A very simple and important relationship between the change in pressure dP and the change in velocity dU in a wavefront can be derived from the method of characteristics solution. {mathematical details}
For forward waves
dP+ = ρc dU+
and for backward waves
dP- = - ρc dU-
where ρ is the density of blood, c is the wave speed and '+' and '-' indicate forward and backward waves.
These equations are called the water hammer equations and they are the result of applying the conservation of mass and momentum to the wave. They show that the pressure and velocity waveforms in the arteries are not independent of each other as is often thought. In unidirectional waves there is a simple linear relationship between P and U.
The difference between the two waveforms measured in the arteries is because of the existence of both forward and backward waves during most of the cardiac cycle. In the main, the forward waves are due to the heart and the backward waves are due to reflections of the forward waves at different sites in the arteries and microcirculation (although there can be forward waves that are re-reflections of backward reflected waves and backward waves that are re-re-reflections of re-reflected forward waves, etc., etc., etc... [or should that be etc., cte., etc...?])