This example models a moving wave in a pool of shallow water.
Although these types of fluid flows are governed by the full
three-dimensional Navier-Stokes equations, they can be simplified with
a two dimensional approximation, where the z-dimension is replaced
with a variable *h* for the unknown free surface height relative to a
mean level *H*. These equations, Saint-Venant shallow water
equations, in the non-conservative form read

\[
\left\{\begin{array}{;'}
\ \frac{\partial h}{\partial t} + (u\frac{\partial h}{\partial x} + v\frac{\partial h}{\partial y} ) + (h+H)(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) = 0 \\\\

\frac{\partial u}{\partial t} + (u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} ) = -g\frac{\partial h}{\partial x} \\\\

\frac{\partial v}{\partial t} + (u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} ) = -g\frac{\partial h}{\partial y}
\end{array}\right.
\]

This model is available as an automated tutorial by selecting **Model
Examples and Tutorials…** > **Classic PDE** > **Shallow Water
Equations** from the **File** menu, viewed as a video tutorial
below. Or alternatively, follow the linked step-by-step instructions.