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a wave enters with a propagation velocity of 100m/s then the travel time for 
a wave enters with a propagation velocity of 100m/s then the travel time for 
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the wave between each node will be 0.01 seconds. The time step, must therefore 
the wave between each node will be 0.01 seconds. The time step, must therefore 
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be significantly less then this. Of the order $10E4$ would be appropriate. 
be significantly less then this. Of the order $10E4$ would be appropriate. 
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This stability criterion is known as the Courantâ€“Friedrichsâ€“Lewy 
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condition given by 
126 

\begin{equation} 
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dt=f\cdot \frac{dx}{v} 
128 

\end{equation} 
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where $dx$ is the mesh size and $f$ is a safety factor. To obtain a time step of 
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$10E4$, a safety factor of $f=0.1$ was used. 
131 


132 
The wave frequency content also plays a part in numerical stability. The 
The wave frequency content also plays a part in numerical stability. The 
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nyquistsampling theorem states that a signals bandwidth content will be 
nyquistsampling theorem states that a signals bandwidth content will be 
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\begin{equation} \label{eqn:samptheorem} 
\begin{equation} \label{eqn:samptheorem} 
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f_{n} \geqslant f_{s} 
f_{n} \geqslant f_{s} 
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\end{equation} 
\end{equation} 
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For example a 50Hz signal will require a sampling rate greater then 100Hz or 
For example, a 50Hz signal will require a sampling rate greater then 100Hz or 
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one sample every 0.01 seconds. The wave equation relies on a spatial frequency, 
one sample every 0.01 seconds. The wave equation relies on a spatial frequency, 
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thus the sampling theorem in this case applies to the solution mesh spacing. 
thus the sampling theorem in this case applies to the solution mesh spacing. 
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This relationship confirms that the frequency content of the input signal 
This relationship confirms that the frequency content of the input signal 
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146 
To accurately model the wave equation with high resolutions and velocities 
To accurately model the wave equation with high resolutions and velocities 
147 
means that very fine spatial and time discretisation is necessary for most 
means that very fine spatial and time discretisation is necessary for most 
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problems. 
problems. This requirement makes the wave equation arduous to 

This requirement makes the wave equation arduous to 

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solve numerically due to the large number of time iterations required in each 
solve numerically due to the large number of time iterations required in each 
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solution. Models with very high velocities and frequencies will be the worst 
solution. Models with very high velocities and frequencies will be the worst 
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affected by this problem. 
affected by this problem. 
302 
function can only be used when the $A$, $B$ and $C$ coefficients of the 
function can only be used when the $A$, $B$ and $C$ coefficients of the 
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general form are zero. 
general form are zero. 
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305 

More information about the lumping implementation used in \esc and its accuracy 
306 

can be found in the user guide. 
307 


308 
To turn lumping on in \esc one can use the command; 
To turn lumping on in \esc one can use the command; 
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\begin{python} 
\begin{python} 
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mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().LUMPING) 
mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().HRZ_LUMPING) 
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\end{python} 
\end{python} 
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It is also possible to check if lumping is set using; 
It is also possible to check if lumping is set using; 
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\begin{python} 
\begin{python} 