## 2 Mathematical Model
### Potential Flow Theory
#### 2.3.2 Second-Order Wave Excitation Force  
In order to predict second-order (2 nd) wave exciting forces, the equations of velocity potential, wave elevation, and the position of a floater can be expanded as follows:  
\[\phi=\varepsilon\phi^{(1)}+\varepsilon^{2}\phi^{(2)}+O\Bigl{(}\varepsilon^{3 }\Bigr{)},\ \text{where}\ \varepsilon\to 0 \tag{8}\]  
\[\zeta=\zeta^{(0)}+\varepsilon\zeta^{(1)}+\varepsilon^{2}\zeta^{(2)}+O\Bigl{(} \varepsilon^{3}\Bigr{)}\,\ \text{where}\ \varepsilon\to 0 \tag{9}\]  
\[\stackrel{{\rightarrow}}{{X}}=\stackrel{{\rightarrow}} {{X}}^{(0)}+\varepsilon\stackrel{{\rightarrow}}{{X}}^{(1)}+ \varepsilon^{2}\stackrel{{\rightarrow}}{{X}}^{(2)}+O\Bigl{(} \varepsilon^{3}\Bigr{)\[ \begin{array}{rl}\overset{\rightarrow}{F}^{(2)}=&-\frac{1}{2}\rho_{w}g\oint\zeta_{r }{{}^{(1)}}\cdot\zeta_{r}{{}^{(1)}}\overset{\rightarrow}{n}dl\\ &+\frac{1}{2}\rho_{w}\iint\limits_{S_{0}}\iint\limits_{S_{0}}[\
abla\phi^{(1)} \cdot\
abla\phi^{(1)}]\overset{\rightarrow}{n}dS+\rho_{w}\iint\limits_{S_{0}} \left[\overset{\rightarrow}{X}^{(1)}\cdot\
abla\frac{\partial\phi^{(1)}}{ \partial t}\right]\overset{\rightarrow}{n}dS+\overset{\rightarrow}{\alpha} ^{(1)}\times\overset{\rightarrow}{F}^{(1)}\\ &+\rho_{w}\iint\limits_{S_{0}}\frac{\partial\phi^{(2)}}{\partial t}\overset{ \rightarrow}{n}dS\\ \overset{\rightarrow}{M}^{(2)}=&-\frac{1}{2}\rho_{w}g\oint\zeta_{r}{{}^{(1)}} \cdot\zeta_{r}{{}^{(1)}}\left(\overset{\rightarrow}{x}\times\overset{ \rightarrow}{n}\right)dl\\ &+\frac{1}{2}\rho_{w}\iint\limits_{S_{0}}[\
abla\phi^{(1)}\cdot\
abla\phi^{( 1)}]\left(\overset{\rightarrow}{x}\times\overset{\rightarrow}{n}\right)dS+ \rho_{w}\iint\limits_{S_{0}}\left[\overset{\rightarrow}{X}^{(1)}\cdot \
abla\frac{\partial\phi^{(1)}}{\partial t}\right]\left(\overset{ \rightarrow}{x}\times\overset{\rightarrow}{n}\right)dS\\ &+\overset{\rightarrow}{\alpha}^{(1)}\times\overset{\rightarrow}{M}^{(1)}+ \rho_{w}\iint\limits_{S_{0}}\frac{\partial\phi^{(2)}}{\partial t}\overset{ \rightarrow}{(x}\times\overset{\rightarrow}{n}\right)dS\end{array}  \tag{12}\]  
where \(\zeta_{r}{{}^{(1)}}\) (\(=\zeta^{(1)}-X^{(1)}_{3\;WL}\)) is the relative wave elevation along the mean undisturbed water line; \(\zeta^{(1)}\) is the wave elevation; \(X^{(1)}_{3\;WL}\) is the vertical distance of the point below the free surface; \(WL\) represents the undisturbed water line; \(\overset{\rightarrow}{\alpha}^{(1)}\) is the 1 st order rotational motion of the center of gravity in the fixed reference axes; \(S_{0}\) is the mean wetted surface; \(dS\) and \(dl\) denote the surface element and the length element of the waterline; \(\overset{\rightarrow}{n}\) is the normal vector of a length element \(dl\); \(\overset{\rightarrow}{F}^{(1)}\) and \(\overset{\rightarrow}{M}^{(1)}\) are the total 1 st order fluid force and moment; and \(\overset{\rightarrow}{\alpha}^{(1)}\) is the 1 st order rotational motion vector. In order to simplify the 2 nd order wave exciting force, the quadratic transfer function (QTF) was applied in the hydrodynamic force analysis. Considering a pair of the regular incident waves with \(\left(a_{jm},\;\omega_{jm},\;x_{m},\;\alpha_{jm}\right)\) and \(\left(a_{kn},\;\omega_{kn},\;x_{n},\;\alpha_{kn}\right)\), the interaction of two waves results in the 2 nd order wave exciting force due to the difference or the sum frequency. The 2 nd order wave exciting force with QTF matrices in the time domain can be expressed as:  
\[