## 4 Results and Discussion
For the flat case (Figure 7a) [PERSON]'s formulation (Equation (3)) seems to be perfectly fitting the data while [PERSON]'s modified version (Equation (4)) diverges for \(h/L>0.2\), as anticipated above in Table 1.  
For the case of the sloped bottom, the experimental values of \(f_{w}\) have been compared to the formulations of [PERSON] [16] represented by Equation (13) and the approach of the wet perimeter proposed by [PERSON] [46] in Equation (12). Both approaches follow a parallel behaviour together with the experimental data up to \(h/L=0.2\), showing a maximum between \(0.15<h/L>0.175\), but [PERSON]'s values (red line) show a fairly big margin of error, probably due to the fact that the empirical equation does not take in consideration the losses from the fluid-structure interactions and frictions. [PERSON]'s Equation (12) results showed a fairly good agreement with the experimental data despite the fact that the approach is based on the hypothesis of small angles of the divergent section of the corner wedge. The circular bottom case comparison is probably the most interesting one since shows two peculiar aspects. On one hand, the experimental \(f_{w}\) values have been compared to [PERSON]'s (also [PERSON]'s) approach for parabolic bottom shapes [16; 45], as in Equation (15), and the result was an overall small error but, as expected, a general trend weakly correlated  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**h/L**} & \multirow{2}{*}{\(f_{w}^{\text{LT}}\) **([PERSON])**} & \(f_{w}^{\text{LW}}\) & \multirow{2}{*}{\(f_{w}^{\text{EXP}}\)} & **Err [\%]** & **Err [\%]** \\  & & & **([PERSON]&Tait)** & & **([PERSON])** & **([PERSON]&Tait)** \\ \hline
0.1 & 0.77 & 0.78 & 0.76 & 1.32 & 2.6 \\
0.125 & 0.85 & 0.88 & 0.85 & 0.11 & 2.4 \\
0.15 & 0.93 & 0.96 & 0.92 & 1.09 & 4.7 \\
0.175 & 0.99 & 1.04 & 1.01 & 1.  
to the obtained data. On the other, [PERSON]'s approach (Equation (12)) with a wet perimeter (\(L^{\prime}\)) _ad-hoc_ modified to consider the circular shape of the corner wedges, shows a very good agreement both in terms of relative error and overall compliance of the trend. In order to estimate the decay function (red line of Figure 5(a)-c) characterising the damping properties, an exponential interpolation of the maxima/minima points was computed for the different bottom geometries and for each water depth, by extracting the coefficients \(C_{1}\) and \(C_{2}\) of the equation  
\[F_{\text{base}}(t)=e^{C_{1}t+C_{2}} \tag{17}\]  
With reference to Equation (16), the damping ratios \(\zeta\) were estimated at each level of water depth and for each cases of bottom geometry, taking into account the natural frequencies previously extracted. Figures 7(a)-c and 8(a)-c summarise the significant evidences of the carried out experiments, as a sort of matrix of the possible responses of the system composed by structure and liquid. More specifically, the _beating phenomenon_ can be related to the slope of the exponential decay functions displayed in Figures 8 and 9. Figures 8 and 9 clearly highlight that the flat bottom is the least responsive to the height change, meaning that its reaction to the change of water mass is slower if compared to the sloped and the circular bottom shapes. Furthermore, the sloped one, for heights lower than 10 cm, substantially keeps performing with the same timing and shows the same delays with respect to the initial impulse. An overall comparison is depicted in Figures 10 and 11, where the trends of only the experimental natural frequencies and damping ratios, with respect to the different water \(h/L\) values are represented, respectively. Figure 11: Damping coefficients vs. water depth ratio for different bottom geometries. Figure 10: Natural frequency values vs. water depth ratio for different bottom geometries. As discussed above, the behaviour of the sloped bottom geometry from the first value of \(h/L\) to \(h/L=0.175\) initially increases while once the water depth equals the height of the wedge, its trend bends to finally keep increasing.