deletions | additions       diff --git a/We_see_from_Figure_4__.tex b/We_see_from_Figure_4__.tex  index ca2140f..7bc0970 100644  --- a/We_see_from_Figure_4__.tex  +++ b/We_see_from_Figure_4__.tex 
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\caption{Theoretical and estimated wave speeds for data calculated at different locations in the vessel during the quasi-steady period.}  \end{table}  It should be noted that the theoretical wave speed $$ is not strictly constant throughout the vessel as indicated here. There is avery a very  small variation in $$ due to the slightly different pressures at the different locations (see Figure 1) but the difference is only in the $6^{th}$ significant figure and for our purposes we can consider $$ to be uniform throughout the vessel. We also notice that the variation in $c_{SS}$ is systematic with $x/L$, seriously underestimating $$ at the inlet of the vessel and overestimating it even more at the outlet. This can be explained by the assumption used to derive the sum-of-squares estimate that there is no correlation between $dP_+$ and $dP_-$. In this single, uniform tube model of the upper thoracic aorta it is obvious that there will be a very high positive correlation between $dP_+$ and $dP_-$ at the major reflection site at the outlet of the vessel.. It also follows that there must be a negative correlation between the two variables at the inlet of the vessel where the boundary condition imposes a known $dU_+$ during systole and the assumption that $dU = 0$ at the inlet during diastole. At the middle of the tube, the forward and backward waves are expected to be less well correlated because of the large number of reflected and re-reflected waves present in the vessel during the quasi-steady period of the calculation.