deletions | additions       diff --git a/In_the_plots_to_the__.tex b/In_the_plots_to_the__.tex  index 98a10e6..6900a37 100644  --- a/In_the_plots_to_the__.tex  +++ b/In_the_plots_to_the__.tex 
...
In the plots to the right of Figue 3 we see the pdfs for $dP_+$ and $dP_-$ which depend on $dP$ and $dU$ and the assumed value of $c$. In both pdfs the salient feature is a distinct peak, this time at approximately -4 Pa, is one half of the value at the peak in $dP$. The peak in the pdf for $dU$ contribues nothing to the these parameters because it is at $dU = 0$. The shape of both of these pdfs will differ for different assumed values of $c$.  The next step of the method is based on the observation that at the 'true' value of the local wave speed we expect $dP = dP_+ + dP_-$ and $\rho cdU = dP_+ - dP_-)$. Using the deterministic equations, these identities and exact for any choice of $c$ and so we cannot use this relationship to determin $c$. From a probabilistic point of view this condition is expressed differently. It says that we expect \textit{on average} that a random sample from our distribution of $dP_+$ values plus a random sample from our distribution of $dP_-$ values should equal a random sample from our distribution of $dP$ values. Since the pdf of the sum of two random samples from different distributions is equal to the convolution of the two pdfs, we look for the value of $c$ that minimises the net distance between $\phi(dP)$ and $\phi(dP_+ + dP_-)$ and the distance between $\phi(\rho cdU)$ and $\phi(dP_+ - dP_-)$. (We have chosen to minimise the arithmetic mean of these distance although virtually identical results are obtained when we minimise the geometric mean.) As discussed above, the distance between two distributions can be measured by the Jensen-Shannon distance $\Delta_{JS}$ which is calculated from the entropies calculated from the distributions and the mean of the distributions. Once we have calculated the distributions shown inf Figure 3, this step is relatively straightforward. It is essential, however, to remember that the range of the two distributions being compared are identical. It is also necessary to realise that $lim_{x \rightarrow 0} x log x \rightarrow 0$ which otherwise can cause problems in calculating the entropies.  Figure 4 shows the Jensen-Shannon distance between $\phi(dP)$ and $\phi(dP_+ + dP_-)$, $\Delta_P$ and between $\phi(\rho cdU)$ and $\phi(dP_+ - dP_-)$, $\Delta_U$ together with the mean of these two distances $\Delta$.