## 4 Mathematical Model
### Governing Equation  
In this study, it is assumed that the seawater around the floating body is a Newtonian fluid, which is homogeneous and isotropic. Specifically, the fluid is assumed to be ideal, incompressible, inviscid and irrotational. While the potential flow \(\phi\) satisfies the Laplace equation and boundary conditions, it can be solved using [PERSON]'s function. According to the source distribution method, the floating body can be separated into numerous grids with evenly distributed source points. The total velocity potential based on the linear wave theory can be written as:  
\[\phi(x,y,z)=\phi_{I}(x,y,z)+\phi_{D}(x,y,z)+\sum_{j=1}^{6}\zeta_{j}\phi_{Rj}(x, y,z), \tag{1}\]  
where \(\phi_{I}(x,y,z)\) is incident velocity potential; \(\phi_{D}(x,y,z)\) is diffraction velocity potential; \(\phi_{Rj}(x,y,z)\) is radiation velocity potential. \(\zeta_{j}\) means the motion displacement; \(j=1,2,\ldots,6\) represents surge, sway, heave, roll, pitch and yaw, respectively.  
Incident velocity potential \(\phi_{I}\) can be described as below:  
\[\phi_{I}e^{-i\omega_{o}t}=\frac{-iga}{\omega_{o}}e^{b_{0}z}e^{i(k_{x}xcos\theta +k_{y}sin\theta)}e^{-i\omega_{o}t}, \tag{2}\]  
where a is the incident wave amplitude; g is gravity; and \(\theta\) the angle of wave incidence (e.g., \(\theta=0^{\circ}\) for heading wave, and \(\theta=90^{\circ}\) for beam wave, respectively).  
The ith mode exciting force \(F_{i}^{ext}\) can be derived from the following equation by adopting the incident and diffraction potentials:  
\[F_{i}^{ext}=-i\rho\omega\iint_{s}[\phi_{1}(x,y,z)+\phi_{D}(x,y,z)]\cdot n_{i} (x,y,z)dS, \tag{3}\]  
where \(\rho\) is seawater density; \(S\) the wetted body surface of the floating body; and \(n_{i}\) the ith mode normal vector on the wetted body surface.  
The added mass coefficient \(\mu_{ij}\) and the radiation damping coefficient \(v_{ij}\) can be derived from the radiation potential and written as:  
Figure 7: The schematic diagram of earth (O\({}_{0}\)-X\({}_{0}\)Y\({}_{0}\)Z\({}_{0}\)) and body-fixed (O-XYZ) coordinate systems.  
\[F_{i}^{R}\zeta_{j}=i\rho\omega\iint\limits_{s}\sum_{j=1}^{6}\zeta_{j}\phi_{Rj} \cdot n_{i}(x,y,z)dS=\sum_{j=1}^{6}\Big{(}\omega^{2}\mu_{\bar{\imath}j}+i\omega v _{i\;j}\Big{)}\zeta_{j},\;i=1,2,\ldots,6 \tag{4}\]  
\[\mu_{\bar{\imath}j}=-\rho\omega\iint\limits_{s}\phi_{Rj}\cdot n_{i}(x,y,z)dS \tag{5}\]  
\[v_{\bar{\imath}j}=-\rho\omega\iint\limits_{s}\phi_{Rj}\cdot n_{i}(x,y,z)dS, \tag{6}\]  
where \(F_{i}^{R}\) is the ith mode radiation force.  
By adopting the 3D source distribution method, \(\phi_{D}\) and \(\phi_{Rj}\) can be expressed as:  
\[\phi_{D}(x,y,z)=\iint\limits_{s}\sigma_{D}(q)G(p,q)ds \tag{7}\]  
\[\phi_{Rj}(x,y,z)=\iint\limits_{s}\sigma_{Rj}(q)G(p,q)ds, \tag{8}\]  
where \(\sigma_{D}\) and \(\sigma_{Rj}\) represent the source strengths for diffraction and radiation, respectively. [PERSON] and [PERSON] [27] implied that both of these two potentials can be determined by solving the related boundary conditions.  
By including the results of wave exciting force, added mass, and radiation damping coefficients, the motion displacements \(\zeta_{j}\) are obtained via the following six coupled equations of motions:  
\[\sum_{j=1}^{6}\Big{[}-\omega^{2}\Big{(}M_{\bar{\imath}j}+\mu_{\bar{\imath}j} \Big{)}-i\omega v_{\bar{\imath}j}+C_{\bar{\imath}j}\Big{]}\zeta_{j}=F_{i}^{ext },\;i=1,\;2,\ldots,\;6, \tag{9}\]  
where \(M_{\bar{\imath}j}\) is the generalized mass of the floating body, and \(C_{\bar{\imath}j}\) the hydrostatic restoring force of the floating body. Finally, the body surface would be divided into numerous panels [19] to solve the velocity potential \(\phi\).