## 2 Mathematical Model
### Potential Flow Theory
#### 2.3.2 Second-Order Wave Excitation Force
\begin{array}{rl}\overset{\rightarrow}{F}^{(2)}=&\sum\limits_{m=1}^{N_{d} }\sum\limits_{n=1}^{N_{d}}\sum\limits_{j=1}^{N_{m}}\sum\limits_{k=1}^{N_{n}}a_ {jm}a_{kn}[\overset{\rightarrow}{P}_{jmm}^{+}\cos\left[\left(\omega_{jm}+ \omega_{kn}\right)t-\left(\alpha_{jm}+\alpha_{kn}\right)\right]\\ &\qquad\qquad\qquad+\overset{\rightarrow}{Q}_{jmm}^{+}\sin\left[\left( \omega_{jm}+\omega_{kn}\right)t-\left(\alpha_{jm}+\alpha_{kn}\right)\right]\\ &\qquad\qquad\qquad+\overset{\rightarrow}{P}_{jmm}^{-}\cos\left[\left( \omega_{jm}-\omega_{kn}\right)t-\left(\alpha_{jm}-\alpha_{kn}\right)\right]\\ &\qquad\qquad\qquad+\overset{\rightarrow}{Q}_{jmm}^{-}\sin\left[\left( \omega_{jm}-\omega_{kn}\right)t-\left(\alpha_{jm}-\alpha_{kn}\right)\right] \end{array}  \tag{13}\]  
where \((\overset{\rightarrow}{P}_{jmm}^{+},\overset{\rightarrow}{Q}_{jkmm}^{+})\) and \((\overset{\rightarrow}{P}_{jkmm}^{-},\overset{\rightarrow}{Q}_{jmm}^{-})\) represent in-phase and out-of-phase components of sum and difference frequency forces/moments. \(\overset{\rightarrow}{P}_{jkmm}^{+^{\prime}}\), \(Q_{jkmm}^{+^{\prime}}\), and \(\overset{\rightarrow}{P}_{jkmm}^{-^{\prime}}\) are symmetric against a pair of waves with \(\left(a_{jm},\omega_{jm},\chi_{jm},\alpha_{jm}\right)\) and \(\left(a_{km},\omega_{kn},\chi_{km},\alpha_{km}\right)\). On the other hand, \(\overset{\rightarrow}{Q}_{jkmm}^{-^{\prime}}\) is skew-symmetric against this pair of waves. \(N_{d}\) is the number of wave directions; and \(N_{m}\) and \(N_{n}\) are the numbers of wave components in the \(m\)-th and the \(n\)-th wave directions. For the purpose of establishing a more simplified equation, the sum and difference frequency components are defined as below:  
\[\overset{\rightarrow}{P}_{jkmm}^{+^{\prime}}=\frac{1}{2}\left(\overset{ \rightarrow}{P}_{jkmm}^{+}+\overset{\rightarrow}{P}_{kjmm}^{+}\right)= \overset{\rightarrow}{P}_{kjmm}^{+^{\prime}} \tag{14}\]  
\[Q_{jkmm}^{+^{\prime}}=\frac{1}{2}\left(\overset{\rightarrow}{Q}_{jkmm}^{+}+ \overset{\rightarrow}{Q}_{kjmm}^{+^{\prime}}\right)=\overset{\rightarrow}{Q}_ {kjmm}^{+^{\prime}} \tag{15}\]\[\overrightarrow{P}_{jbmm}^{-^{\prime}}=\frac{1}{2}\bigg{(}\overrightarrow{P}_{jbmm}^ {-^{\prime}}+\overrightarrow{P}_{kjmn}^{-^{\prime}}\bigg{)}=\overrightarrow{P}_{ kjmn}^{-^{\prime}} \tag{16}\]  
\[\overrightarrow{Q}_{jbmm}^{-^{\prime}}=\frac{1}{2}\bigg{(}\overrightarrow{Q}_{ jbmm}^{-^{\prime}}-\overrightarrow{Q}_{kjmn}^{-^{\prime}}\bigg{)}=Q_{kjmn}^{-^{ \prime}} \tag{17}\]  
Applying the above definitions, Equation (13) can be rewritten as below:  
\[ \begin{array}{rl}\overrightarrow{F}^{(2)}=&\sum\limits_{m=1}^{N_{d}}\sum \limits_{n=1}^{N_{d}}\sum\limits_{j=1}^{N_{p}}\sum\limits_{k=1}^{N_{n}}a_{jm} a_{kn}\overrightarrow{P}_{jbmm}^{-+^{\prime}}\cos\Big{[}\Big{(}\omega_{jm}+ \omega_{bn}\Big{)}t-\Big{(}\alpha_{jm}+\alpha_{bn}\Big{)}\Big{]}\\ &\qquad\qquad\qquad+\overrightarrow{Q}_{jbmm}\sin\Big{[}\Big{(}\omega_{jm}+ \omega_{bn}\Big{)}t-\Big{(}\alpha_{jm}+\alpha_{bn}\Big{)}\Big{]}\\ &\qquad\qquad\qquad+\overrightarrow{P}_{jbmm}\cos\Big{[}\Big{(}\omega_{jm}- \omega_{bn}\Big{)}t-\Big{(}\alpha_{jm}-\alpha_{bn}\Big{)}\Big{]}\\ &\qquad\qquad\qquad+\overrightarrow{Q}_{jbmm}\sin\Big{[}\Big{(}\omega_{jm}- \omega_{bn}\Big{)}t-\Big{(}\alpha_{jm}-\alpha_{bn}\Big{)}\Big{]}\end{array}  \tag{18}\]