# The Effect of Porous Media on Wave-Induced Sloshing in a Floating Tank
## 2 Methodology
### Equivalent Mechanical Model  
In addition to the EL method, the mechanical model can be useful to solve the sloshing problem in porous media by means of a series of motion equations. For the rectangular tank, the mechanical model consists of a fixed mass \(m_{f}\) located at the height \(z_{f}\), and infinite mass-dashpot-spring systems (denoted by \(m_{n}\), \(c_{n}\), and \(k_{n}\)) located at the heights \(z_{n}\) as shown in Figure 2. The equivalent masses, stiffness, damping ratios, and location heights can be expressed as [24]:  
\[m_{n}=\frac{8\gamma\rho B}{a_{n}^{3}L}tanh\;a_{n}h \tag{30}\]  
\[k_{n}=m_{n}\omega_{n}^{2}=\frac{8\gamma\rho gB}{a_{n}^{2}L}tanh^{2}a_{n}h \tag{31}\]  
\[\xi_{n}=\frac{\alpha_{1}}{2\omega_{n}}=\frac{\gamma\mu}{2k\rho\omega_{n}}= \frac{\gamma\mu}{2k\rho}\sqrt{\frac{cotha_{n}h}{a_{n}g}} \tag{32}\]  
\[z_{n}=h+\frac{2}{a_{n}sinha_{n}h}-\frac{1}{a_{n}tanha_{n}h} \tag{33}\]  
\[m_{f}=m_{w}-\sum_{n=0}^{\infty}m_{n}=\gamma\rho BLh\left[1-\sum_{n=0}^{\infty }\frac{8}{a_{n}^{3}L^{2}h}tanha_{n}h\right] \tag{34}\]  
\[z_{f}=\frac{1}{m_{f}}\left[\gamma\rho B\left(\frac{Lh^{2}}{2}+\frac{L^{3}}{1 2}\right)-\sum_{n=0}^{\infty}m_{n}z_{n}\right] \tag{35}\]  
where \(a_{n}=(2n+1)\pi/L\), \(n=0,1,2\cdots\), \(m_{w}=\gamma\rho BLh\) denotes the total water mass in the tank, and \(B\) is the tank width. The above equivalent parameters are obtained when the tank is subject to horizontal excitations. However, it is not sufficient to describe the angular motion studied in this paper. The moment of inertia should be appended to satisfy dynamic equilibrium. As the tank is subjected to a harmonic pitch motion, its angular displacement is expressed by:  
\[\theta(t)=\theta_{0}sinovt \tag{36}\]  
where \(\theta_{0}\) is the amplitude of angular displacement. The velocity potential on the walls and bottom should satisfy the impermeable boundary condition as:  
\[\frac{\partial\phi}{\partial x}=\theta_{0}\omega zcos\omega t,\text{ on wetted walls} \tag{37}\]\[\frac{\partial\phi}{\partial z}=-\theta_{0}\omega xcos\omega t,\text{ on bottom} \tag{38}\]  
Hence the velocity potential can be expressed as:  
\[ \begin{array}{ll}\phi=\theta_{0}\omega cos\omega t&\sum\limits_{n=0}^{\infty} 4(-1)^{n}\bigg{[}\frac{sinh_{n}\big{(}z-\frac{h}{2}\big{)}}{b_{n}^{2}h}\frac{ sinh_{n}x}{cosh_{n}\frac{1}{2}}+\frac{sin_{n}x}{a_{n}^{2}L}\frac{cosh_{n}(z-h)}{ simha_{n}h}\bigg{]}\\ &+\theta_{0}\omega\sum\limits_{n=0}^{\infty}(D_{1n}cos\omega t+D_{2n}sin\omega t )sin_{n}x\text{ }cosh_{n}z\end{array}  \tag{39}\]  
where \(b_{n}=(2n+1)\pi/h\). On the free surface, the linear kinematic and dynamic boundary conditions are combined and applied at \(z=h\) as:  
\[\frac{\partial^{2}\phi}{\partial t^{2}}+\alpha_{1}\bigg{(}\frac{\partial\phi} {\partial t}-\frac{\partial\psi}{\partial t}\bigg{)}+g\frac{\partial\phi}{ \partial z}=0 \tag{40}\]  
Substituting Equation (39) into Equation (40), the coefficients \(D_{1n}\) and \(D_{2n}\) can be solved as:  
\[D_{1n}=\frac{-p_{n}}{p_{n}^{2}+q_{n}^{2}}\frac{4\omega^{2}}{sina_{n}xcosh_{n} h}\left[\frac{sinh_{n}x}{b_{n}^{3}hcosh_{n}\frac{L}{2}}+\frac{(-1)^{n}sina_{n}x}{ a_{n}^{3}Lsinh_{n}h}\right] \tag{41}\]  
\[D_{2n}=\frac{q_{n}}{p_{n}^{2}+q_{n}^{2}}\frac{4\omega^{2}}{sina_{n}xcosh_{n} h}\left[\frac{sinh_{n}x}{b_{n}^{3}hcosh_{n}\frac{L}{2}}+\frac{(-1)^{n}sina_{n}x}{ a_{n}^{3}Lsinh_{n}h}\right] \tag{42}\]  
where \(p_{n}=\omega^{2}-ga_{n}tanh_{n}h\) and \(q_{n}=\gamma a_{1}\omega\). The overturning moment due to the sloshing fluid can be obtained by integrating the pressure along the wetted boundary as:  
\[