The more small bubbles a cloud holds, the later it turns fast

Above their resonance, bubbles make water faster solved the dispersion relation for water carrying air bubbles of one size and found three regimes. Below the bubbles’ resonance the mixture is slow — 312 metres a second at a void fraction of a thousandth, the value Wood’s formula gives with the gas isothermal. In a band above the resonance no wave propagates. Above that band the mixture carries sound faster than water does, because bubbles driven above their resonance move against the pressure and stiffen the liquid rather than softening it.

The essay ended on the obvious objection. No real bubbly flow has bubbles of one size. The bubble layer under a breaking wave, the wake of a ship, the cloud behind a cavitating propeller and the froth in a pipe all carry a spread of sizes over two or three decades, and every size has its own resonance. This essay sums the relation over such a spread and asks which of the three regimes survive.

The relation for many sizes #

For a dilute cloud the bubbles act independently, so the bubble term in the dispersion relation becomes a sum over sizes:

$$k^2 = \frac{\omega^2}{c_l^2} + 4\pi\omega^2\int \frac{R\,n(R)\,dR}{\omega_0(R)^2 - \omega^2 + 2i\,b(R)\,\omega},$$

each size with its own natural frequency $\omega_0$ and damping $b$ — viscous, thermal and radiation — from the single-size calculation. The distributions here are power laws, $n(R) \propto R^{-q}$, from ten micrometres to one millimetre, scaled so the void fraction is a thousandth in every case. Measured distributions under breaking waves are close to $q = 3/2$ for bubbles smaller than about a millimetre and close to $10/3$ for larger ones; $q = 0$ puts the gas mostly in the largest bubbles and $q = 10/3$ puts more of it in the smallest, so the three span what a cloud can be.

The two measured slopes have explanations of different standing. Above the size at which turbulence can split a bubble — the Hinze scale, about a millimetre under a breaking wave — the $-10/3$ follows from a cascade of fragmentation in which each bubble breaks at a rate set by the turbulent eddies of its own size, a dimensional argument Garrett, Li and Farmer gave in 2000. Below it the $-3/2$ is measured rather than derived, and its origin, in the jets and drops thrown up at the crest that entrain small bubbles directly, is argued rather than computed. The calculation here takes the exponent as given and asks only what it does to the sound; that is the part that does not depend on how the bubbles were made.

Slow, then smeared, then fast #

At low frequency nothing changes. Every bubble, of whatever size, is driven far below its resonance and simply follows the pressure, as a signal in any medium follows the medium’s compressibility, so the cloud’s compressibility is the gas’s total compressibility and its speed is Wood’s 312 metres a second whatever the distribution. The low-frequency speed of a bubbly liquid measures its void fraction and nothing else, which is why Wood’s formula has no bubble size in it.

The single size then drops through its stop band, from about 3 to 13 kilohertz, and emerges fast. The distributions do something different. They have resonances spread from 3.2 kilohertz for the millimetre bubbles to 301 kilohertz for the ten-micrometre ones, and across that range the speed climbs without a gap: there is no frequency at which no wave propagates, because at every frequency some bubbles are below resonance and some above. The stop band has been smeared into a band two decades wide. Above it every cloud comes out faster than water — but the more of its gas is in small bubbles, the further up the frequency axis it comes out.

Why: a tug of war at every frequency #

The figure takes one frequency, thirty kilohertz, and adds up the bubble term size by size from the smallest. Bubbles smaller than the resonant radius at that frequency, 105 micrometres, are driven below resonance; their term is positive and makes the cloud softer. Bubbles larger than it are driven above resonance; their term is negative and makes it stiffer. The resonant size itself, where the denominator nearly vanishes, contributes mostly to the imaginary part — to attenuation.

What the cloud does at thirty kilohertz is the sign of the total. With the gas spread evenly in radius, $q = 0$, the large bubbles win and the total is −0.39: faster than water. At $q = 3/2$ it is −0.68, faster still, because the large bubbles’ stiffening is concentrated nearer resonance. At $q = 10/3$ the small bubbles hold so much of the gas that their softening wins outright, +1.8, and the cloud is far slower than water at a frequency where the single-size cloud and the flatter spreads are fast. The single-size calculation could not show this contest, because a single size is all on one side of resonance or the other.

The attenuation is the distribution #

The imaginary part tells the same story more directly. A single size attenuates sound at one frequency, its resonance. A distribution attenuates over the whole band its sizes resonate in, and at each frequency in roughly the proportion of bubbles at the size resonant there. Spread evenly in radius, the cloud’s attenuation peaks at 89 nepers a metre near 4.9 kilohertz, where its largest bubbles resonate. At $q = 10/3$ the peak is 610 nepers a metre near 273 kilohertz, among the smallest bubbles, and it is higher because the gas is divided among more bubbles, each a separate resonator.

So the attenuation spectrum of a bubbly layer is close to a picture of its size distribution, read through the resonance. That is the basis of acoustic bubble sizing, and it is why a breaking wave’s bubble layer, which is opaque to sonar over a wide band of frequencies, is opaque over exactly the band its bubbles resonate in and transparent above it.

Where the fast band starts #

A fast phase speed is not much use if the wave dies in a wavelength, so the figure asks for both: faster than water, and losing less than about a quarter of its amplitude per wavelength. Millimetre bubbles alone reach that at 18.1 kilohertz. Spread evenly in radius, at 27.8; at $q = 3/2$, 54.1; and then the edge climbs steeply, to 379 kilohertz at $q = 10/3$ — beyond the resonance of the smallest bubbles in the cloud, at 301. Once small bubbles dominate the gas, the fast band begins only where every bubble in the cloud is driven above resonance, because only there have the softeners all changed sides.

That is the answer to the question the previous essay left. The fast band survives a size distribution. Its lower edge is set by the small end of the distribution, not by the void fraction or the mean size, and for the distributions measured under breaking waves — three-halves below a millimetre — it sits in the tens of kilohertz. A sonar or an acoustic gauge working above that frequency in a bubble layer sees water that is faster than water, and one working inside the band sees an absorber.

A metre of bubbly water, in decibels #

The numbers become concrete for a sonar looking through a metre of bubble layer at a void fraction of a thousandth, with sizes spread as $q = 3/2$ — the slope measured under breaking waves below a millimetre. At one kilohertz the layer is slow, 343 metres a second, and nearly transparent, five decibels a metre. At ten kilohertz it is in the middle of the resonance band: 751 decibels a metre, which is to say opaque — nothing gets through a metre of it. At thirty kilohertz the attenuation has fallen to 254 decibels a metre while the speed has risen to 2,430 metres a second, faster than water but still too lossy to use. At a hundred kilohertz the layer carries sound at 1,535 metres a second, 3.6 per cent faster than the water around it, and loses 43 decibels a metre. A high-frequency sonar looking through the layer therefore sees a layer of fast water a metre thick, which bends its beam away from the layer, where a low-frequency one sees a slow layer that bends it towards it — with an absorber between the two frequencies.

That reversal of the layer’s sign with frequency is the most practical consequence of the fast band, and it depends entirely on the distribution’s small end. A layer with the same void fraction but most of its gas in ten-micrometre bubbles is still inside its resonance band at a hundred kilohertz, slow and lossy.

Why the speed and the loss go together #

There is a reason the speed climbs exactly where the attenuation is large, and it is not particular to bubbles. A medium that responds causally — that cannot answer a pressure before the pressure arrives — has a speed and an absorption that are not independent: the change in its refractive index across a frequency band is fixed by an integral of its absorption over all frequencies, weighted towards the low ones, the Kramers–Kronig relations of optics. A cloud that goes from 312 metres a second below its resonances to faster than 1,481 above them must absorb, somewhere between, by an amount set by that change, and a distribution that spreads the resonances spreads the absorption without removing it.

That is why the single size’s clean stop band, with no propagation at all, and the distribution’s broad lossy band are the same object: the integral of the attenuation across the band is fixed by the speeds on either side, which are fixed by the void fraction — Wood’s speed below, and above, a speed close to the liquid’s. Spreading the sizes lowers the peak loss and widens the band, as the attenuation figure shows, and cannot make the band disappear.

The same physics at a valve and at a crown #

Bubbles in a pipe are the same cloud at much lower frequencies. Air at a valve softens the hammer only in quantity treated entrained air as a softening of the column, which is the low-frequency, Wood end of this curve: a water hammer’s frequencies are tens of hertz, far below any bubble’s resonance, and the column’s wave speed there is the void fraction’s alone. The air a cavity releases is capped by the cavity followed where that air comes from. At the other end of the scale the nuclei at a siphon’s crown are bubbles of a few micrometres, whose resonances are in the megahertz, and a cavitating flow’s acoustic signature in that range is the attenuation spectrum of exactly this calculation — which is one of the ways cavitation nuclei are counted in the water tunnels where a body tears the water.

Reading the sizes back #

The obvious use of the attenuation is to invert it. If each bubble’s resonance is narrow in radius, the integral’s imaginary part reduces to the bubbles at resonant size, and

$$n(R_{\rm res}) \approx \frac{k_i\,k_r}{\pi^2 R_{\rm res}^2},$$

which gives the number of bubbles of each size from a measured speed and attenuation at the frequency that size resonates at. The rule is the standard first step of acoustic bubble spectroscopy, and the figure tests it on clouds whose distributions are known.

It works where small bubbles dominate. For $q = 10/3$ the recovered numbers are within twelve per cent of the truth at every size. For flatter distributions it fails at the small end: at $q = 3/2$ it overestimates the number of 12-micrometre bubbles by a factor of 3.9. The reason is the contest above. At a high frequency, where few small bubbles resonate, the many large bubbles driven above their resonance still add a little attenuation each through the tails of their responses, and the rule — which assumes every bit of the attenuation belongs to the bubbles at resonance — books it all to the few small ones. The inversion has to be done on the whole integral, which makes it an ill-conditioned problem rather than a reading; that is why practical bubble sizers solve it with regularisation and report error bars that grow towards small radii.

What was checked #

The sum over sizes is a trapezium rule in the logarithm of the radius, on enough points to resolve every resonance — each a Lorentzian whose width in log radius is its damping ratio, a few per cent. Three checks pin it down. A distribution a millionth wide at one millimetre reproduces the single-size relation, speed and attenuation, to two parts in ten thousand million at four frequencies on both sides of the band. Halving the quadrature step changes the speed and attenuation inside the band by at most three parts in ten thousand. And at five hertz every distribution gives the single-size Wood speed, 312 metres a second, to seven parts in ten thousand — the residual being the distributions’ smallest bubbles, whose surface tension stiffens their gas by $2\sigma/R$.

The convention: phase speed, and loss per wavelength #

Speeds are phase speeds, $\omega/k_r$, and attenuation is the imaginary part of the wavenumber in nepers per metre — the amplitude falls by a factor $e$ in $1/k_i$ metres. The usable fast band asks that $k_i/k_r$ be below 0.05, which loses $1 - e^{-2\pi\cdot 0.05}$, about 27 per cent of the amplitude, per wavelength. The distributions are of number per unit radius, $n(R)\,dR$, normalised to a void fraction of a thousandth; the exponent $q$ is the slope of $n$ against $R$ on logarithmic axes.

What the picture cannot show #

The relation is linear and dilute. At a void fraction of a thousandth the bubbles are about ten radii apart and interact weakly; at a few per cent, under the crest of a breaking wave, they do not, and multiple scattering changes both the speed and the attenuation in ways a sum over independent bubbles cannot reproduce. A strong sound drives bubbles nonlinearly — the bubble that hammers is the extreme case — and a cloud driven near the resonance of most of its bubbles can rectify gas into them and change its own distribution. The bubbles here do not rise, dissolve or coalesce, which real ones do on time scales of seconds, so the distribution measured acoustically is a snapshot. And the power laws are idealisations of measured distributions that bend and cut off at both ends. The water is also at atmospheric pressure. Ten metres down the ambient pressure has doubled, every bubble’s stiffness has doubled with it, and every resonance has moved up by the square root of two: the same distribution there has its band and the start of its fast band about forty per cent higher in frequency, and a sonar’s view of a deep bubble plume differs from its view of a shallow one for that reason alone.

Who found it, and when #

Wood wrote down the mixture’s low-frequency speed in 1930. Foldy’s 1945 theory of multiple scattering put the bubbly-liquid relation on a footing for many sizes, and van Wijngaarden in 1972 and Commander and Prosperetti in 1989 gave the form used here, with the thermal behaviour of the gas inside. Acoustic bubble sizing by inverting attenuation, beginning with the resonant-bubble approximation and then solving the full integral, was developed through the 1980s and 1990s by Medwin, by Commander and Moritz, and by Duraiswami and others. Deane and Stokes’s measurements of 2002 established the two power laws under breaking waves, separated by the size above which turbulence can fragment a bubble.

Still open: a cloud too dense to be a sum #

Every number here treats the cloud as a sum of bubbles that do not see one another. The next calculation keeps the distribution and raises the void fraction past where that holds, replacing the independent sum with the effective-medium relation in which each bubble is driven by the field its neighbours scatter, and asks at what void fraction the usable fast band closes: whether the dense layer under a breaking crest, at several per cent of gas, still has frequencies at which sound outruns water, or whether multiple scattering fills the band with loss and leaves a layer that is an absorber at every frequency a sonar uses.