3.2 A simple example of an isolated wave

We can illustrate the main features of wave intensity analysis using a very simple example. Consider a syringe pump (heart) connected to a long, uniform elastic tube (aorta). The piston of the syringe is accelerated over one time interval to some speed V, which is maintained for two time intervals and then decelerated back to zero velocity over the next time interval. The results are shown in the animation and sketched in the figure below.

In the first time interval the fluid displaced by the piston flows into the elastic tube distending it. The velocity of the fluid in the tube depends on the instantaneous velocity of the piston and the relative areas of the piston and the tube.

In the next time interval the fluid continues to flow into the tube distending the tube more and more. The change in velocity and pressure in the tube generates a forward wavefront that propagates along the tube at the wave speed determined by the distensibility of the tube and the density of the fluid.

In the next time interval there is no change except that the wavefront has advanced further along the tube. The total fluid displaced by the piston (denoted by red vertical hatching) is equal to the total volume needed to distend the tube (denoted by blue horizontal hatching). This is required by the conservation of mass since the fluid is incompressible.

In the next time interval the piston is decelerated back to zero velocity and the velocity of the fluid moving into the tube decreases. This change in velocity and pressure causes a forward travelling decompression wave.

In the next time interval the piston has stopped moving so that there is no new fluid moving into the tube. The two wavefronts continue to propagate along the tube and produce a 'wave'. This wave continues to propagate along the tube as shown in the successive time intervals.

The wave intensity produced by this simple wave at the last time interval is shown at the bottom of the figure. Note that both the compression wavefront at the front of the wave and the decompression wavefront at the back have positive wave intensities, indicating that they are forward waves. Also note that there is no wave intensity in the middle of the wave since the pressure and velocity there are constant.

Looking at the wave, we see that the fluid displaced at the back of the wave as the distended tube returns to its original diameter (the decompression wavefront at the back of the wave) is convected forward to the front of the wave where it distends the tube (the compression wavefront at the front of the wave). In front and behind the wave the fluid velocity is zero. A particle of fluid in the tube will be accelerated as the front of the wave passes and then decelerated to zero as the back of the wave passes. The displacement of the particle of fluid is just equal to the displacement of the piston times the ratio of piston and tube areas. We see, therefore, that the movement of the wave and the movement of the fluid are very different.

In the arteries the wave speed is generally an order of magnitude larger than the fluid velocities. The pulse typically travels from the heart to the radial artery in approximately 0.1 s. The blood displaced by one contraction of the heart typically moves approximately 20 cm into the ascending aorta in 1 s. (The stroke volume is accommodated in about this distance in the ascending aorta.)