## 2 Theoretical Aspects
The Laplace Equation (2) given in Cartesian coordinates \(Oxy\) is still valid, by denoting with flat length the projection of the sloped length to the horizontal axis \(L^{\prime}=L/\cos\theta\). The coordinate transfer functions are described by  
\[x =\sum_{i}^{4}N_{i}(r,s)x_{i} \tag{5a}\] \[y =\sum_{i}^{4}N_{i}(r,s)y_{i} \tag{5b}\]  
where \((x_{i},y_{i})\) are the coordinates of four nodes in the left reference system of Figure 2. The shape functions of the four nodes are as follows  
\[N_{1} =(s/2L^{\prime}h)(2r-L^{\prime}) \tag{5}\] \[N_{2} =(s/2L^{\prime}h)(-L^{\prime}-2r)\] \[\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2 }}=0 \tag{2}\]0 \[\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2 }}=0 \tag{2}\]1  
The Laplace Equation (2) can be transformed to the reference coordinate system \(Ors\) by differentiation of the potential \(\phi\) with respect to coordinate \(r\) and \(s\) as written in the following equation  
\[ \begin{split}\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{ \partial^{2}\phi}{\partial y^{2}}&=\frac{\partial^{2}\phi}{ \partial r^{2}}\left[\left(\frac{\partial r}{\partial x}\right)^{2}+\left( \frac{\partial r}{\partial y}\right)^{2}\right]+\frac{\partial^{2}\phi}{ \partial s^{2}}\left[\left(\frac{\partial s}{\partial x}\right)^{2}+\left( \frac{\partial s}{\partial y}\right)^{2}\right]+\\ &\ldots+\frac{\partial\phi}{\partial r}\left[\frac{\partial^{2}r }{\partial x^{2}}+\frac{\partial^{2}r}{\partial y^{2}}\right]+\frac{\partial \phi}{\partial s}\left[\frac{\partial^{2}s}{\partial x^{2}}+\frac{\partial^{ 2}s}{\partial y^{2}}\right]+\frac{\partial^{2}\phi}{\partial s\partial r} \left[2\frac{\partial s}{\partial x}\frac{\partial r}{\partial x}+2\frac{ \partial r}{\partial y}\frac{\partial s}{\partial y}\right]\end{split}  \tag{6}\]  
Figure 2: Scheme of the coordinate transformation for the study of the Laplace equation in a sloped bottom tank. The right-hand terms of Equation (6) are determined through shape functions differentiation, as follows  
\[\left(\frac{\partial s}{\partial x}\right)^{2}+\left(\frac{\partial r }{\partial y}\right)^{2}=\left(\frac{1}{\cos\theta}\right)^{2},\,\partial^{2}r/ \partial x^{2}+\partial^{2}r/\partial y^{2}=0 \tag{7}\] \[\left(\frac{\partial s}{\partial x}\right)^{2}+\left(\frac{ \partial s}{\partial y}\right)^{2}=\left(\frac{2h}{\left(2h-L\tan\theta-2r \sin\theta\right)}\right)^{2},\,\partial^{2}s/\partial x^{2}+\partial^{2}s/ \partial y^{2}=0\] (8) \[2\left(\frac{\partial s}{\partial x}\right)\left(\frac{\partial r }{\partial x}\right)+2\left(\frac{\partial r}{\partial y}\right)\left(\frac{ \partial s}{\partial y}\right)=0 \tag{9}\]  
By substituting Equations (7)-(9) into Equation (6), the Laplace equation in the reference system \((r,s)\) is obtained  
\[\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2 }}=\left[\frac{1}{\cos^{2}\theta}\right]\frac{\partial^{2}\phi}{\partial r^{ 2}}+\left[\frac{2h}{2h-L\tan\theta-2r\sin\theta}\right]\frac{\partial^{2}\phi }{\partial s^{2}}=0 \tag{10}\]  
For a small value of \(\theta\), the right-hand terms of Equation (10) can be simplified into  
\[\frac{\partial^{2}\phi}{\partial r^{2}}+\frac{\partial^{2}\phi}{\partial s^{ 2}}=0 \tag{11}\]  
Equation (11) is the Laplace equation in the new reference system. Given that it is identical to Equation (2), the natural frequency of the water sloshing motion can be determined directly from Equation (3) by replacing the length with the projection of bottom length to the horizontal axis, which is the contact length between the water and the tank bottom. Generally speaking Equations (2) and (11) can be simplified for a specific case neglecting the inactive components of force and motion, through a proper definition of boundary conditions.