deletions | additions       diff --git a/results.tex b/results.tex  index c574d29..eef1ea7 100644  --- a/results.tex  +++ b/results.tex 
...
The direct measurement of local wave speed by the foot-to-foot method, i.e. measuring time of travel of the foot of a waveform between two measurement sites a known distance apart, is problematic in practice. The wave speed varies with axial distance along an artery because of anatomical and structural changes in the artery wall and so the 'local' wave speed must be measured over relatively short distances. Small distances between measuring sites means that travel times are small and it is difficult to make measurements with sufficiently high temporal resolution with currently available transducers. Wave speed (i.e. pulse wave velocity) measured clinically by the transit time (e.g.foot-to-foot method) is actually a weighted average measure of many local wave speeds over the arterial path length, typically carotid-to-femoral, and is not a good estimate of the local wave speed at a particular location.  Local wave speed is usually measured by the simultaneous measurement of pressure and velocity (or volume flow rate). The measurement of the local characteristic impedance, which is closely related to the local wave speed, involves averaging the impedance $Z Z  = $\widetilda{P}$/$\widetilde{Q}$ over a certain range of harmonics, where $\widetilde{P}$ and $\widetilde{Q}$ are the Fourier transforms of the measured pressure, $P$, and flow rate, $Q$. [ref. Segers] This method relies upon the assumption that reflections of the higher frequency components will die away due to viscous effects. However, the amplitude of the Fourier components of $P$ and $U$ also decrease rapidly with frequency and their ratio becomes sensitive to noise. The success of this method of calculating the characteristic impedance relies upon the judicious choice of frequencies that are free from reflections and free from the effects of noise. In wave intensity analysis, the most common method for determining the local wave speed in arteries is the PU-loop method where $P$ is plotted against $U$ and the slope of the straight line during early systole is taken as the product $\rho c$. This is based on the assumption that there are only forward waves during early systole and so the forward water hammer equation holds during this period  \[