The Fast Fourier Transform of initial water level for five various initial water levels (first plot) allows to know that the initial water level has an influence on the natural frequency. Thanks to these Fast Fourier Transforms, the natural frequency can be extracted for all the cases and we obtain the second plot.

This plot represents the natural frequency (obtained after a fast fourier transform) for four initial water levels. Blue points are the Fluent results. A curve fitting returned a square root function (red points). So the natural frequency is proportional to the squared root of the initial water level : $f_0 \propto \sqrt{h} $. Therefore the characteristic length is proportional to the inverse of the initial level : $ L^* \propto \frac{1}{h} $. We can suppose that the characteristic length is proportional to the squared of tank's lenght over the intial water level : $ L^* \propto \frac{L^2}{h} $. We will verify this expression studying the influence of tank's lenght on the natural frequency.