Originally \(P_r\) was identified with the uniform Windkessel pressure which can be calculated from the measured volume flow rate into the arteries, \(Q_{in}\left(t\right)\), with two assumptions: 1) the net compliance of the arteries \(C\)  is constant and 2) the flow out of the arteries through the microcirculation is governed by a form of Ohm's law (\(Q_{out}=\frac{P-P_{\infty}}{R}\), where \(R\) is the resistance of the microcirculation and \(P_{\infty}\) is the pressure at which outflow ceases; not necessarily equal to venous or mean filling pressure because of a Starling resistor effect \cite{5647639}). With these assumptions the conservation of overall arterial blood volume is
\(\frac{dP_r}{dt}=Q_{in}-k_d(P_r-P_{\infty})\)
where \(k_d=\frac{1}{RC}\) is a rate constant. The equation can be solved by quadrature for any \(Q_{in}\left(t\right)\)
\(P_r\left(t\right)\ -\ P_{\infty}=e^{-k_dt}\left(\int_0^tQ_{in}\left(t'\right)e^{k_dt'}dt'\ +\left(P_r\left(0\right)-P_{\infty}\right)\right)\)
During diastole when \(Q_{in}=0\), the solution is the decreasing exponential
\(P_r\left(t\right)-P_{\infty}=\left(P_r\left(t_n\right)-P_{\infty}\right)e^{-k_d\left(t-t_n\right)}\)
where \(t_n\) is the time of the start of diastole. We call \(k_d\) the diastolic rate constant which is the reciprocal of \(\tau=RC\), the time constant of the diastolic fall-off of pressure.
Because the Windkessel pressure defined by Otto Frank is uniform throughout the arterial compartment, it cannot propagate. Experimental data showed clearly that the reservoir pressure did propagate and consequently in later papers the reservoir pressure was defined as a pressure that propagated along the arteries like the measured pressure waveform \(P\left(x,t\right)\), where we have included the axial distance \(x\) as an independent variable because we observe that the pressure waveform changes significantly as it propagates. We then assumed that the reservoir pressure is uniform throughout the arteries but is delayed by a wave propagation time dependent on location. The reservoir wave hypothesis can be expressed mathematically
\(P\left(x,t\right)=P_r\left(t-T\left(x\right)\right)+P_x\left(x,t\right)\)
where \(T\left(x\right)\) is the wave propagation time and \(P_x\left(x,t\right)\) is the excess pressure; simply the difference between \(P\) and \(P_r\).
Using the calculus of variations we showed that \(P_r\), so defined, is the pressure waveform that results in the minimum hydraulic work that the unassisted ventricle must do to produce a given volume flow \(Q_{in}\left(t\right)\).\cite{Parker_2012}
There is currently some controversy about whether or not \(P_r\) is a wave which we believe reduces to different definitions of 'wave'. One school defines a wave as anything that propagates. By this definition \(P_r\) is a wave. Another school observes that anything that is initiated by a propagating wave will appear to be propagating; an example might be harbour masters ringing a bell to mark high tide where it would appear that bell ringing was propagating along the coast with the tide. This school would view \(P_r\) as a local phenomenon triggered by the propagating \(P\). Another, more mathematical school defines a wave as any solution of a wave equation. Most relevant to arterial hemodynamics, this definition would define any function composed of elementary waves, such as the sinusoidal waves of impedance analysis or the successive wave fronts of wave intensity analysis, as a wave. These different schools are based on definitions which cannot be proved or disproved. Given these different views, it is important not to misinterpret the meaning of 'wave' in any particular work.
The reservoir pressure as defined above is of limited use clinically because it requires the simultaneous measurement of \(P\left(t\right)\) and \(Q\left(t\right)\). An assumption based on experimental observations in the dog provides a means of calculating \(P_r\left(t\right)\) from \(P\left(t\right)\) alone. We assume that the excess pressure at any position is proportional to the flow into the aortic root,