77 
diagonal in \autoref{eqn:gradrank1}. 
diagonal in \autoref{eqn:gradrank1}. 
78 


79 
Thus when solving for equations containing the Laplacian one of two things must 
Thus when solving for equations containing the Laplacian one of two things must 
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be completed. If the object \verb!p! is less then rank 1 the gradient is 
be completed. If the object \verb!p! is less than rank 1 the gradient is 
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calculated via; 
calculated via; 
82 
\begin{python} 
\begin{python} 
83 
gradient=grad(p) 
gradient=grad(p) 
119 
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 
This stability criterion is known as the Courant\textendash 
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condition given by 
Friedrichs\textendash Lewy condition given by 
124 
\begin{equation} 
\begin{equation} 
125 
dt=f\cdot \frac{dx}{v} 
dt=f\cdot \frac{dx}{v} 
126 
\end{equation} 
\end{equation} 
128 
$10E4$, a safety factor of $f=0.1$ was used. 
$10E4$, a safety factor of $f=0.1$ was used. 
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130 
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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accurately represented when an equispaced sampling rate $f _{n}$ is 
accurately represented when an equispaced sampling rate $f _{n}$ is 
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equal to or greater then twice the maximum frequency of the signal 
equal to or greater then twice the maximum frequency of the signal 
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$f_{s}$, or; 
$f_{s}$, or; 