deletions | additions       diff --git a/Figures_2_and_3_shows__.tex b/Figures_2_and_3_shows__.tex  index 58446b6..ce2416e 100644  --- a/Figures_2_and_3_shows__.tex  +++ b/Figures_2_and_3_shows__.tex 
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Figures 2 and 3 shows the mutual information for the separated forward and backward changes in pressure for two cases: in Figure 1 2  we have used all of the measured pressure and velocity $P$ and $U$, in Figure 2 3  we have used only the ensemble average data $pe$ and $ue$. We see that the wave speeds calculated using the sum-of-squares method and with $\rho = 1040 \; kg/m^3$ are very different for the two cases: $c_{ss} = 16.3 \; m/s$ when calculated from the measured data and $c_{ss} = 43.9 \; m/s$ when calculated from the ensemble averaged data. The minimum of the mutual information occurs at a wave speed that is fairly similar for both cases: $c_{MI} = 64.1 \; m/s$ for the measured data and $c_{MI} = 70.6 \; m/s$ for the ensemble averaged data. These calculations are sensitive to the way that the probabilities $p(dp_+)$, $p(dp_-)$ and $p(dp_+,dp_-)$ are calculated. In particular, the results are sensitive to the choice of the number of bins used to calculate the histograms. Further work is required to find an algorithm for selecting the number of bins, and hence the bin sizes, that minimises the sensitivity of the method.