$$\frac{\partial u}{\partial t} - f\nu + g \frac{\partial \eta}{\partial x} = 0$$

$$\frac{\partial \nu}{\partial t} + fu + g\frac{\partial \eta}{\partial y}=0$$

$$\frac{\partial \nu}{\partial t}+H\big(\frac{\partial u}{\partial x} + \frac{\partial \nu}{\partial y}\big)=0$$

https://people.sc.fsu.edu/~jburkardt/m_src/shallow_water_1d/shallow_water_1d.html

http://personalpages.manchester.ac.uk/staff/paul.connolly/teaching/practicals/shallow_water_equations.html

http://sanders.eng.uci.edu/matlabcodes.html

Those were 3 of the first 9 hits on google, and they all have code examples. There is no close option due to lack of effort, but I'll give it a downvote.