Influence of the characteristic length

Thanks to the previous studies, the expression of the characteristic length is known as $ \displaystyle L^* = \frac{L^2}{h} $. All the studied cases are gathered on the following plot : 

Blue points are Fluent Results. Red points are the mathematical approximation :

$ \displaystyle f_0 = \frac{2.73}{2 \pi} \sqrt{\frac{g}{L*}} $

Results and approximation match for almost all the cases. For the first one, there is a difference which can be explained by the non-validity of the long waves hypothesis for this case (L=1.5m).

The characteristic length and so the expression of the natural frequency were groped for but we find the waves celerity. In fact, with the long waves hypothesis, the waves' celerity is equal to : $ c = \sqrt{gh} $

So the new expression of the natural frequency is :

$ \displaystyle f_0 = 0.53 \frac{c}{L} $

$\Rightarrow$ To conclude this part about the natural frequency, we found an expression function of a characteristic length and it does not depend on the fluid's viscosity.