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

The purpose of this test is to check the ability of the model to represent runup and rundown of non-breaking long waves. To that end, a comparison was made with the analytical solution of the non-linear shallow water equation (NSWE) by \citet{CarrierGreenspan1958}, which describes the motion of harmonic, non-breaking long waves on a plane sloping beach without friction. 

A free long wave with a wave period of 32 seconds and wave amplitude of half the wave breaking amplitude ($a_{in} = 0.5 \cdot a_{br}$) propagates over a beach with constant slope equal to 1:25. The wave breaking amplitude is computed as $a_{br} = 1/\sqrt{128} \cdot \pi^3} \cdot s^{2.5} \cdot T^{2.5} \cdot g^{1.25} \cdot h_0^{-0.25} = 0.0307 m$, where $s$ is the beach slope, $T$ is the wave period and $h_0$ is the still water depth at the seaward boundary. The grid is non uniform and consists of 160 grid points. The grid size $\Delta x$ is decreasing in shoreward direction and is proportional to the (free) long wave celerity ($\sqrt{g \cdot h}$). The minimum grid size in shallow water was set at $\Delta x = 0.1 m$. 

A comparison of surface elevation and velocity snapshots is shown in \autoref{fig:CarrierGreenspan_fig}. The maximum and minimum values of the analytical solution and the XBeach computations are shown in \autoref{tab:CarrierGreenspan_tab}

${tools.image('CarrierGreenspan/', 'fig1.eps', 'Snapshots of water level and velocity','CarrierGreenspan_fig')}


${tools.table('CarrierGreenspan/', 'table.tex', 'CarrierGreenspan_score','CarrierGreenspan_tab')}

% if interpret:

The two panels in \autoref{fig:CarrierGreenspan_default_fig1} compare the XBeach results with the analytical solution. The agreement is reasonably well, though there are small deviations in the water level near the water line and the flow velocities seem to lag slightly on the analytical solution during the second part of the run down. Since the analytical solution is stationary, numerical output over multiple waves is shown in \autoref{fig:CarrierGreenspan_default_fig1}, verifying that also the numerical solution is reasonably stationary.

% endif