Slower than either of them

The speed of sound is a property of a medium, and this collection has already established what it is a speed of: the rate at which a small disturbance propagates, which is the square root of the ratio of a stiffness to an inertia.

Both of those are ordinary quantities and both mix in ordinary ways. Put the two together and the answer is not ordinary at all.

Two mixing rules, going opposite ways #

The density of a mixture is the volume-weighted average of the densities. That is not a model; it is what density means, and it is exact for any mixture whose components are not reacting.

The compressibility of a mixture is the volume-weighted average of the compressibilities. That takes one line: apply a pressure change to the whole mixture, each component shrinks by its own compressibility times its own volume, and the total volume change divided by the total volume is the weighted mean.

So

$$\rho_m = \sum \alpha_i\rho_i, \qquad \frac{1}{\rho_m c_m^2} = \sum \frac{\alpha_i}{\rho_i c_i^2},$$

and dividing the second into the first gives the sound speed of the mixture. That is Wood’s formula, and it has been known since 1930.

The trouble it makes is visible in that figure. Density is dominated by the heavy phase — a mixture that is ninety per cent air by volume still has ten per cent of water’s density, which is a hundred times air’s. Compressibility is dominated by the light phase — a mixture that is one per cent air by volume has a compressibility a hundred and sixty times water’s, because air is fifteen thousand times more compressible and one per cent of that is still enormous.

The mixture takes the water’s inertia and the air’s springiness. A heavy spring-mounted thing oscillates slowly, and a medium with a large density and a large compressibility carries sound slowly.

There is a way of seeing it that needs no algebra. Imagine squeezing a sealed bag containing a litre of water and a litre of air. Almost all of the volume change comes from the air, because the water barely moves; so the bag as a whole is nearly as squashy as the air alone. Now shake the bag. Almost all of the mass being accelerated is the water, because the air weighs nothing; so the bag as a whole is nearly as heavy as the water alone.

A wave is a squeeze followed by an acceleration, repeated. Squashy and heavy is the slow combination, and the mixture is both at once.

The same picture says why the effect is strongest in the middle and why it is so sharp at the light end. Adding the first per cent of air to water multiplies the compressibility by a hundred and sixty and changes the density by one per cent, so it does nearly all of its damage immediately; adding the last per cent of water to air multiplies the density by nine and barely touches the compressibility, so it too does its damage immediately. The curve is steep at both ends and flat in between, which is the shape in the first figure.

The number #

For air and water the minimum is at 0.5006 by volume — almost exactly half, because the two mixing rules are almost exactly opposed — and the sound speed there is 23.8 metres a second.

That is fourteen times slower than air and sixty-two times slower than water. It is slower than a cyclist. A mixture of two of the fastest-conducting media in everyday life carries sound at the speed of a moderate breeze, and nothing about the calculation is subtle: it is two averages and a square root.

Nor is it peculiar to those two. Steam and water give 23.1, air and oil 25.5, steam and oil 24.7, and air and mercury — the largest density contrast easily available — gives six and a half metres a second. Every mixture is slower than both of its constituents at every void fraction between the ends, and the proof is one line: the density is bounded below by the lighter phase and the compressibility below by the stiffer one, the two bounds are attained at opposite ends, so their product is larger in the middle than at either end.

The part that matters in practice #

The minimum is the striking number and the small void fractions are the useful ones.

A hundredth of a per cent of air by volume takes water from 1,481 metres a second to 929. A tenth of a per cent takes it to 365. One per cent takes it to 119 — a factor of twelve, in a mixture that looks exactly like water, in which the density has fallen by one per cent and the volume by nothing worth measuring.

The curve is steepest where the void fraction is smallest, which is the awkward property. A quantity that changes by a factor of twelve when a barely detectable amount of gas is present is a quantity nobody can rely on being what the handbook says.

A remark about how the minimum is found, since it matters to whether the number can be trusted. Differentiating Wood’s formula with respect to the void fraction gives a linear equation, because the quantity to be extremised is a product of two linear functions of the void fraction. So the minimum has a closed form and is not searched for, and the scan over six hundred void fractions is a check on the closed form rather than the source of the answer. The two agree to four decimal places.

That the answer is so near a half is a coincidence of air and water, and an informative one. The minimising void fraction depends on the ratio of the two constituents’ densities and on the ratio of their compressibilities, and for air and water those two ratios are nearly reciprocal — which puts the optimum in the middle. For air and mercury it is not, and the minimum sits at a different place.

What moves with it #

The sound speed is not usually wanted for its own sake. It is wanted because a great many other quantities are built on it, and all of them move.

The Mach number. A ten-metre-a-second flow of water is at Mach 0.007 and is incompressible by any standard anybody applies. The same flow with one per cent of air in it is at Mach 0.084, and with twenty per cent it is at Mach 0.34 — past the point where compressibility stops being negligible, in a liquid, at walking pace. A pump inlet, a hydrofoil, a valve: all of them can be transonic in water without anybody having gone fast.

The water-hammer pressure. Joukowsky’s result is $\Delta p = \rho a\,\Delta V$, so it is proportional to the sound speed. Stopping two metres a second of pure water costs 29.6 bar; with one per cent of air it costs 2.4; with ten per cent, 0.71. This collection has an essay on why stopping water costs more than moving it, and this is the same arithmetic used deliberately: an air chamber or an accumulator on a pipeline is a device for putting compressibility into a liquid line, and the factor it buys is this one.

And choking. A nozzle chokes when the throat reaches the local sound speed, and a bubbly liquid can do that at a few tens of metres a second. Cavitating pumps, steam-water mixtures in a relief line, and the discharge of a flashing liquid are all choked flows in a medium that a reader would call a liquid.

There is a fourth consequence that is less obvious and is the one most likely to catch somebody out. A pressure measurement in a bubbly liquid is a measurement in a compressible medium, so the relation between a stagnation pressure and a velocity is not the incompressible one. At Mach 0.34 the incompressible dynamic pressure under-reads the stagnation rise by about three per cent, which is not large; but the reader has arrived at a Mach number of 0.34 without believing themselves to be anywhere near compressible flow, and which correction to apply is a question they will not have asked.

And there is a fifth that is not a nuisance but a mechanism. A medium whose sound speed drops by a factor of twelve when a little gas appears has a feedback: a pressure drop makes gas come out of solution or a cavity open, which lowers the sound speed, which raises the local Mach number, which lowers the pressure further. That is one of the ways a cavitating line becomes unstable, and it is arithmetic rather than a resonance.

Where the formula stops #

Wood’s formula has no bubble size in it. That is not an omission to be repaired; it is the formula telling a reader what it assumes, and the assumption is equilibrium: the bubbles have time to reach whatever pressure the wave imposes, so the mixture responds as one medium.

Above the bubbles’ own resonance they cannot. A bubble is a mass of surrounding liquid on a spring of enclosed gas, and Minnaert’s frequency for it is about three kilohertz for a millimetre bubble and a megahertz for a micron one. Below that the mixture is soft and slow; near it the medium is violently dispersive and strongly attenuating; above it the bubbles cannot follow at all, the mixture stiffens towards the liquid, and the sound speed climbs back.

There is a second limit in the same place. For a mixture to have a sound speed at all, the wave has to be long compared with the bubbles. At the resonance of a millimetre bubble the wavelength in a one per cent mixture is thirty-six bubble radii — a separation, but not a comfortable one, and it is where the continuum description starts to be an approximation rather than a limit.

So the honest statement of the formula’s range is: low frequency, small bubbles, well mixed, in equilibrium. Every one of those is violated somewhere in the applications above, and the failures are not small — near resonance the attenuation per wavelength can exceed everything else in the problem.

Where the mixture actually is #

Three ordinary situations sit in the range this page is about, and it is worth naming them so that the numbers do not read as a curiosity.

A pump inlet with a little dissolved gas coming out of solution as the pressure falls. The void fraction needed to halve the sound speed is a few hundredths of a per cent, which is far below what would be visible and far below what any instrument in a pump loop would report.

A steam-water line in a power plant, where the void fraction is not a trace at all and the mixture is squarely in the slow region. Two-phase flow in such a line chokes, and the choking is computed with exactly this sound speed.

And a breaking wave or a ship’s wake, where entrained air fractions of several per cent are ordinary. The acoustic properties of the top metre of a rough sea are dominated by bubbles rather than by water, which is why sonar looks downward through a layer it cannot see through.

In every one of them the fluid would be described by anybody looking at it as a liquid, and in every one the number that decides the behaviour is closer to a gas’s than to a liquid’s.

What is unusual about this result #

It is worth saying why a mixture behaves so unlike its parts, because the general form recurs.

Two properties are being combined, and they combine by different averages. The density is an arithmetic mean weighted by volume; the compressibility is also an arithmetic mean weighted by volume, which makes the stiffness a harmonic mean. A harmonic mean is dominated by its smallest member and an arithmetic mean by its largest, so a mixture inherits the worst of both — and the quantity built from their ratio falls off a cliff.

That is the same structure as several results in this collection. The permeability of a porous medium is a fourth moment over a second while its surface is a first over a second, and the two do not determine each other. A parallel electrical network takes the conductance of its best branch and a series one takes the resistance of its worst. Whenever two quantities of a mixture average with different weights, their combination is not between the parents’ values, and may be far outside.

The general rule is worth carrying: mixtures do not interpolate. A reader who expects a mixed property to lie between its constituents’ values is assuming a single averaging rule, and there is usually more than one in play.

Reading it as a regime rather than as a number #

The most useful form of all this is not the sound speed at all. It is that the regime a flow is in is not a property of the fluid.

This collection’s standing position is that one dimensionless number decides which physics applies, and that the number has to be evaluated rather than assumed from the material. Wood’s formula is the sharpest available demonstration: the Mach number of a flow of water depends on a void fraction that nobody measured, and it moves by a factor of fifty across void fractions that all look like water.

The same is true of the other indices. A bubbly liquid’s effective compressibility changes its Weber number’s meaning, because the pressure that resists a deformation is different; its water-hammer response changes the timescale a valve closure has to be compared against; and the relation between a pressure and a velocity, which is not the same equation in every circumstance, acquires a compressible correction in a flow nobody thought was compressible.

So the practical instruction is to compute the sound speed of the actual medium rather than to look up the sound speed of the liquid. It takes two lines. It requires a void fraction, which is usually not known — and the fact that it is usually not known, in a quantity that moves the answer by a factor of twelve, is itself the most useful thing on this page.

The two-phase Mach number, and what it is not #

One clarification is worth making because it is the most likely misreading of this page.

The sound speed computed here is the equilibrium mixture sound speed, and the Mach numbers built from it are Mach numbers of the mixture as a continuum. They are the right numbers for asking whether a nozzle chokes, whether a wave steepens, or whether the pressure and velocity fields are coupled the way a compressible flow’s are.

They are not the right numbers for asking about the individual phases. The liquid is still nearly incompressible and the gas is still nearly ideal; nothing about either has changed. What has changed is the response of the mixture as a whole, and the low sound speed is a property of the mixture rather than a claim that water has become compressible.

That distinction matters when a calculation has to be set up. A two-fluid model that tracks the phases separately will produce the low mixture sound speed as an emergent property, provided it couples them correctly; a homogeneous model has to be given it. And a model that treats the liquid as incompressible and the gas as a passive void fraction will not produce it at all, which is a common and quiet failure.

The general point is one this collection makes about what “incompressible” is a statement about. It is a statement about a flow rather than a fluid, and here it is a statement about a mixture rather than either of its constituents — which is one level further out than the usual warning, and in the same direction.

How to notice it in a system that has it #

There are three symptoms of an unexpectedly low sound speed, and they are worth naming because none of them looks like a compressibility problem.

A pressure wave that arrives late. Water hammer arrives at $L/a$ after a valve closes, and with a per cent of air that is twelve times longer than a handbook calculation predicts. A pipeline whose measured transient timing does not match its computed one is being told its sound speed.

A surge that is far weaker than predicted. The same factor of twelve applies to the pressure rise, in the other direction, and a system that survives a closure the calculation said would burst it has gas in it.

And a flow that will not exceed a certain rate. A choked two-phase line has a maximum mass flux set by the mixture’s sound speed at the throat, and it is reached at velocities that look far too low to choke anything.

All three are usually attributed to something else — an instrument error, a conservative design, a partly closed valve — because the possibility that a liquid line is a compressible-flow problem is not one a reader arrives with. The arithmetic on this page is two lines and settles it.

What is not claimed #

The formula is not derived here. The two mixing rules are stated and their consequences computed; the derivation of each is a paragraph and neither is in dispute. What is computed is the size of the result across five pairs of phases and the location of its minimum in closed form.

Nothing here describes a real bubbly flow’s dynamics. The bubbles are assumed present, uniformly distributed and unchanging. A real bubbly flow has bubbles that grow and collapse with the wave, migrate relative to the liquid, coalesce and break; all of that adds dispersion and attenuation that this description does not contain.

The Minnaert frequency is the isothermal-to-adiabatic simplification. It assumes a spherical bubble in an infinite liquid with no surface tension and no thermal damping, which is enough to locate the resonance and not enough to say how sharp it is.

And the water-hammer numbers assume a rigid pipe. A real pipeline’s wall elasticity already reduces the effective sound speed — that is the standard Korteweg correction — so the ratios quoted here are the additional effect of the gas rather than the whole of it.