Analysis:



Looking at this data, we were unable to make very many generalized conclusions.  Based on the data from the grid-vertex location graphs, we produced additional graphs of the distance a given vertex moved from its rest position to its deformed position, as compared with its initial distance from the point of impact of the BB.  Not surprisingly, we found that pixels closer to the impact site had a strong tendency to move a greater distance within the first few microseconds of the impact.  This effect is illustrated nicely in Figure \ref{fig:grid_test1_dist_v_dist}, which shows two nodes, one at around $80 pixels$ ($1.3''$) and one at around $140 pixels$ ($2.8''$).  The first node appears to be the expected edge of the deformation as the bullet pushes the glass inwards.  We are not certain what causes the second node.

\begin{figure}[h!]
\centering
\includegraphics[width=6.5in]{grid_test1_dist_v_dist}
\caption{Graph of Point Displacement vs. Point Radius}
\label{fig:grid_test1_dist_v_dist}
\end{figure}

With our two-frame trial run, we were able to compare two consecutive sets of position changes.  As can be seen in Figure \ref{}, in the first $13 \mu s$ after the impact of the BB, the glass deforms as in the first trial; but in the second $13 \mu s$, the glass near the BB doesn't show much deformation (probably because it has begun to crack), but there is a ring of deformation at a radius of around $100 pixels$ ($2''$).  We do not know if this is caused by a shock wave propagating outwards at some velocity or by a secondary local deformation caused by the cracking of the glass.

\begin{figure}[h!]
\centering
\includegraphics[width=2.25in]{grid_test1_graph1}
\caption{Graphs of the grid vertices from Trial 1, as shown in Figure \ref{fig:grid_test1_frames}}
\label{fig:grid_test1_graphs}
\end{figure}

If we make the assumption that the second peak in the second chart is caused by a shock wave propagating outwards, we can attempt to estimate that shock wave's velocity.  In Figure \ref{}, we take the radial distances of the pixels that moved the greatest distance within the specified time windows, and take their distance from the impact site to be the radius of the shock wave.  Making this assumption, we have two radii (measured at $23$ and $87$ pixels, respectively), and a time delta ($13 \mu s$).  This gives us a velocity of $1,530 m/s$.

As noted in Table \ref{table:glass_wave_propagation_rates}, waves propagate at speeds that are on the same order as $1,530 m/s$.  The slowest propagation rate is roughly $1.5\times$ the number that we calculated for a possible propagation velocity.  This is far enough off that it is unlikely that what we are seeing is strict wave propagation, but it is close enough that what we are seeing is likely a related phenomenon.

\begin{table}
\centering
\caption{Glass Wave Propagation Rates}
\begin{center}
\small
\begin{tabular}{c|c|c}
\emph{Glass Type}	&	\emph{velocity of longitudinal waves (m/s)}	&	\emph{velocity of shear waves (m/s)} \\
Fused Silica					&	5968	&	3764 \\
Glass, heavy silicate flint	&	3980	&	2380 \\
Glass, light borate crown	&	3980	&	2380 \\
Glass, pyrex					&	5640	&	3280
\end{tabular}
\label{table:glass_wave_propagation_rates}
\end{table}


Sources of Error:

There are a number of possible sources of error in this data.  One error source is the resolution, both pixel resolution and time resolution, of the Phantom HSV camera.  Capturing video of an event as quick as the deformation that we are interested in is at the edge of the Phantom's range; we had to use a frame rate near its maximum, and as a result a resolution near its minimum, to get any deformation imagery at all.  This has two effects:  First, we are only able to view at most two frames of deformation, so we have very few frames and relatively little timing data to work with.  Second, there are only about 10 pixels between each vertex in the grid that we are watching, so an error of a single pixel has a substantial impact.

Additionally, the software that we used to find the pixel locations of the grid points was quite simplistic, and depended on a lot of human interaction.  It was easy for a point measurement to be off by one or more pixels.

Based on these factors, we expect the position error to be on the order of $\pm2 pixels$, or $\frac{1}{12}''$.  We are interested in time estimates on the order of two or three frames, so time error may as much as half a frame duration, or $7 \mus$.

We feel that the experiments that we ran were approximately as thorough as possible given the limitations of the Phantom HSV camera.  Were we to repeat the experiments, we would follow the same procedure.  However, there would be substantial benefit to a future group with more sophisticated analysis software and an HSV camera capable of higher frame rates at a higher resolution, re-running these experiments to attempt to gain data of greater accuracy.

