<%namespace name="tools" file="tools.tex"/>

The purpose of the this test is to check whether the NSWE numerical scheme is not too dissipative and that it does not create large errors in propagation speed. 

A long wave with a small amplitude of 0.01 m and period of 80 s is sent into a domain with a length of 1 km, a depth of 5 m and a grid size of 5 m. Since only long waves are modelled a grid size of 5 m can be applied (more than 100 points per wave length). At the end, a fully reflecting wall is imposed. The wave length in this case should be $\sqrt{g \cdot d} \cdot T = \sqrt{9.81 \cdot 5} \cdot 80 = 560m$. The velocity amplitude should be $\sqrt{g/h} \cdot A = \sqrt{9.81/5} \cdot 0.01 = 0.014m$, because the these waves are shallow water waves. After the wave has reached the wall, a standing wave with double amplitude should be created. 

The computed surface elevation and velocity snapshots before the waves reach the end of the domain is shown in \autoref{fig:long_wave_propagation_snapshotshort}. The surface elevation and velocity snapshots with the standing wave pattern are shown in \autoref{fig:long_wave_propagation_snapshotlong}. The computed and analytical wave amplitudes and wave lengths are shown in \autoref{tab:long_wave_propagation_table}. Note that the maximum velocity and surface elevation amplitude is found at the wall.

${tools.image('long_wave_propagation/', 'snapshotshort.eps', 'Water levels and velocities from the start of the experiment until the wave just reaches the end of the flume. The amplitude of the analytical solution is shown with a red line.')}

${tools.image('long_wave_propagation/', 'snapshotlong.eps', 'Snapshots of water levels and velocities showing a standing wave pattern. The amplitude of the analytical solution is shown with a red line.')}

${tools.table('long_wave_propagation/', 'table.tex')}

% if interpret:

\autoref{fig:long_wave_propagation_default_snapshotshort} and \autoref{fig:long_wave_propagation_default_snapshotlong} show that the model accurately represents this situation. There is hardly any dissipation, the wave length is very close to what it should be and there is no reflection off the seaward boundary.

% endif
