf{q}(\xi,\eta)\) is the source point, \(\xi\), \(\eta\) is the coordinate of \(\mathbf{q}\) at \(ox\) and \(oy\) axis, respectively, \(\tau\) is the time when \(\mathbf{q}\) pulses, and \(\epsilon\) is the non-dimensional viscosity coefficient. According to the definition of [PERSON]'s function, the free-surface memory term should satisfy the following definite conditions:  
\[\left\{ \begin{array}{ll}\
abla_{4}^{2}\widetilde{G}=0,&\mathbf{p},\mathbf{q}\in \Omega,t>\tau\\ \frac{\partial^{2}\widetilde{G}}{\partial\tau^{2}}+g\frac{\partial\widetilde {G}}{\partial\eta}-2\epsilon\omega_{0}\frac{\partial\widetilde{G}}{\partial \tau}=0,&\eta=0\\ \
abla_{q}\widetilde{G}=0,&\mathbf{q}\in S_{\infty}\\ \widetilde{G}=0,&t=\tau\end{array} \right. \tag{4}\]  
where the subscript \(\mathbf{q}\) from \(\
abla_{q}\), \(\
abla_{q}^{2}\) means that the operation is taken with respect to variable \(\mathbf{q}(\xi,\eta)\). In the second equation of (4), the sign of the viscous term changes to minus due to the relation \(\partial\widetilde{G}/\partial\tau=-\partial\widetilde{G}/\partial t\). Obviously, the definite conditions given in Equation (4) are the same as those in [PERSON] et al. [21], i.e., they are not related to the body condition. Thereby, according to [PERSON] et al. [21], the expression of the free-surface memory term \(\widetilde{G}(\mathbf{p},t;\mathbf{q},\tau;\epsilon)\) is  
\[\widetilde{G}=2\int_{0}^{\infty}\sqrt{\frac{\mathsf{g}}{k}}\mathrm{e}^{(k+e^{ 2}k_{0})(y+\eta)}\mathrm{e}^{-\epsilon\omega_{0}(t-\tau)}\cos\Bigl{(}\Bigl{(} k+e^{2}k_{0}\Bigr{)}(x-\xi)\Bigr{)}\sin\Bigl{(}\sqrt{\mathsf{g}k}(t-\tau) \Bigr{)}\mathrm{d}k \tag{5}\]  
where \(k_{0}=\omega_{0}^{2}/\mathsf{g}\) is the reference wave number in deep water conditions. On the other hand, the definite conditions for the instantaneous term \(\overline{G}\) are associated with the free-surface memory term \(\widetilde{G}\). The derivation of definite conditions for \(\overline{G}\) is detailed in Appendix A,and the final results are shown in Equation (A8), which are the same as those in [PERSON] et al. [21]. According to Appendix B of [PERSON] et al. [21], the unknown constant is \(\beta=1\). Therefore, the final definite conditions for \(\overline{G}\) are as follows:  
\[\left\{ \begin{array}{ll}\
abla_{\overline{q}}^{2}\overline{G}=\delta(\mathbf{p}- \mathbf{q}),&\mathbf{p},\mathbf{q}\in\Omega,t>\tau\\ \overline{G}=0,&\eta=0\\ \overline{G}\sim O(\frac{1}{r_{pq}}),&r_{pq}\to\infty\\ \frac{\partial\overline{G}}{\partial\eta}=-\frac{1}{8}\frac{\partial}{ \partial\tau},&t=\tau,\eta=0\end{array} \right. \tag{6}\]  
where \(r_{pq}\) is the distance between field point \(\mathbf{p}\) and source point \(\mathbf{q}\). Since the definite conditions for the instantaneous term \(\overline{G}\) and the expression of free-surface memory term \(\widetilde{G}\) are the same as those in [PERSON] et al. [21], the expression of the instantaneous term \(\overline{G}\) must also be the same as that in [PERSON] et al. [21], i.e.,  
\[\overline{G}(\mathbf{p},\mathbf{q};\mathbf{\epsilon})=\text{Re}\Big{\{}E_{1}\Big{(}- \epsilon^{2}k_{0}R_{p\overline{q}}\Big{)}-E_{1}\Big{(}-\epsilon^{2}k_{0}R_{pq }\Big{)}\Big{\}} \tag{7}\]  
where  
\[\left\{ \begin{array}{ll}R_{p\overline{q}}=y+\eta+\mathrm{i}(x-\xi)\\ R_{pq}=-|y-\eta|+\mathrm{i}(x-\xi)\end{array} \right.