Above their resonance, bubbles make water faster

Slower than either of them computed one of the stranger numbers in fluid mechanics. Water and air carry sound at 1481 and 343 metres a second, and a mixture of the two can carry it at 23.8: slower than either. Wood’s formula explains why. A mixture’s density is nearly the water’s and its compressibility nearly the air’s, and the speed of sound is the square root of one over their product, so the worst of each combines. A single per cent of air brings the speed down to 119 metres a second.

That essay was careful about where the formula stops. Wood’s formula has no bubble size in it, because it assumes the bubbles follow the wave’s pressure instantly; above the bubbles’ own resonance — Minnaert’s frequency, about three kilohertz for a millimetre bubble — they cannot, and it said the mixture would stiffen towards the liquid there. It drew the resonance and did not compute the passage through it. This essay computes it, and the passage has three surprises in it. The low-frequency speed is not Wood’s as usually evaluated. Above the resonance there is a band in which sound does not propagate at all. And above that band the mixture does not merely stiffen towards water: it overshoots, and carries sound faster than water does.

A wave driving a field of springs #

Each bubble is a spring and a mass: the gas inside is the spring and the water it must push aside is the mass, about three times the volume of water the bubble displaces. A wave’s pressure drives it. When the wave is slow the bubble breathes in time with the pressure and adds its compliance to the water’s — Wood’s mixture. When the wave is near the bubble’s natural frequency the bubble’s response is large and lags by a quarter cycle; above it the bubble moves against the pressure, expanding as it rises, which is a negative compliance.

For a dilute suspension of $n$ bubbles of radius $R$ per unit volume, Commander and Prosperetti wrote this as a dispersion relation for the complex wavenumber,

$$k^2 = \frac{\omega^2}{c_l^2} + \frac{4\pi n R\,\omega^2}{\omega_0^2 - \omega^2 + 2ib\omega},$$

the first term the water and the second each bubble’s driven response. The natural frequency $\omega_0$ and the damping $b$ are not constants. Both depend on what the gas inside does when squeezed, and that depends on the frequency.

The gas inside #

A gas squeezed quickly heats up and resists more than one squeezed slowly, which has time to give its heat to the water. Its stiffness lies between the isothermal, pressure times volume constant, and the adiabatic, with the exponent γ = 1.4. Prosperetti solved the heat conduction inside an oscillating bubble and wrote the result as a complex polytropic function $\Phi$: its real part is three times the effective exponent, its imaginary part the energy lost to the water each cycle. The change from isothermal to adiabatic happens where the time heat takes to diffuse across the bubble, $R^2/D$, equals the period — near 1.4 kHz for a 50 µm bubble, 14 Hz for a 0.5 mm one and a seventh of a hertz for a 5 mm one.

That has a consequence for the low end. Wood’s formula as usually evaluated uses the gas’s own sound speed, which is the adiabatic value. But at the low frequencies where Wood’s formula is meant to apply, millimetre bubbles are isothermal. For a void fraction of a thousandth the computed low-frequency speed is 312 metres a second, the isothermal Wood value; the adiabatic Wood value is 366. The formula is right; the number usually put into it is the wrong gas’s.

What the isothermal gas does to a water hammer #

The correction at the slow end is not academic, because the slow end is where stopping water costs more than moving it lives. A valve closing on a column of water raises its pressure by the density times the sound speed times the change of velocity, and the earlier essay used Wood’s adiabatic speed to show that one per cent of air cuts a two-metre-a-second hammer from 29.6 bar to 2.4. With the gas isothermal, as millimetre bubbles are below a few hertz, the mixture’s speed is 101 metres a second rather than 119, and the same hammer is 2.0 bar. A closure lasting a few milliseconds excites frequencies near a hundred hertz, where the gas is partly adiabatic and the speed is 110: 2.2 bar. The hammer in a gassy line depends on how fast the valve shuts in a second way, through the thermodynamics of its bubbles, besides the familiar one through the wave’s travel time.

Slow, blocked, fast #

The phase speed against frequency is the essay’s main figure. At a void fraction of a thousandth the speed starts at 312 metres a second and holds it through the audible range. As the frequency approaches the bubbles’ resonance, 3.24 kHz for millimetre bubbles at one atmosphere, the bubbles respond more than their static compliance and the speed falls further, to about 140 metres a second. Then it jumps. Between the resonance and 13.3 kHz the wave hardly travels and its phase speed, set by what little real part the damping leaves the wavenumber, swings up to tens of kilometres a second. Above 13.3 kHz the wave propagates again, and faster than in water: 4.4 kilometres a second at 14 kHz, 2.0 at 20 kHz, 1640 metres a second at 30 kHz, and down to water’s own 1481 only far above, from above.

The overshoot follows from the sign. Above the resonance each bubble moves against the pressure, its contribution to $k^2$ is negative, and the wavenumber is smaller than the water’s alone: a longer wavelength at the same frequency, a faster phase. The mixture is stiffer than water, because the bubbles push back with the wave’s pressure instead of yielding to it. More bubbles widen the band and raise the overshoot; at 10⁻² the slow end is down at about 100 metres a second, the band runs to 43 kHz, and the fast side lies above that.

The very large speeds in and at the top of the band are phase speeds where the wavenumber is near zero, and they carry no signal faster than anything: the group speed, which carries energy, stays finite and the wave there is strongly damped. The essay below what a signal travels at explains why a phase speed is allowed to do that. What is physical at 30 kHz is a mixture through which a sonar ping travels eleven per cent faster than through clear water.

Where the sound goes #

Damping is what turns the band from a mathematical forbidden zone into a physical absorber. At a void fraction of a thousandth the attenuation peaks at 210 nepers a metre at 3.25 kHz: the amplitude falls by a factor of e in under half a centimetre. Across the whole band it stays above ten nepers a metre. Below the band, in the slow regime, the wave travels many metres; above it, at 100 kHz, the attenuation has fallen to 1.3 nepers a metre.

That is the physics behind two practical facts. A ship’s wake, full of bubbles, is close to opaque to sonar near its bubbles’ resonances — the wake screen exploited deliberately by bubble curtains around underwater construction, where a curtain of air protects marine animals from pile-driving noise. And breaking waves, which fill the surface layer with bubbles of every size, absorb and scatter sound across the band their size distribution spans.

A mixture stiffer than either part #

The overshoot deserves a plain statement, because it contradicts an intuition the slow end teaches. A mixture of a soft thing and a stiff thing is normally between them, and Wood’s mixture at low frequency is softer than both. Above the band the same mixture is stiffer than the stiffer of its parts. Nothing about the materials has changed; what has changed is the phase of the bubbles’ motion. A spring driven above its resonance moves opposite to the force, and a bubble driven above its resonance shrinks while the pressure falls and grows while it rises. In the average over many bubbles that is a negative compressibility, and added to the water’s positive one it makes the total smaller than the water’s alone.

This is the oldest example of what is now designed deliberately as an acoustic metamaterial: a medium whose effective modulus is negative in a band of frequencies because it is threaded with local resonators. The bubble is the resonator and water supplies both the matrix and the resonators’ mass. Its relatives in this collection are every damped oscillator driven through resonance — a drop that rings like a bell is the same mathematics with surface tension as the spring, and a damper that turns into a spring is the same change of character with frequency in a squeezed film of air: what absorbs at one frequency stores at another.

The band’s two edges also give the measurement its handle. The lower edge depends only on the bubble’s size and the ambient pressure, through its resonance; the upper edge depends on how many bubbles there are. Sweep a sound’s frequency through a bubbly layer, find where it stops getting through and where it starts again, and the two numbers give radius and void fraction separately — which is how bubble populations under breaking waves and in ships’ wakes are sized acoustically, without a probe in the water that would itself make bubbles.

The band #

Without damping the band is exact: $k^2$ is negative from the resonance to $\omega_0\sqrt{1 + 4\pi n R c_l^2/\omega_0^2}$, and a wave in it is evanescent. The lower edge is the bubble’s resonance, 3242 Hz for millimetre bubbles, whatever the void fraction. The upper edge rises with the number of bubbles: 5.38 kHz at 10⁻⁴, 13.3 kHz at 10⁻³ and 43 kHz at 10⁻². A tenth of a per cent of air opens a band two octaves wide. The square root has a simple meaning: it is the ratio of the mixture’s stiffness above resonance, water plus the bubbles pushing back, to the bubbles’ own stiffness, and it grows with how many bubbles share the wave’s compression.

The same frequency, different bubbles #

Turn the question round and hold the frequency instead. At a fixed void fraction a sonar at 1 kHz sees bubbles smaller than 3.3 mm as a slow medium — they resonate above its frequency — and bubbles a little larger than that put its frequency in their stop band. At 20 kHz the dividing radius is 0.16 mm. A cloud of bubbles of mixed sizes is therefore three media at once to a single sound: slow through its small bubbles, opaque through those near resonance, and fast through its large ones. That is why a measured sound speed in bubbly water is a statement about a size distribution, not about a void fraction — and why acoustic measurement of a bubble population, by sweeping the frequency and watching where the band falls, works.

What damps a bubble #

The three dampings divide the sizes among them. Viscous damping, $2\mu/\rho R^2$, falls fastest with size and dominates below 3.4 micrometres. Thermal damping — the gas giving its heat to the water through each cycle, the imaginary part of $\Phi$ — dominates from there to 3.8 millimetres, which covers every bubble in the bubble that hammers and in most cavitation. Above that, the bubble loses its energy mainly by radiating sound, at a nearly constant fraction of its resonant frequency. A millimetre bubble’s resonance has a quality factor of about twenty-five, which is why its band edge is sharp and why the hum of a single bubble, set ringing by a drip, lasts several cycles.

Five checks on the relation #

Prosperetti’s $\Phi$ must tend to three when the gas is slow and to $3\gamma$ when it is fast, and its imaginary part — a loss — must be positive at every frequency; it is 3.0000000 at a hundredth of a hertz, 4.198 at ten megahertz, and never negative across nine decades. At low frequency the relation must give Wood’s speed with an isothermal gas: it does to five parts in ten thousand, checked without surface tension, which raises a millimetre bubble’s stiffness by two-tenths of a per cent and is otherwise kept. Without damping, $k^2$ changes sign exactly at the computed band edges. And at ten megahertz the phase speed is water’s to a part in a billion, from above. The tests also refuse a frequency of zero, a negative radius and a void fraction too large for a dilute theory.

What a linear, dilute theory cannot say #

Large amplitude. The relation is linear, and when a body tears the water is where it stops being so. A bubble driven near resonance by a loud sound oscillates nonlinearly, radiates harmonics, and above a threshold collapses violently, which is cavitation; the linear band is then an underestimate of what is absorbed.

Crowding. The bubbles are assumed far apart compared with their size and to feel only the wave, not one another. At a void fraction of a per cent they are seven or eight radii apart and their near fields interact, shifting the resonance.

One size. Real bubble populations span decades of radius, and the relation is then an integral over the distribution: the band smears into a broad absorption and the overshoot is diluted by the slow contribution of the small bubbles.

Bubbles that grow. The population is fixed for the length of a calculation, and in a sound field it is not. Over many cycles a bubble takes in a little more gas while it is large, because its surface is larger and the diffusion layer round it thinner, than it gives out while it is small. Above a threshold amplitude, which Eller and Flynn worked out in 1965, this rectified diffusion makes bubbles grow towards resonance, so a sustained loud sound moves the band it is travelling through.

Bubbles that move. The bubbles are held in the liquid, and the liquid itself is treated as incompressible except for its sound speed. In a real flow they rise, drift and change size with depth, and a wave long enough to span a change in hydrostatic pressure sees bubbles of different resonances along its path.

The convention: phase speed, and nepers #

Speeds are phase speeds, $\omega/\mathrm{Re}\,k$, and attenuations are $|\mathrm{Im}\,k|$ in nepers a metre, so an amplitude falls by a factor of e over a distance of one over the attenuation. The time dependence is $e^{i\omega t}$, which fixes the sign of the damping term. Void fraction is the volume of gas per volume of mixture. Water is at 20 °C and one atmosphere, with a surface tension of 0.0728 N/m, and the gas is air with a thermal diffusivity of $2.2\times10^{-5}\,\mathrm{m^2/s}$.

Wood, Minnaert, and the people who joined them #

Wood gave the mixture formula in his 1930 textbook. Minnaert measured and explained the resonance of a single bubble in 1933. Foldy in 1945 and van Wijngaarden in the 1960s and 70s put the bubbles into a wave equation as driven oscillators, Prosperetti solved the gas’s heat conduction in 1977, and Commander and Prosperetti in 1989 assembled the dispersion relation used here and compared it with every set of measurements then available, finding it good below and above the band and poor within it, where the bubbles interact and the amplitude is rarely small.

Still open: a cloud with a size distribution #

The relation here holds one bubble size. A real bubbly flow — a wake, a breaking wave, a cavitating pump — has a distribution, typically a power law in radius between a few micrometres and a few millimetres. The next calculation integrates the bubble term over such a distribution and asks how much of the overshoot above the band survives: whether a cloud whose small bubbles keep it slow at every frequency can still carry a fast band anywhere, and how the measured speed and attenuation of a breaking wave’s bubble layer can be inverted for the distribution that made it.