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Kim H. Parker added From_the_upper_right_plot__.tex
about 8 years ago
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From the upper right plot in Figure 2, we see that the outstanding feature of the pdf for $dP$, $\phi(dP)$ is a distinct peak at $dP \approx - 7$ Pa. This corresponds to the nearly constant pressure change during the diastolic period when the pressure is falling at an almost constant rate. There are a number of smaller peaks that correspond to other features on the $P$ waveform, but these are relatively less important. In the lower right plot in Figure 2, there is a similar peak in $\phi(\rho cdU$ at approximately 0 Pa. This reflects the very small velocity in the middle of the vessel during diastole. Again there are a number of small peaks at negative values of $zro cdU$ corresponding to other features of the $U$ waveform.
The proposed method for calculating the local wave speed is based on the observation that while there is no direct theoretical relationship between the measured (or in this case calculated) $dP$ and $dU$ both of these parameters are equal to the sum of contributions from a forward (+) and a backward (-) wave and that the water hammer equations provide a theoretical relationship between these separated waves, $dP_\pm = \pm \rho cdU_\pm$.
From a statistical point of view, this problem can be thought of a an example of 'structural equation modelling' where the measurable manifest variables $dP$ and $dU$ are functions of the unmeasurable, latent variables $dP_+$ and $dP_-$ through a network of equations and the unknown parameter $c$. This interpretation of the method has not been exploited but may cast some light on the empirical method proposed here. [This comment should be moved to the Introduction??]
The next step in the method is to calculate the variables $dP_+$ and $dP_-$ for the assumed value of $c$.
