## Pressure Pulse and Flow Wave Reflection

### Straight to the Heart

• Pressure and flow oscillations ( sinusoids ) travel the conduits of the circulation as waves.  At a given frequency, the pressure and flow wave both propagate at the same velocity that is characteristic of the physical properties of the blood and conduit. As a consequence, the ratio of the pressure oscillation to  flow oscillation ( the characteristic impedance ) is a constant for that particular frequency and conduit properties.
• If (incident) pressure and flow waves encounter a change in characteristic impedance (an impedance discontinuity), a portion is transmitted into the downstream system, but a portion is also reflected.
• In consequence to reflected pressure and flow waves, we find that impedance varies not only with frequency but with location – the exact position in the conduit where the pressure and flow measurements are determined.  The measured pressure is no longer the result of a simple traveling wave, but is due to the summation of both and antegrade and retrograde wave. This is true of the flow wave also.
• The results of this phenomenon can be VERY counterintuitive although the videos and schematics on this page will help you to grasp the the reasons and anticipate seemingly complicated observations.

### Reflections

Why do we have to talk about reflected waves in relation to the circulation?  Is this really necessary to understand how the thing works??

$$\LARGE \text{YEP}$$

To motive the discussion, take a look at the following pressure and flow wave recordings made at the aortic root (human, from Fourier data in Hemodynamics by WR Milnor).

Like the figure in Milnor's book, the pressure and flow have been scaled to be approximately the same size.  Now consider:  The characteristic impedance of the aorta is a ratio of sinusoidal pressure to flow and is nearly constant over a wide range of frequencies.  If we regard the flow as the input to this system, this means that we are essentially multiplying each sinusoid of the flow input by the same constant to get the pressure - the output.   We should see that the pressure and flow tracings look the same (same shape); we would just multiply the flow input by a constant (with physical units of pressure/flow) to get the pressure.  This is not what we observe as is certainly clear from the figure and your daily experience.  The flow signal looks quite a bit like the normal Doppler aortic flow signal you observe daily (since we could get the flow rate simply by multiplying the velocity by the effective cross-sectional area of the aorta), but nothing like the pressure signal.   There might be more than one reason for this, but the fact is that much of the disparity in shape between the pressure and flow signals is due to the presence of reflected waves (both pressure and flow waves).   And the disparity isn't subtle, is it?  So .. we're not going to understand the circulation very well without delving into this.

It was shown in an earlier demonstration that pressure and flow disturbances are propagated through the circulation as waves.  If the system is linear, then we can consider a wave (pressure or flow) of any complexity as being composed of sinusoids of varying frequency.  Also, each frequency can be considered separately; the different frequencies don't interact with each other. Consequently we can consider a sinusoid of a single frequency, knowing that we will be able to do the same for each frequency that makes up a more complicated wave.

The situations shown below don't occur in the circulation, but the examples are a good way to start getting a handle on wave reflections and how they affect observations in the circulation. We're not attempting to understand the circulation yet; we're just learning about the background physics so we can tackle the circulation at some level. We are again considering a long uniform elastic conduit with a sinusoidal input at the left hand side.  You can consider the input to be either a pressure wave or a flow wave.   In the next few instances, the fluid viscosity has been removed so none of the variation in wave magnitude is due to wave attenuation.

The next video depicts the propagation of a single sinusoid along an elastic conduit - a long tube.  The "long" aspect of the description has to do with the fact  that the conduit is significantly  longer than the wavelength of the sinusoid.   We don't have any long straight elastic conduits in the circulation of course, but the concepts being illustrated still apply, even for the much "shorter" tubes and the fantastic complexity of the branching circulatory network.  Being a mathematical model, we can watch the wave propagation in "real-time".  The pressure ( upper tracing ) and flow ( lower tracing ) are being shown at all locations in the conduit simultaneously with the video showing the progress of time; this is not like a cardiac cath where we are only able to see the pressure and flow at a limited number of specific locations.

The model appears rather simplistic and results from a situation where the wave continues to propagate to the right, "forever" without encountering any change in geometry or physical attributes of the conduit.  Technically the reason we would see this is that the impedance at the downstream (right) end of the conduit is "matched" to the conduit itself.  When impedances match, no reflected waves occur.

The frozen tracing below is more like what you'd see in the cath lab with time along the x-axis (like a physiograph); we have the added advantage of easily depicting the instantaneous flow rate as well.   The "measurement" in this case was made at the far upstream (left) end of the tube of the above figure, but we'd see the same thing regardless of measurement location within the conduit (for this oscillation frequency).  There's no point in showing this as a video; it's just like the frozen tracing you'd show at a cath conference.  Here's a quick recap of some things discussed previously:

• The pressure sinusoid is accompanied by a corresponding flow sinusoid of the same frequency.
• The relationship between the flow and pressure amplitudes at this particular frequency is governed by physics through the characteristic impedance.
• In this special case with NO viscosity, the pressure and flow sinusoids are (perfectly) in phase; peaks and troughs of pressure and flow coincide exactly.

The situation is dramatically altered from the above by a (virtual) clamp on the distal end (far right) of the conduit.  This imposes a drastic discontinuity in the impedance at that location.  Whatever iimpedance the conduit may have, the clamped end has infinite impedance; conceptually there's no pressure you could apply there that would cause flow.

For this contrived circumstance with a clamped conduit on the far right-hand side, the flow is forced to zero at that location.   We'll soon understand that this is due to the presence of both antegrade (blue) and retrograde (gray) waves for both pressure and flow.  While we know that these antegrade and retrograde waves exist, it would require techniques not usually employed in a clinical setting to prove it. The measured (actual) pressure and flow are depicted in red.

The only way that the flow can be held at zero at the clamp is for the retrograde flow wave to precisely cancel out the antegrade wave.   The video shows how the antegrade and retrograde flow waves at the far right-hand side have the same flow value at every instant in time.  But these 2 flows are in opposite directions within the tube and cancel each other out.  The pressure waves at the clamped end, however, are additive there and the pressure envelope (peak pressure) exhibits a doubling at the clamped end.  This is an extreme example of a closed reflection site, one where the downstream impedance (terminal or termination impedance) is greater than the characteristic impedance of the conduit under consideration.

The video also shows regular variation of the pressure and flow wave envelopes with location.  Below is a freeze frame from the above video, paused at a moment in time when the antegrade and retrograde pressure waves exhibit maximal reinforcement.  The flow waves are canceling each other out at this moment and we see the rather bizarre situation of zero flow (momentarily) at every location in the tube.  We see that over each wavelength within the conduit, the envelope exhibits 2 locations that remain pegged at zero (nodes), and also 2 that exhibit peak (envelope) values that are twice the amplitude of the antegrade wave (anti-nodes).  To couch this in other terms you may be familiar with, the waves exhibit a pattern attributable to constructive and destructive interference.

With a closed termination like this one, the pressure envelope exhibits a (global) maximum at the clamp (terminus) and a node $$1/4$$ wavelength proximal to this reflection site.  Subsequently, the envelope alternates between node and anti-node each quarter wavelength from the terminus.  The flow wave envelope alternates at the same spatial rate, but exhibits a node where the pressure anti-node occurs and vice versa.  The pressure peaks where flow wave exhibits a minimum; the flow peaks at the pressure nodes.  Don't forget however; we're only looking at one frequency so far.

The Linear Hemodynamics Active Figure allows us to place measuring devices at various / multiple locations within the system (single conduit in this case).  In the next video a measurement location has been added, for a total of 2, and the positions of the measurements in the conduit are shown as vertical, color-coded lines. The blue (upstream) line has been placed at a pressure anti-node (flow node) and the red (downstream) line is located at a pressure node (flow anti-node).

The next figure depicts the physiograph recordings (time for x-axis) of simultaneous measurements at these sites.

Color coding for the measured pressure and flow correspond to the measurement sites from the real-time recording.  I hope it will intrigue you that the large pressure oscillation at the pressure anti-node (blue) results in no flow whatsoever at that location; similarly the flow anti-node (magenta) is a location of constant (zero) pressure.  The marked oscillation in pressure results in no flow; HOW CAN THAT BE?  I hope this will serve as the cement for a concept that was discussed in an earlier article.  Pressure per se has nothing to do with driving flow through the circulation;  it's the pressure gradient that does this - the fact that pressure varies from one location to another.  More specifically, pressure gradient is one of the forces (the main one) that acts on the blood to cause acceleration ( $$\mathbf{F} = m \mathbf{a}$$).  I don't have it to refer to anymore but I think that D.A. McDonald referred to this fact in his classic monograph, Blood Flow in Arteries,  as "a blinding glimpse of the obvious".

The following video shows the pressure waves (antegrade, retrograde, measured, and envelope) on the upper tracing and the negative pressure gradient (the pressure force) on the lower.  Yes, the pressure gradient is a wave with antegrade and retrograde components also; I'm only showing the "measured" (actual) pressure gradient for this 2 Hz wave.  Although the pressure changes dramatically at the anti-node (blue vertical line), the pressure gradient there is fixed at zero.  There's NO FORCE TO MOVE THE FLUID AT THIS LOCATION.  This is also apparent by observing the video of the measured pressure wave at this location (in red).  The pressure rises and falls, but the slope of the measured pressure with respect to location in the conduit remains zero throughout; there's no pressure gradient.

We would find equally perplexing phenomena for the flow and flow gradient; you can explore this with the Active Figure if you like.

Might as well show the other kind of termination next.  If we chop the end of the conduit off, instead of clamping the tube at the end, we get an open reflection site.  In essence we've "clamped" the pressure of the conduit end at atmospheric (room) pressure.  This does NOT cause the terminal impedance to match the conduit; it's another kind of impedance mismatch and causes marked wave reflection.  Forcing the pressure to zero (open to air) at the end causes the flow wave to reinforce and a doubling of flow wave as it exits the tube; it spurts big time!!   Below is the corresponding video.

Next is the corresponding physiograph recording made at the 2 vertical lines.  Again we have locations with zero flow but double the pressure, and vice versa.  You get the picture.

### Wave Phase Relationships

We saw above, prior to clamping the end, that the pressure and flow sinusoids were perfectly in phase; the "perfect" qualifier is due to the absence of viscosity as we'll see. What's the relationship between the sinusoids with the end clamped (and still no viscosity)?

Next is a freeze frame of the video ..

.. and physiograph recordings from the 2 measurement sites.

What we're looking for is the phase relationship between pressure and flow so you're comparing the shift in the blue waves, from upper to lower trace, and also the magenta waves.

What we see is that the pressure sinusoid is now $$90^{\circ} = \pi/2$$ radians out of phase with the flow.  This is true for both locations but in fact, we'd see the same thing at every location.  Now look again at the last figure and you'll see that the pressure peak precedes the flow peak for the magenta tracings (measurement site) but the pressure peak follows the flow peak at the blue site.  The clamp has created 2 kinds of zones due to reflected waves.  The pressure sinusoid "leads" the flow sinusoid in the zone just upstream of a pressure node; pressure "lags" flow just downstream of the pressure node.  In fact we get an abrupt shift in this relationship every quarter wavelength along the conduit (the shift occurs at each node and anti-node and it won't matter if you look at pressure vs flow).

Yes this is still the contrived situation of no viscosity and nobody's got a clamp on their aorta (I hope).  But the circulation exhibits this exact phenomenon of pressure and flow sinusoids either lagging or leading each other. (Depending on wave frequency and measurement location .. read on!)  If it didn't occur to you that the flow would start to rise before the pressure does, then I hope you're intrigued because this is exactly what occurs at the root of the aorta for the fundamental frequency, i.e. the heart rate!  And it's due to reflected waves!

For a preview of things to come, we'll now input a wave of slightly different frequency than the one above.  It's still the same conduit, measurement locations are identical to above, and we've put the clamp back on the end.

Changing the oscillation frequency has brought about a change in the wavelength.  Measurement sites previously chosen to correspond to nodes and anti-nodes no longer fulfill that specification.  We'll delve into this again, but you can quickly imagine here that:

• Locations of nodes and anti-nodes for one frequency are not the same as another (any other) frequency.
• A higher frequency corresponds to a shorter wavelength and closer physical spacing between nodes and anti-nodes.
• So keep it in mind that we could position our measurement device(s) at a pressure node for a particular input frequency. But changing the input frequency appropriately will change the pressure node into an anti-node!!
• You can now imagine that wherever we put the measurement site, that location will correspond to a node for some particular frequency (and an anti-node for some other frequency).   We'll investigate the frequency dependence of the wave interaction in the frequency domain (with a spectrum) -- the input impedance.

### More on the Formation of Wave Reflections

We've focused thus far on a rather simplistic source of reflection - either a clamped or open-ended conduit.  With luck, this is rarely encountered in the circulation (OK, all too often).  Let's look however at the physics of a wave reflection.  When 2 conduits meet that have differing characteristic impedance, we seem to encounter a paradox.  Each conduit "insists" by physical law that the ratio of pressure to flow will be a specific value.  Yet somehow, at the location where the conduits meet, the ratio of pressure to flow needs to be 2 different values (2 different characteristic impedances).  Ma Nature circumvents this apparent paradox with ease, by allowing for the generation of reflected waves.  Obviously we are all familiar with the phenomenon of wave reflection and spend our days reflecting ultrasound off of structures we'd like to know more about.  The antegrade and reflected waves traveling near an impedance discontinuity must occur in such a way that characteristic impedance is "observed" in each ( every ) conduit, and yet the actual pressure and flow at the junction is a single value; pressure and flow are continuous across the junction.

The Linear Hemodynamics Active Figure allows us to add multiple conduits in series and place multiple measurement sites in each.  The next video shows the result of joining 2 conduits of different impedance; the tubes meet at the x-center of the figure with each being 200 cm in length.  The downstream conduit however has a much greater elastic modulus than upstream; the material is stiffer so that it takes greater pressure to distend the vessel.  This translates into the downstream conduit having a greater characteristic impedance and greater wave velocity.  The downstream conduit terminates in a matched impedance as if the conduit continues forever without any change in physical characteristics (no reflected waves in the downstream conduit).  (A little bit of viscosity has been added also.)

This one allows us to see how the antegrade and retrograde waves develop in the upstream conduit.  (The downstream conduit has no reflected waves by design.  The measured pressure wave is the same as the antegrade wave.)  The physical necessity is that the pressure and flow at the junction of the 2 conduits is identical.  However we must also have the condition imposed by the characteristic impedance; the ratio of pressure to flow is specified by physical aspects of the conduit.  The paradox is resolved by the creation of reflected waves at the junction.  Each of the antegrade and retrograde wave obeys the pressure-flow prescription of the characteristic impedance;  the sum of the antegrade and retrograde waves allows the pressures and flow at the junction of the conduits to be equal.  The downstream conduit could have reflected waves in it, but the sum of antegrade and retrograde must be identical at the junction (both pressure and flow).

The last video also leads us to the observation of relatively closed and relatively open reflection sites.  The downstream conduit has greater impedance than the upstream one, but it isn't infinite like the clamp was.  Part of the wave is reflected, but the rest is transmitted into the downstream conduit.  Consequently the magnitude of the reflected waves (pressure and flow) isn't great enough to completely cancel out the incident (antegrade) waves.  We have relative nodes, locations where destruction wave interference decreases the wave envelope magnitude, and relative anti-nodes; the envelope is greater at these locations but the reflected wave isn't great enough to result in the full doubling of wave amplitude.  The relationship between nodes and anit-nodes is the same however, alternating at quarter-wavelength intervals and with flow nodes at pressure anti-nodes (and vice versa).

We could also have a situation where the downstream conduit has a lower characteristic impedance.  This leads to a relatively open reflection site at the junction (not shown).  We'd see a diminution of the transmitted pressure wave and augmentation of the flow wave

### Phase Velocity Digression

We'll digress momentarily now to a topic discussed previously, wave velocity or phase velocity.  We examined the situation where there are no reflected waves as shown below:

The above video depicts the pressure and flow waves of a specific frequency traveling through an elastic conduit (no reflected waves).  We want some ideas as to how we could measure i.e. quantify the velocity of the wave transmission.  How about we put 2 transducers in the conduit at known separation distance and measure how much time it takes for the peak (or trough) to get from the upstream to downstream site.  Here's that recording:

The above tracing is color coded to correspond with colored vertical lines on the video.  Obviously there is a time delay for the pulse (pressure and/or flow) to pass from the upstream site (blue) to the downstream on (magenta).  We can readily pick off a $$\Delta t$$ from the tracing ( time between 2 peaks ) and the $$\Delta z$$ from the video (or measured directly during the experiment) to compute a velocity.  Note however that there's at least one assumption we need to be aware of that relates directly to a topic you're familiar with - aliasing.  If we've only got the 2 presure (or flow) transducers, we're assuming that the magenta peak immediately following the blue one corresponds to the same pressure peak of the spatial wave.  In actuality, there's nothing about these last pressure or flow tracings alone that tells us this!  If all we've got are the (physiograph) pressure and flow tracings, we have no idea how many spatial pulse peaks there are between the 2 measuring sites. There is a spatial Nyquist frequency for this situation, JUST as there is a temporal one that applies to analyzing Doppler flows.  We must somehow determine, from other information, that the measurement sites are no farther apart than $$1/2$$ wavelength.  The video gives us that information in this case, but it's important to understand the concept (and things are going to get a lot more confusing soon).

Also noted in the previous article, there is a better way to determine the wave velocity than picking off peaks - one that would be necessary in the absence of a simple, single frequency wave.  First we would need to perform a Fourier transform on each of the upstream and downstream recordings.  For each frequency, we would find a difference in phase angle that would be measured either in radians or degrees.  Then we would readily determine a propagation coefficient referred to as $$\beta$$, commonly reported in radians (or degrees) per cm.  Since we know the frequency ($$\omega$$) of the wave, e.g. radians/sec,  we can compute the velocity as $$\omega / \beta$$ which will have units of length/time, e.g. cm/sec.

The 2 following videos depict this concept, showing the same 2 Hz wave traveling through 2 different conduits (no reflections).  The frequency of the wave is the same, but the faster wave is more spread out over a greater distance within the conduit.  (It might help your visualization to activate both videos at the same time.)

The next 2 figures are frozen frames from the videos - slow wave above, fast wave below. Take a moment to consider and you'll realize that a rapidly moving sinusoid has a smaller number of radians per cm within the conduit.  This is how the value of $$\beta$$ is used to express the wave velocity for an oscillation of specified frequency.

### Apparent Wave Velocity with Reflections

And now we put the clamp back on the distal (right) end of the conduit.  The next video is a recap of one shown above.  Please take a moment to watch the measured (actual, in red) pressure and flow waves to try to get a handle on how fast they are moving

You'll observe that the conduit is partitioned into segments defined by the nodes and that our previous thoughts on how to measure the wave speed are out the window.  Previously, there was a delay for a pressure (or flow) wave peak to move from one location to the next.  Now we see that the wave peak arrives at the same time  as long as the measurement sites lie between any 2 nodes; there's no delay whatsoever.  The pressure oscillation rises and falls in perfect synchrony (in phase) with every other location.   This pressure is also perfectly in phase with alternating segments ( every other inter-nodal segment ) but is $$180^{\circ}$$  ($$\pi$$ radians) out of phase with all the other inter-nodal segments.  The flow wave exhibits the same phenomenon,  but of course the inter-nodal segments don't coincide spatially with the segments due to the pressure wave.

Below is a freeze frame from the video, and also the physiograph recordings at the color-coded locations.  The pressure recording locations lie on opposite sites of a pressure node and the pressure signals are $$180^{\circ}$$ out of phase between these 2 location.  But note too that we get this same result regardless of the location(s) of measurement as long as the measurement transducers are on opposite sides of a node.  We can't affect the apparent wave velocity except by crossing a node but more importantly the apparent velocity we determine has nothing to do with actual wave velocity.  For either of the 2 methods outlined in the section above, we would find an apparent wave velocity of $$\infty$$ within the region bounded by nodes.

Having the tube clamped (and with minimal friction) has resulted in standing waves for both pressure and flow.   You probably recognized this earlier, since you learned about them in high school, but maybe you hadn't thought about them with regard to the circulation.  While this doesn't occur in the circulation, there was a time when it was thought they might, and reflected waves certainly occur and confound our interpretations.

Another interesting demonstration of our inability to measure wave velocity in the presence of strong wave reflections is seen when we start looking at more complex waves - with more than one frequency.  The next video shows a pressure wave (no flow shown this time), this time with two harmonics in it.

There's a freeze frame from the video below. Try to watch the red (measured / actual) spatial wave peak in the vicinity of the 2 measurement sites, blue and magenta.  My impression is that the wave peak appears to move downstream in this locale, but the wave trough appears to move upstream - as if the wave is moving in 2 directions at the same time! It almost looks like the red wave bounces back and forth between the 2 pressure nodes.

This impression isn't remedied by observing the physiograph recordings from the 2 measurement sites (below).  The upstream site (blue) shows the (maximal) pressure peak ahead of the downstream (magenta), but the (maximal) magenta trough precedes the blue trough.   It's as if the magenta wave somehow skips ahead of the blue somewhere between the 2 measurement sites.

Seeing the antegrade and reflected waves in the video allows us to understand exactly how this is coming about but remember - that won't help you in a clinical setting. You've never been able to see this in the cath lab unless you're doing hemodynamics research.  Wave reflections in the circulation aren't as strong as shown here; the above demonstrations have been devised simply to demonstrate how erroneous your impressions can be.

(Just in case you've been irritated by the omission thus far ..)  I'll spill the beans that we've got at least one more issue to get a handle on with apparent wave velocity, but we need some viscosity to visualize it so we'll hit it a little farther down the page.  The spatial peak of the wave passing a point doesn't (in general) coincide at all with the temporal peak at that location.   So the notion that the time for the spatial wave peak to pass between 2 sites is the same thing as the time between temporal peaks at the 2 sites is completely erroneous!  That's even if we've only got one frequency in the wave. (Don't freak out yet!)

One instance where an understanding of the above comes up clinically is when the femoral sheath pressure is used as a surrogate for an aortic root recording.  In the past, the procedure for measuring the aortic gradient has involved "shifting"  the femoral pressure wave tracing in time for a presumed interval to (sort of) align it with aortic tracing; then the pressure difference between the 2 tracings was assumed to constitute the gradient.  One problem with this practice is that the pressure wave is traveling in both directions.  The pulse is transformed as it appears to propagate from upstream to downstream, but the unidirectional aspect of propagation is an illusion.  (Another problem with this is that the presence of aortic stenosis causes a highly nonlinear situation and this whole section on linear hemodynamics goes out the window.)    Sure, you will find that a particular wave shifting practice would give a best estimate of the gradient on a statistical basis (relative to other practices).  But the whole notion of shifting the recording in time is non-physical in the presence of (strong) reflected waves.

BTW.  We actually can determine the wave velocity in elastic conduits, but it's not easy.  One method involves 3 high fidelity (micromanometer) simultaneous pressure recordings at equal spacing along the conduit.  One then computes Fourier transforms on the recordings and the complex pressure at all three sites goes into an equation involving the hyperbolic inverse cosine of complex numbers (yup, really).  I was going to try to do this for my graduate studies at one time and had to call the Millar company to ask for a specially made catheter with the required 3 piezo crystals.  They thought I was nuts (I was).

### And Now, with Viscosity ..

We've been looking at some pretty unrealistic systems thus far and it's time to start adding widgets.   The video shows the wave propagation for a fluid that has a certain amount of viscosity – the property of the fluid that results in friction between fluid layers and consequent loss of energy.  As a result, the pressure and flow waves both attenuate exponentially as they are propagated left to right along the tube.  However we would also find that the ratio of magnitudes, pressure to flow, remains a constant regardless of location in the tube (for a given frequency).  This was seen previously and the ratio is known as the characteristic impedance.

The time-domain (physiograph) tracing below yields no surprises in getting from the blue upstream measurement to the magenta downstream site.  There's a time delay for the pressure wave peak to traverse from the blue to the magenta measuring sites and an appropriate decrease in the magnitude of the wave due to the friction.  The green site recording is a little perplexing.   Although the cause of decreased wave magnitude is obvious enough from the video (attenuation), we're not seeing a delay in the peak relative to the magenta site.   That's because the green site is a full wavelength downstream of the magenta site; we've exceeded the spatial Nyquist frequency and this is the spatial aliasing problem I alluded to above.

When the end of the tube is clamped we induce a marked degree of reflected waves.

However, you see readily that we no longer have nodes where pressure and flow are zero.  Instead we have relative nodes and anti-nodes; pressure and flow waves interact in such a way as to produce local maxima and minima, but the magnitude of the reflected wave is no longer great enough to fully cancel out the antegrade wave; maxima cannot be fully twice the magnitude of the antegrade wave.  You can also see that the effect of reflected waves diminishes the farther the measurement is made from the reflection site in accordance with the degree of attenuation and the distance it acts over. The wave travels twice the distance to the reflection site (round trip distance) and is attenuated according, with a smaller reflected wave resulting. If the measurement site were far enough away, the effects of the reflected wave would be negligible (and your ultrasound image gets pretty bad).  We also discussed previously that higher frequency waves are attenuated to a greater degree.

The physiograph recording alerts you to some potentially confusing observations (below).  We've already seen that the amplitude of the measured wave varies with location for the above non-viscous examples.  Now that we've got a viscous flow, you might expect that the amplitude of the wave would diminish continuously in the downstream direction.  However we're still seeing that the magenta wave is larger than the measured wave at the upstream site.  This very phenomenon is quite apparent in the circulation and is due to wave reflection!  The pulse amplitude in the femoral artery is greater than at the aorta. Really!

As promised above, we'll take one more look at the apparent wave velocity - now with viscosity added.  The next video shows the pressure wave only, in slow-motion, and allows you to focus on the red (measured) pressure wave in comparison with the blue (antegrade) wave.  With a clamp on the end of the conduit at the right end, reflected wave effects are most prominent there and diminish towards the left.

We see that antegrade (blue) wave moves at a constant wave speed to the right as always.  But the measured (red) wave appears to "gyrate".  You will see the spatial peak of the measured (red) wave seem to pass the spatial peak of the blue wave near the (relative) pressure nodes, only to lag behind the blue peak in the anti-node regions.  In the absence of viscosity, this effect is not as obvious; the measured wave actually appears to move at infinite speed.  In the presence of viscosity however, the fact that apparent wave velocity depends on location (relative to nodes and anti-nodes) is readily apparent.   Once again, it's clear that attempting to measure wave velocity by simple means can get you into a lot of trouble.

The next bit is going to seem very counter-intuitive (does to me anyway), so pay very close attention.  The arrival of the spatial peak of the wave at a specific site (the peak that your eye follows along the conduit on the video), does NOT (in general) coincide with the temporal peak that we record on the physiograph at that site.  The 2 freeze frame figures below demonstrate this fact.  The first shows the moment that the spatial peak (red wave) reaches the magenta measurement site (peak of the red wave touching the magenta line).

Now the next figure shows the red wave when the temporal peak occurs at the magenta site, i.e. the red wave touches the wave envelope at the magenta site.  This is not the same thing as passage of the spatial peak at the site!

When we watch the red spatial wave in "real time" on the video, the peak appears to move quickly in the regions near the relative nodes and slowly near the anti-nodes.    Prove it to yourself.  Say "now" when the peak of the red wave passes the blue site, the magenta, and the green (which are equidistant apart).  You'll see that there's a much shorter time interval from blue to magenta (moving quickly) vs magenta to green (moving slowly).

If we look at the physiograph recordings for the sites, however, the story is entirely different.  There's quite a larger delay between the temporal peaks of the blue and magenta sites vs magenta to green.  It's the exact opposite conclusion than we'd get from watching the wave move in real time. (OK to freak out now.)

In fact, determination of the wave speed by observing the passage of some feature of the wave (peak, trough, wiggle) is fraught with potential error and basically ill-conceived in the presence of significant wave reflection.  As you  see from the latter, it doesn't even help us to be able to see the whole wave as we do in the videos.

You may have come across recommendations in the literature that it's preferred to look at the "foot-to-foot" time interval, the earliest upstroke of the temporal recording,  when attempting to estimate wave velocity.  This recommendation is based on some of the principles we've just considered.   That sharp upstroke includes a relatively high frequency component of the wave and the higher frequencies are attenuated to a greater extent than the low.  So this recommendation stems from an attempt to minimize the effect of reflected waves -- good idea!!

### Reflection Coefficient

Ah, the time has come for some math.  We'll let $$\large P_a$$ be a complex number (Fourier domain) that stands for the sinusoidal antegrade pressure wave; $$\large P_r$$ is the complex retrograde wave.  Traditionally we use a capital Greek gamma,  $$\large \Gamma$$ to indicate the reflection coefficient, the ratio of the reflected wave over the antegrade.  The fact that the quantity is complex allows it to embody information about both the magnitude relationship and the phase relationship between the antegrade and retrograde waves.

$$\Large \Gamma = \frac{P_r}{P_a}$$

Consequently the measured pressure is just the sum of the antegrade and retrograde waves:

$$\Large P_m = P_a + P_r = P_a (1 + \Gamma)$$

We're going to see shortly that most of the quantities under discussion will vary with the axial coordinate, $$\large z$$.  For the moment however, all I'm saying is that the measured pressure is the sum of an antegrade and retrograde wave; inventing $$\large \Gamma$$, as you see, is just a different way of saying the same thing.

Now there is something of a dichotomy in the literature as to how to represent the reflected flow waves.  I'll take the approach that pressure and flow of both the antegrade and retrograde waves are related through the same characteristic impedance, $$\large Z_0$$, which happens to be a quantity that does not vary with location in a uniform conduit (physical properties don't depend on $$\large z$$:

$$\Large P_a = Z_0 Q_a$$

$$\Large P_r = Z_0 Q_r$$

However unlike the pressure, the measured flow is the difference of the antegrade and retrograde waves:

$$\Large Q_m = Q_a - Q_r = Q_a(1-\Gamma)$$

Consider now the junction between an upstream conduit with characteristic impedance, $$\large Z_0$$, and a downstream system with input impedance, $$\large Z_T$$ (a "termination" impedance).  To be sure, $$\large Z_T$$ is simply the input impedance for whatever lies downstream of the junction. The termination impedance can be any simple (e.g. clamp, open to air) or complicated (entire circulation) system so long as it is linear and can be described as an input impedance.  By definition, $$\large Z_T$$ prescribes the ratio of the pressure and flow sinusoids at the junction:

$$\Large Z_T = \frac{P_m}{Q_m}$$

Again we're working in the frequency (Fourier) domain; $$\large P_m$$ (measured or actual pressure), $$\large Q_m$$ (measured or actual flow), and $$\large Z_T$$ are all complex with both modulus and phase.  $$\large P_m$$ and $$\large Q_m$$ must be the same at the distal end of the conduit, even though the characteristic impedance prescribes a different $$\large P/Q$$ relationship than $$\large Z_T$$ does (in general).  For $$\large P_m$$ and $$\large Q_m$$  to match across the junction, we end up having reflected waves in the upstream conduit.  In terms of the upstream system:

$$\Large P_m = P_a + P_r = Z_0 (Q_a + Q_r) = Z_0 \; Q_a(1+\Gamma)$$

$$\Large Z_T = \frac{P_m}{Q_m} = \frac{Z_0 Q_a (1+\Gamma)}{Q_a(1-\Gamma)} = \frac{Z_0 (1+\Gamma)}{1-\Gamma}$$

This last equation is readily rearranged to solve for the reflection coefficient in terms of $$\large Z_0$$ and $$\large Z_T$$:

$$\Large \Gamma = \frac{Z_T - Z_0}{Z_T + Z_0}$$

In English, the equation indicates that the meeting of 2 unequal impedances ( $$Z_0$$ and $$Z_T$$ ) dictates a nonzero reflection coefficient; there must be a reflected  for there to be pressure and flow continuity across the junction. This is the reflection coefficient at the junction of the conduit and the downstream system.   Note however that this can be defined at any location; the location could still be somewhere in the middle of a uniform conduit.

Let's do a little reality check; if we've clamped the the end of the conduit, then $$\large Z_T$$ is essentially $$\infty$$ (very large relative to $$Z_0$$).  $$\large \Gamma$$ approaches $$\infty/\infty \approx 1.0$$ if you plug in those numbers.  If $$Z_T$$ is $$0$$ (open to air), we end up with $$\large \Gamma = -1$$; the latter is the same as $$\large \Gamma = 1$$ (modulus of $$1$$) but with a phase of $$\pi$$, i.e. the reflected wave is as large as the incident (antegrade) wave but $$180^{\circ}$$ out of phase with it.  These results are exactly as indicated graphically above.

Now we'll look at how these variables and attributes change within the conduit.  In a previous article, we also so how the antegrade and retrograde waves transform as they move along the conduit.  If we know what $$\large P_a$$, $$\large P_r$$, $$\large Q_a$$, and $$\large Q_r$$ are at any specific location, defined arbitrarily as ($$z = 0$$), we find that the measured pressure and flow change / are transformed along the conduit:

$$\Large P_m(z) = P_a e^{-\gamma \; z} + P_r e^{+\gamma \; z}$$

$$\Large Q_m(z) = Q_a e^{-\gamma \; z} - Q_r e^{+\gamma \; z}$$

$$\Large Z(z) = \frac{P_m(z)}{Q_m(z)} = \frac{P_a e^{-\gamma \; z} + P_r e^{+\gamma \; z}}{Q_a e^{-\gamma \; z} - Q_r e^{+\gamma \; z} }$$

I'm now using $$\large Z(z)$$ to stand for the complex input impedance, the ratio of a measured pressure sinusoid to the corresponding (same frequency) flow sinusoid.  (Various authors use symbols for this like $$\large Z_{in}$$ and $$\large Z_x$$). Here, the lower case Greek gamma, $$\large \gamma = \alpha + j \beta$$, is the complex propagation coefficient that describes the attenuation ($$\alpha$$) and wave velocity ($$\beta$$) of both the pressure and flow waves.  $$\large z$$ increases in the downstream direction and both $$\large \alpha$$ and $$\large \beta$$ are positive and real, so $$\large \gamma$$ describes the exponential decay of the wave in the direction of travel and a linear shift in phase with distance.  $$\large \beta$$ is literally the number of radians per cm that we picked off from the experiment above (in the absence of reflections).   It's clear that the input impedance is a function of axial position and we'll soon see that it's a function of frequency too (next article).

$$\Large -\frac{\partial P_m}{\partial z} = \gamma \left[ P_a e^{-\gamma \; z} - P_r e^{+\gamma \; z} \right]$$

$$\Large -\frac{\partial Q_m}{\partial z} = \gamma \left[ Q_a e^{-\gamma \; z }+ Q_r e^{+\gamma \; z} \right]$$

In the following, we can see that the reflection coefficient actually varies with location, $$\large \Gamma (z)$$.  We'll let $$\large \Gamma_0$$ be the value that corresponds to the location of specification for the antegrade and retrograde waves.

$$\Large \frac{P_r(z)}{P_a(z)} = \Gamma (z) = \frac{P_a e^{-\gamma \; z} }{P_r e^{+\gamma \; z} } = \Gamma_0 \frac{e^{-\gamma \; z}}{e^{+\gamma \; z}} = \Gamma_0 e^{-2 \gamma \; z}$$

$$\large \Gamma_0$$ is defined at the same location as $$\large P_a$$ and $$\large P_r$$ and varies with spatial location as indicated.  By diligent application / rearrangement of the equations we can also express the input impedance within the conduit in terms of the reflection coefficient.  :

$$\Large \frac{Z(z)}{Z_0} = \frac{1 + \Gamma \; e^{-2\gamma z}}{1 - \Gamma \; e^{-2\gamma z}}$$

We saw that the apparent wave velocity varies depending on the magnitude of reflected waves and the location of the determination.  We can compute an apparent propagation coefficient that expresses how the wave appears to be changing with the axial coordinate.  For this situation, the apparent coefficient is different for pressure ($$\large \gamma_P$$) than for flow ($$\large \gamma_Q$$).

$$\Large \gamma_P(z) = - \frac{ \frac{\partial P_m}{\partial z} }{P_m(z)}$$

$$\Large \gamma_Q(z) = - \frac{ \frac{\partial Q_m}{\partial z} }{Q_m(z)}$$

I've expressed derivatives wrt $$z$$ as partial derivatives since all the quantities of interest are also functions of frequency ($$\omega$$).  The mathematical expressions indicate that we're determining the rate of change of the wave with respect to the axial coordinate ($$\large \partial P_m / \partial z$$ for example) but then dividing by the wave itself; the latter normalizes for the wave size.  Physical units of both propagation coefficients are $$L^{-1}$$, e.g. per cm.  For wave velocity, these expressions determine the apparent value of $$\large \beta$$ e.g. in radians per cm between 2 measurement sites as the distance between them approaches zero.  We get a value that is specific to both the wave frequency and location of measurement since, once again, the apparent velocity depends on location relative to nodes and anti-nodes.  Apparent wave speed is fast near an anti-node and slow near a node.   This is in keeping with the limiting case examined previously with zero viscosity where we saw infinite apparent wave speed between nodes, but a $$180^{\circ}$$ in phase across the node (indicating a large change in phase over a very short, i.e. infinitesimal distance.

### Variations on Reflections

Maybe I gave the impression that wave reflections are due either to an open-ended tube OR one with a clamped end.  Actually ANY impedance discontinuity results in  wave reflections as suggested by the math.  The next video shows an example where there's an impedance discontinuity in the middle of the tube, due in this case to a 10 fold increase in elastic modulus of the material. It's a big change - I didn't want it to be subtle for ya.

Next is a freeze frame from the video.  This change means that the downstream section has a markedly greater characteristic impedance than the upstream section.  This is a relatively closed junction and we see relative nodes and anti-nodes upstream of the reflection site.  We saw relative nodes and anti-nodes when there was viscosity, but this model has none.  The reflected wave isn't as great in magnitude as the antegrade because of the nature of the reflection site itself.  In this model and the one that follows, the distal (right) side terminates in a matched impedance so there are no reflections from the far right and no reflected waves in the right segment of the conduit.  The characteristic impedance of the distal section is higher than upstream and the pressure sinusoid is large relative to the flow sinusoid.  Obviously the wave velocity in the section with the stiffer (downstream) tube is much greater than in the upstream section.

Next is a relatively open reflection site.  The right side of the tube has an elastic modulus 10 time lower than the upstream side.  The impedance of the downstream section is relatively low with a large flow sinusoid relative to the pressure sinusoid.  The wave (phase) velocity is much lower in the downstream section than upstream.

We hope you don't have any clamps on your arteries, and also that none of them are pumping into open air.  But pressure and flow wave reflections are produced at every impedance discontinuity in your circulation.  Among other things, that's every single arterial bifurcation!!  (Although we'll see that you're put together so as to minimize the reflections at these branch points.)