The pulse that has to travel

Where the parabola goes followed Womersley’s number through a rigid tube and found it honest about the flow rate and dishonest about the phase. Everything in that essay happened everywhere at once: a pressure gradient was imposed along the whole tube, and the whole tube responded to it together.

An artery does not work that way, and nobody who has taken a pulse at the wrist thinks it does. The heart ejects into the aorta, the aorta’s wall stretches to take the blood, and the stretch travels outwards as a pressure wave at a few metres a second — fast enough that the pulse at the wrist arrives a tenth of a second after the heartbeat, slow enough that the delay is measurable with a stopwatch and two fingers. The rigid-tube essay ended by naming exactly this case: the elastic tube, where “the pressure pulse becomes a wave and the impedance becomes the object of study”.

Taking it up changes two things. A second dimensionless number appears, and the first one acquires a job it did not have in the rigid tube.

The wave, and where the Womersley number enters it #

An elastic tube filled with an inviscid fluid carries a pressure wave at the Moens–Korteweg speed

$$c_0 = \sqrt{\frac{Eh}{\rho D}},$$

which is the springiness of the wall — its elastic modulus times its thickness, over its diameter — against the inertia of the fluid. It is the same construction as a signal’s speed in any medium: a stiffness over an inertia, under a square root, with the tube’s wall standing in for the fluid’s own compressibility. For a human aorta with a Young’s modulus of a few hundred kilopascals it comes to about five metres a second.

Put viscosity back and the fluid no longer moves as a plug under each wave crest. It moves with the Womersley profile — parabolic at small α, a plug with an annular jet at large α — and the momentum the wave has to shift is no longer all of the fluid’s. John Womersley worked out in 1955 what that does to the wave, and the answer is a single factor:

$$c = c_0\,\sqrt{1 - F_{10}(\alpha)}, \qquad F_{10}(\alpha) = \frac{2 J_1(\Lambda)}{\Lambda J_0(\Lambda)}, \quad \Lambda = i^{3/2}\alpha.$$

$F_{10}$ is the same function that sets the flow in the rigid tube; it is the ratio of the mean velocity to the velocity an inviscid fluid would have under the same gradient. So the wave speed is complex. Its real part sets how fast a crest travels, and its imaginary part how fast the wave dies.

The Bessel functions of complex argument were computed from their ascending series, which the derivative relation $J_0' = -J_1$ confirms to $7\times10^{-10}$ at complex arguments up to twenty. Above a Womersley number of twenty the series loses its digits to cancellation, so it hands over to Hankel’s asymptotic expansion, $F_{10} \approx 2i/\Lambda + 1/\Lambda^2$, and at the join the two agree to three parts in ten thousand.

A wave that slows and a wave that dies #

The speed behaves as the rigid-tube essay would lead one to expect. At large α the viscous layer is thin, nearly all of the fluid moves as a plug, and the wave travels at nearly the inviscid speed — 96 per cent of it at α = 15, 94 at 10, 88 at 5. As α falls the viscous layer fills the tube, more of the fluid is held back by the wall, and the wave slows: 73 per cent at α = 2, 46 at 1.

At small α something qualitative happens. The expansion $1 - F_{10} \approx i\alpha^2/8$ gives a speed $c_0\,\alpha/2$, and α is proportional to the square root of the frequency. A wave whose speed goes as the square root of its frequency is not a wave. It is a diffusion. The same dependence holds for heat soaking into a wall, and for the depth an oscillating plate reaches into the fluid above it.

The attenuation settles it. A wave in the aorta loses a little over a quarter of its amplitude in each wavelength it travels, which over the length of an aorta — an eighth of a wavelength, as the next section shows — is a few per cent. At α = 3 a wave keeps 13 per cent of itself per wavelength, and at α = 1 half a per cent. A disturbance that loses more than ninety-nine per cent of itself before it has completed one oscillation in space is not usefully described as travelling.

What that looks like along a tube #

Along the tube the difference is visible. At α = 15 there are clear crests at the expected spacing and an envelope that slopes gently down. At α = 5 the second crest is a shadow of the first. At α = 2 the first trough is shallow and there is no second crest to speak of, and at α = 0.5 the pressure simply falls away from the entrance, oscillating in time but not in space in any way the eye can find.

That is the structural fact this essay adds to the Womersley number’s account. In a rigid tube, α decides the shape of the flow. In an elastic tube, it also decides whether the tube is a transmission line or a diffusion line. The boundary between them is the same boundary — around α of two or three — and for the same reason: whether the viscous layer is thin compared with the tube.

A quarter of the wave per wavelength, in closed form #

The aorta’s numbers have a closed form that is worth having, because it says how they scale. At large α Hankel’s expansion gives $1 - F_{10} \approx 1 - 2i/\Lambda$, and taking the square root to first order,

$$\frac{c}{c_0} \approx 1 - \frac{1}{\sqrt2\,\alpha} + \frac{i}{\sqrt2\,\alpha}.$$

The real part says the wave travels slower than the inviscid one by $1/(\sqrt2\,\alpha)$: 4.8 per cent at α = 14.7, against the 4.6 per cent the full Bessel functions give. The imaginary part, turned into a decay over one wavelength, says the wave keeps $e^{-\pi\sqrt2/\alpha}$ of its amplitude per wavelength: 0.739 at α = 14.7, against 0.728 from the full solution.

Both corrections go as $1/\alpha$, which is the thickness of the Stokes layer at the wall over the tube’s radius. That is the whole physics at large α: the wave feels viscosity only through a thin annulus of fluid near the wall, whose share of the cross-section is proportional to its thickness. Halve the thickness — quadruple the frequency, or double the radius — and the wave loses half as much per wavelength.

Five vessels, and which of them carry waves #

The aorta, with a radius of a centimetre, has α = 14.7 and carries its fundamental at 4.77 metres a second, a wavelength of 3.98 metres, losing a quarter of the wave per wavelength. The carotid and femoral arteries, with α of 4.4 and 5.9, carry waves at 6.1 and 8.1 metres a second but lose 70 and 58 per cent per wavelength. A small artery with a radius of a millimetre has α = 1.5 and keeps just over one per cent per wavelength. An arteriole, at α = 0.15, keeps two thousandths.

So the arterial tree is a transmission line that becomes a diffusion line as it branches. The pulse a clinician feels is carried by the large arteries as a wave, and it is gone — smoothed into a steady flow — by the time the vessels are small enough to supply tissue. The smoothing is not done by some separate damping mechanism in the small vessels. It is the same viscosity that shapes the profile, acting in tubes narrow enough that it dominates.

The radii, wall speeds and lengths in that table are borrowed typical values, and real vessels taper, branch and stiffen with distance from the heart. The statement that does not depend on them is the dependence on α.

Heart rate, and animals #

The Womersley number contains the frequency, so the boundary between wave and diffusion is not a fixed vessel. At a resting 1.2 Hz, α = 2.5 falls in a vessel of radius 1.7 mm. At an exercising 3 Hz it falls at 1.1 mm: a faster heart rate pushes the wave further down the tree before viscosity turns it into a diffusion. The aorta itself moves the other way along the axis, from α = 14.7 at rest to 23 during exercise, where it keeps 82 per cent of its wave per wavelength instead of 73.

The same arithmetic across animals is more striking. A mouse’s heart beats about ten times a second and its aorta has a radius of about 0.6 mm. Its Womersley number is 2.5 — the value of a small human artery — and with a wave speed of 4 m/s its aortic pulse keeps only 8 per cent of itself per wavelength, travels at 80 per cent of the inviscid speed, and has a wavelength of 32 centimetres. A rat, at α = 3.9, keeps a quarter. An elephant’s aorta, beating once every two seconds through a radius of four centimetres, sits at α = 38 and keeps 89 per cent.

Heart rate and body size scale together across mammals in a way that keeps the aortic Womersley number within an order of magnitude — faster hearts in smaller bodies — but not within a factor of two. A mouse’s large arteries are at the edge of carrying waves at all, which is one reason results about pulse-wave reflection in humans do not transfer simply to the animals most experiments are done on. The radii, rates and wave speeds are borrowed typical values; the dependence on α is computed.

The second number: length in wavelengths #

In the rigid tube the only number was α. The elastic tube has a second, the tube’s length measured in the wavelength of the wave it carries:

$$\frac{L}{\lambda} = \frac{fL}{c}.$$

A tube much shorter than a wavelength behaves as a single compliant chamber: its pressure rises and falls together along its length, and the lumped model Otto Frank called the Windkessel describes it. A tube a quarter of a wavelength or more carries visibly different pressures at its two ends at the same instant, and nothing lumped describes it.

The complication is that a pulse is not one frequency. A pressure waveform with a sharp systolic upstroke and a dicrotic notch needs ten or more harmonics to describe, and each harmonic has its own wavelength — and its own Womersley number, since α grows with the square root of frequency. For its fundamental, a 50-centimetre aorta is an eighth of a wavelength long and very nearly a Windkessel. For its second harmonic it is a quarter of a wavelength. For its ninth it is more than a whole one.

So the same vessel is lumped for the slow part of the pulse and a transmission line for the fast part, which is why both descriptions of the circulation — the Windkessel and the wave — have survived a century of argument about which is right. Each is right for a different band of the same waveform. The harmonics also sit at different Womersley numbers, from 14.7 for the fundamental to 46 for the tenth, so the higher ones travel closer to the inviscid speed and lose less per wavelength: the fast part of the pulse is both the part that is most wave-like and the part that is least damped.

Which speed a clinic measures #

The speed of the pulse is a routine clinical measurement. Pressure sensors at the neck and at the groin time the arrival of the pulse’s foot, the distance between them is divided by the delay, and a carotid-to-femoral pulse wave velocity above about ten metres a second is read as a sign of stiffened arteries. The Moens–Korteweg formula is what connects the number to the wall: the speed goes as the square root of the wall’s stiffness, so a doubling of the speed means a fourfold stiffer wall.

The dispersion computed here says which speed that measurement returns. The foot of the pulse is its sharpest feature, built from its highest harmonics, and the high harmonics sit at the largest Womersley numbers and travel closest to the inviscid speed. In the aorta the fundamental travels at 95.4 per cent of Moens–Korteweg’s speed, the second harmonic at 96.7, the fifth at 97.9 and the tenth at 98.5. The foot-to-foot method therefore measures something within a couple of per cent of the wall’s own speed, and a method based on the phase of the fundamental would read three per cent lower from the same artery. In a femoral artery, at a lower Womersley number, the gap between the fundamental and the tenth harmonic is larger: 90 against 96 per cent.

That is a small correction next to the physiological spread, and it is not usually the one that matters. But it has a clean direction — viscosity makes every harmonic slower than the wall alone would, and the slow ones slowest — and it is a case of the Womersley number reaching into a measurement that is normally thought of as purely about the wall.

The checks #

The last group is the one that tests the idea rather than the arithmetic. A harmonic sum and a time march share no code: one multiplies complex exponentials, the other sends characteristics along a grid and reflects them off a load. They agree on the pulse at the far end to 0.01 per cent with a partly reflecting load and to half a per cent with a strongly reflecting one, where the grid’s own dispersion begins to show.

What the uniform tube cannot show #

Taper and branching. Arteries narrow and branch, and each change of cross-section or stiffness reflects part of the wave. A uniform tube has no internal reflections, which is the case this essay needs in order to isolate what viscosity does; the reflections are the subject of their own essay.

Wall viscoelasticity. The wall here is purely elastic. Arterial walls dissipate energy when they stretch, which adds a second attenuation — often larger than the fluid’s at high frequencies — and makes the Moens–Korteweg speed itself frequency-dependent.

Longitudinal wall motion and tethering. Womersley’s full theory lets the wall move along the tube as well as outwards, and the answer depends on how firmly surrounding tissue tethers it. The radial-only form used here is the tethered limit.

Non-Newtonian blood. Blood’s viscosity rises at low shear rates, which matters most in exactly the small vessels where the wave has already become a diffusion.

Nonlinearity. The wave’s amplitude is small against the tube’s pressure scale. A real pulse raises the pressure by a third of its mean, stiffens the wall as it does, and steepens slightly as it travels.

Who worked it out #

Thomas Young estimated the speed of the pulse in 1808, in the lecture in which he also set out the mechanics of the circulation as a problem of elastic tubes. The formula for the speed was published independently by Adriaan Isebree Moens and Diederik Korteweg in 1878. Otto Frank’s Windkessel, the lumped model, is from 1899. John Womersley, working at St Bartholomew’s Hospital with Donald McDonald, published the oscillating-flow solution in 1955 and its extension to an elastic tube over the following two years. McDonald’s Blood Flow in Arteries of 1960 is where the two numbers — α and the length in wavelengths — became the working vocabulary of the subject.

Still open: where the wave comes back #

A uniform tube carries a wave outwards and never returns it. Every real vessel ends — in a branch, a narrowing, a bed of arterioles — and each ending reflects part of what arrives. The reflected wave travels back towards the heart and adds to the outgoing one, and whether it adds to the pressure or to the flow depends on the kind of ending.

The next calculation terminates the same tube in a load with a stated reflection coefficient and asks what happens to the pressure and flow pulses along it. The prediction worth testing is a sharp one: that a single number, the reflection coefficient, decides whether the pressure pulse grows towards the periphery while the flow pulse shrinks, or the reverse.