## 2 Governing Equations of Motion  
### Modeling of Waves and Currents  
A schematic layout of the submerged floating tunnel (SFT) is shown in Figure 1. The \(X\)-, \(Y\)- and \(Z\)-axes are pointed along the axis of the tunnel, traverse and vertical to the tunnel axis, respectively. The tunnel is supported by vertical and inclined mooring cables. Figure 1: Schematic layout of the submerged floating tunnel (SFT): (**a**) side view of the SFT and (**b**) cross-sectional view of the SFT. Most of the wave theories used for the evaluation of wave forces on offshore structures are based on three parameters, i.e., water depth, wave height and wave period. The Airy wave theory, which holds for wave heights up to 10-15 m, covers a wide range and can consider significant waves of any actual sea states [13]. As the SFT is located at locations where the significant wave height is smaller than the water depth, the Airy wave theory can be used to model the sea states. According to the Airy wave theory, the first-order velocity potential \(\Phi\) is given as follows [26]:  
\[\Phi=\frac{gH}{2\omega}\frac{\cosh[k(Z+h)]}{\cosh(kh)}\sin\alpha \tag{1}\]  
where \(\alpha=k(Y\pm ct)=kY\pm\omega t\); \(k(=2\pi/\lambda)\) is the wave number; \(\omega\)\((=2\pi/T)\) is the wave frequency; \(\lambda\)\((=gT^{2}/2\pi)\) is the wavelength selected on the basis of deep-water criteria; \(H\) is the wave height; \(c\) is the wave speed; \(Y\) and \(Z\) are the vertical and transverse coordinates, respectively, having their origin at the free water surface; and \(h\) is the water depth. From the first-order velocity potential \(\Phi\), the water particle velocities in the transverse (\(\gamma\)) and vertical (\(Z\)) directions can be obtained as follows:  
\[\dot{u}=\frac{d\Phi}{d\gamma}=\frac{gkH}{2\omega}\frac{\cosh[k(Z+h)]}{\cosh(kh )}\cos\alpha \tag{2}\]  
\[\dot{v}=\frac{d\Phi}{dZ}=\frac{gkH}{2\omega}\frac{\sinh[k(Z+h)]}{\cosh(kh)}\sin\alpha \tag{3}\]  
Differentiating the above equations with respect to time, the water particle accelerations in the transverse and vertical directions can be given as follows:  
\[\ddot{u}=\frac{gkH}{2}\frac{\cosh[k(Z+h)]}{\cosh(kh)}\sin\alpha \tag{4}\]  
\[\ddot{v}=-\frac{gkH}{2}\frac{\sinh[k(Z+h)]}{\cosh(kh)}\cos\alpha \tag{5}\]  
The calculation of hydrodynamic forces on the SFT is one of the primary tasks in the design of the structure. It is also one of the most challenging tasks during numerical simulations due to the involvement of the fluid-structure interaction phenomena [27]. [PERSON]'s equation can be used to evaluate the hydrodynamic forces on the SFT and gives reliable estimates as compared to experimental data [11; 12]. The hydrodynamic forces from ocean waves and currents, acting per unit length of the SFT, are given by the modified [PERSON]'s equation as follows [26]. \[\{f(q_{i},t)\}=\frac{1}{2}C_{D}\rho_{w}D\Big{\{}\Big{|}\dot{w}_{i}\pm\dot{w}_{ i}-\dot{q}_{i}\Big{|}\Big{(}\dot{w}_{i}\pm\dot{W}_{i}-\dot{q}_{i}\Big{)} \Big{\}}+C_{M}\rho_{w}\frac{\pi D^{2}}{4}\big{\{}\ddot{w}_{i}\big{\}}-C_{A} \rho_{w}\frac{\pi D^{2}}{4}\big{\{}\ddot{q}_{i}\big{\}}\ (i=1,2) \tag{6}\]  
where subscript \(i\) denotes the \(Y\) or \(Z\) direction; \(\dot{w}_{i}\) is the water particle velocity (\(\dot{u}\) or \(\dot{v}\)); \(\ddot{w}_{i}\) is the water particle acceleration (\(\ddot{u}\) or \(\ddot{v}\)); \(\dot{W}_{i}\) is the velocity (\(\dot{U}\) or \(\dot{V}\)) of water currents acting at the centerline of the tunnel and is obtained by the linear equation: \(\dot{U}=[(h-Z)/h]\dot{U}_{c}\) or \(\dot{V}=[(h-Z)/h]\dot{V}_{c}\); \(\dot{U}_{c}\) and \(\dot{V}_{c}\) are the velocities of surface currents in the \(Y\) and \(Z\) directions, respectively. \(\dot{q}_{i}\) and \(\ddot{q}_{i}\) are the structural velocity and acceleration, respectively; \(\rho_{w}\) is the density of water; \(D\) is the external diameter of SFT; \(C_{D}\) is the drag coefficient; \(C_{M}\) is the inertia coefficient; and \(C_{A}=C_{M}-1\) is the added mass coefficient.