\label{euler_basico}
    \frac{\partial \mathbf{v}}{\partial t}
    + (\mathbf{v} \cdot {\boldsymbol{\nabla}}) \mathbf{v}
    &= - \frac{1}{\rho} {\boldsymbol{\nabla}}p - g \hat{z}, \\ 
    {\boldsymbol{\nabla}}\cdot \mathbf{v} &= 0 .
    \label{eq:incompressible}\end{aligned}$$ Under certain assumptions, to be discussed in the following paragraphs, the evolution of the free surface can be adequately described by means of a vector equation for the two horizontal components of the velocity at the interface, plus an equation for the local height of the fluid column.

Linear dispersion relation
--------------------------

Considering the case of very small amplitude waves, one can linearize the system of Eqs.  and (see, e.g., [@landau_fluid_2004]). The solutions of the resulting equations are gravity waves with the following dispersion relation $$\label{rel_disp_tot}
    \omega^2 = gk \frac{1 - e^{-2kh_0}}{1 + e^{-2kh_0}} .$$

We are interested in the shallow water case, i.e., when $h_0 \ll
L \Rightarrow h_0 k \ll 1 $. In that limit the following dispersion relation results $$\label{rel_disp_sw}
    \omega = \sqrt{g h_0} k = c_0 k ,$$ where $c_0 = \sqrt{g h_0}$ is the phase velocity. Note in this case waves are not dispersive, unlike the general case given by Eq. .

Shallow water model
-------------------

It is possible to derive a set of non-linear equations for the surface height and the 