fluid dynamics - What does the Froude number represent?

While reading on Wikipedia, I read the following

The Froude number is defined as:

$$\mathrm{Fr} = \frac{v}{c}$$

where $v$ is a characteristic velocity, and $c$ is a characteristic water wave propagation >velocity. The Froude number is thus analogous to the Mach number. The greater the Froude >number, the greater the resistance.

While reading the following paper on shallow water waves, I could not understand how/why the Froude number was given to be

$$\mathrm{F} = \frac{gt_0^2}{L}$$

Where $t_0$ is the time scale, and $L$ is the length scale along the X and Z axes. Note that the scale for the amplitude of the surface above the mean depth is some other $A$ and not $L$.

Please explain how the author has defined the Froude number. Is it the same as the one given on Wikipedia? If so, then please provide a step-by-step reasoning to show that they are indeed the same. If not, then please explain what the Froude number mentioned in the paper represents.

EDIT: I'd like to add another minor question as it is related to the abovementioned paper. It mentions that the surface tension is accounted for but I do not understand how that is possible given that there is no term containing the surface tension $\gamma$ explicitly.