The distance that takes the treble out

Two things happen to a sound on its way across a field, and they are usually lumped together as “it gets quieter”. They are quite different processes with different arithmetic, and only one of them changes what the sound is.

The second is absorption, and it is not.

The distinction matters because the two losses have completely different arithmetic. Spreading is a power law in distance: six decibels per doubling, for ever, and always six. Absorption is an exponential in distance: a fixed number of decibels per kilometre, so it is negligible near the source and eventually beats any power law. And because that fixed number depends on frequency, absorption reshapes the sound while spreading only scales it.

Why the exponent is two #

The mechanism is easiest to see per cycle rather than per metre, which is how a damped oscillator’s loss is always counted. A viscous fluid being compressed and expanded loses a small fraction of the elastic energy on each cycle, and the fraction depends on the ratio of the relaxation time of the medium to the period. For a Newtonian fluid well below any resonance that fraction is proportional to the frequency, and the number of cycles in a metre is also proportional to the frequency, so the loss per metre carries two powers of it.

The exponent is two because the loss is a loss per cycle. In the simplest system that has one, a ringdown loses the same fraction of its amplitude every period — so the envelope is exponential in the number of cycles rather than in the time. Convert cycles to distance at a fixed wave speed and the absorption per metre comes out proportional to the frequency squared: one factor because higher frequencies fit more cycles into a metre, and a second because each cycle’s loss itself rises with frequency.

Writing the same thing as a formula, the classical absorption coefficient is

$$\alpha = \frac{\omega^2}{2\rho c^3}\left[\frac{4}{3}\eta + \zeta + \frac{(\gamma-1)\kappa}{c_p}\right],$$

where η is the shear viscosity and κ the thermal conductivity. Both of those are transport coefficients of the gas, computed by the same mean-free-path argument that gives the viscosity in the first place, and neither has anything to do with sound.

The viscosity in that expression comes from molecules carrying momentum a mean free path sideways — and the remarkable consequence is that it does not depend on the density. A thinner gas has fewer carriers and each one goes further, and the two effects cancel exactly. So the absorption at a given frequency is nearly independent of pressure, which is why sound carries no better on a mountain than at sea level.

The other quantity in the denominator is the sound speed, and it too is built from the molecular motion. A gas’s stiffness and its density both come from the same molecules, so the speed of sound lands close to the molecular speed — which means the whole absorption coefficient is expressible in molecular quantities, and the frequency-squared law is a statement about how a molecular medium handles a wave rather than about air in particular.

And the exponent is not two #

Having derived the f² law carefully, the honest next step is to compare it with a measurement, and it fails by a factor of thirty.

The reason is a mechanism the transport calculation has no place for. A nitrogen or oxygen molecule has a vibrational state a long way above its ground state — far too high to be excited by room-temperature collisions in any number — a frozen-out degree of freedom — — but the rate at which the small population there equilibrates is slow, milliseconds in dry air, and the sound wave’s compressions are pumping energy into and out of that state on a timescale of milliseconds too. Energy goes in during compression and comes out late, and out of phase, and that is a loss.

The exponent is not two whenever the medium has a relaxation time. A viscoelastic material under a sudden strain relaxes its stress over a characteristic time, and driving it faster than that time changes its response completely — so the absorption stops rising as the square and flattens off. Air has exactly such a mechanism, which is why the simple law fails in the middle of the audible range rather than at its edges.

What that mechanism gives is not f². A relaxation process contributes an absorption per wavelength that peaks where ωτ = 1, so the absorption per metre rises as f² below the relaxation frequency and levels off above it. Air has two such processes — nitrogen’s and oxygen’s — at frequencies that depend strongly on the water vapour present, because collisions are what equilibrate anything, at the rate a mean free path sets, and a water molecule is enormously more effective at exchanging energy with a vibration than another nitrogen molecule is.

Humidity is the variable because the relaxation frequency is set by how often a molecule meets something that can take the energy. Nitrogen and oxygen store energy in vibrations that are slow to exchange with translation, and water molecules are far better at accepting it — so a few per cent of water vapour moves the relaxation frequency by an order of magnitude, and the absorption at 4 kHz changes by a factor of several between a dry day and a humid one.

There is one more twist, and it is the reason the mechanism is so easy to miss. The vibrational states involved are barely populated at room temperature — the fraction of nitrogen molecules with a vibrational quantum is a few parts in ten thousand — so the heat capacity they contribute is tiny, and any calculation that asks how much energy the mode holds concludes that it cannot matter. What matters is not how much energy sits there but how slowly it gets in and out, and a small store filled and emptied out of phase dissipates far more than a large store that keeps up.

The fitted exponent tells the story without any of the mechanism. Over the audible range the measured curve goes as f^1.42 rather than f². An exponent between one and two is the signature of a process sitting on top of its own relaxation frequency, and it is enough on its own to say that the classical calculation is not the operative one.

The two mechanisms, priced against each other #

It is worth putting the classical and the relaxation contributions on the same scale before leaving them, because the ratio between them is not constant.

At sixty-three hertz the measured absorption is 0.1 dB per kilometre and the classical calculation gives 0.0005 — a factor of two hundred. At sixteen kilohertz the measured value is 400 and the classical is 1.9, still a factor of two hundred. Between them the ratio dips to about thirty at a kilohertz, which is where the oxygen relaxation is doing least. So the classical mechanism is never the dominant one anywhere in the audible range, at any humidity, and it does not become dominant at high frequency either: above about a hundred kilohertz the relaxations have saturated and the f² law reasserts itself, but by then the absorption is so large that the question is academic.

The honest summary is that the classical calculation predicts a law of the right form for the wrong reason, and that the coincidence is not lucky — the relaxation contribution goes as f² below its own relaxation frequency for exactly the reason the viscous one does, which is that the number of cycles per metre is proportional to the frequency. Two mechanisms, one counting argument, and only the coefficient differs.

What it sounds like #

There is one more reason the effect is under-appreciated: it is almost invisible indoors. A room is a few tens of metres across at most, and at that range absorption removes a fraction of a decibel at every audible frequency. Everything that dulls a sound in a room is the surfaces, not the air — and above about four kilohertz in a large hall the air does begin to contribute, which is why the reverberation time of a concert hall falls at the top of the spectrum in a way no absorption coefficient of the seats explains.

That is the whole account of thunder. A lightning channel makes a shock that decays to an acoustic pulse a few milliseconds long, which is a spectrum with content well above a kilohertz. Heard from three hundred metres it is a crack. Heard from ten kilometres — the same event, the same spectrum at the source — it is a rumble lasting many seconds, because the high frequencies have been removed by the air and what is left is the low-frequency tail, arriving over the spread of path lengths from different parts of a channel kilometres long.

The same effect sets the character of every large space. Standing waves in a room decay the same way: a cathedral’s reverberation is bright at first and dulls as it decays, because the late arrivals have travelled further through the air as well as bounced more often; the sound of a distant orchestra through an open window is missing its top; and an outdoor loudspeaker aimed at a crowd two hundred metres away has to be equalised for what the air will do on the way. None of that is a property of the source.

And it is why distance is a cue. A listener has no ruler, and yet a rifle shot at fifty metres and one at a kilometre are immediately different — not merely quieter, but duller. Absorption is the only propagation effect that leaves a signature in the sound itself rather than in its level, which is why it works as a distance cue when a level does not.

The same two relaxations, in the sea #

The strongest evidence that the mechanism above is about relaxation rather than about air is that sea water does the same thing, with different molecules, at different frequencies, and with the same shape.

Sound in the ocean is absorbed far less than in air — a kilohertz tone loses something like a twentieth of a decibel per kilometre, against air’s 3.7 — which is why the sea is the medium of choice for anything that has to be heard a long way off. But the frequency dependence has exactly the structure this essay has been describing: two relaxation processes, one near a kilohertz and one near a hundred, sitting on a classical viscous background that is negligible below both. The molecules responsible are dissolved boric acid for the lower and magnesium sulphate for the upper, each with a chemical equilibrium that the passing pressure wave shifts and that takes time to shift back.

The consequences are the same consequences, scaled. A hundred-kilohertz sonar loses tens of decibels per kilometre and is a short-range instrument with fine resolution; a hundred-hertz one loses almost nothing and can be heard across an ocean basin. Whale song sits at the bottom of that range for the same reason, and so does every long-range detection system anybody has built.

What makes the parallel worth drawing is that nothing about the two media is alike. Air is a gas with a vibrational mode; sea water is a liquid with a chemical equilibrium. Their densities differ by a factor of eight hundred and their sound speeds by a factor of four. The absorption curves have the same shape because the shape belongs to the idea of a relaxation, and the identity of the thing relaxing enters only through two numbers.

The trade every ultrasound scan makes #

The exponent turns into an engineering constraint most directly in medical imaging, where the absorbing medium is soft tissue and the loss is close to linear in frequency rather than quadratic — roughly half a decibel per centimetre per megahertz.

That single figure decides the whole design. Resolution improves with frequency, because the finest detail a beam can separate is of order a wavelength; penetration worsens with frequency, at half a decibel per centimetre per megahertz, and the sound has to go there and come back. A three-megahertz probe looking twelve centimetres deep loses about thirty-six decibels on the round trip, which is manageable. Doubling the frequency to improve the resolution doubles that to seventy-two, which is not. So a scan of something deep is done at two or three megahertz and a scan of something just under the skin at fifteen, and no amount of transmit power fixes the difference, because the noise the echo competes with rises with the power too.

The reason the exponent matters rather than merely the coefficient is that it sets how sharp the trade is. Under an $f^2$ law the same doubling would cost four times the loss instead of twice, and the useful band would be narrower than it is. Tissue’s near-linear behaviour is a piece of luck that imaging depends on, and it is a fingerprint of a broad distribution of relaxation times rather than one — a whole spectrum of processes, each contributing its own peak, adding up to a slope between the two ideal cases.

What this model still does not have #

Nothing here is about a solid. A sound wave in steel or in rock loses energy far more slowly per wavelength than in air, and the frequency dependence is often close to linear rather than quadratic — because the mechanisms are grain boundaries and dislocations rather than a molecular relaxation, and their loss per cycle is nearly independent of how fast the cycle is. A linear law and a quadratic one are different statements about the microscopic process, and reading the exponent off is one of the few ways to tell which is at work.

Nothing above has any weather in it. Real outdoor sound is dominated by refraction long before absorption matters: a temperature gradient bends sound rays down or up, exactly as a gradient of index bends light, a wind gradient does the same asymmetrically, and the difference between hearing a source at a kilometre and not hearing it at all is usually which way the sound was bent. Absorption sets a ceiling that refraction rarely lets anything reach.

Nor is any of it about water droplets. Fog and rain change atmospheric absorption very little at audible frequencies — the droplets are far smaller than the wavelength, and the scattering is negligible. What fog does change is the temperature profile, and therefore the refraction, which is why sound seems to carry further on a foggy morning: not the fog, the inversion under it.

And nothing here is loud. Every expression above is linear in the amplitude, so the absorption per metre is a property of the air and not of the sound in it. A disturbance large enough to steepen its own front — a blast, a shock, the first few hundred metres of a thunderclap — loses energy at the front itself, by a mechanism that is not in any of these curves and that acts hardest on precisely the sharpest features.

And the ground is not in it either. A source and a listener both near the ground have a direct path and a reflected one, and their interference removes particular frequencies entirely — an effect that is far larger than absorption at low frequencies and has nothing to do with the air.

The same exponential turns up in a medium with no acoustics anywhere. A beam of anything passing through matter is attenuated as $e^{-n\sigma x}$, with a number density and a cross-section in the exponent — and the reason is the one this essay has been using throughout: a constant probability of loss per unit of path gives an exponential in path. What differs between the cases is only what is being lost and to what.

What a quality factor is, said once #

Absorption, ringdown, linewidth and loss per cycle are four names for one number, and the essay has used all four without saying so.

A resonator’s quality factor is 2π times the energy stored divided by the energy lost per cycle. A travelling wave has no resonator, but it has the same ratio: the fraction of its energy lost per cycle is 2π/Q, and the distance in which it falls by a factor of e is Q/2π wavelengths. So an absorption coefficient in nepers per metre and a quality factor are the same statement in different units, and the conversion contains only the wavelength.

That is why the figures above could be drawn with a damped oscillator standing in for a medium: they are the same equation. It is also why the exponent question is so sharp. A Q that is independent of frequency gives α ∝ f, which is what most solids do; a loss per cycle proportional to frequency gives α ∝ f², which is what a viscous fluid does; and a relaxation gives a Q with a minimum at one particular frequency, which is what air does. The exponent is a fingerprint of the microscopic process, readable without knowing anything about it.

Putting a number on it makes air seem a much better resonator than it feels like. At a kilohertz, where the absorption is 3.7 decibels per kilometre, a wavelength is 34 centimetres, so the energy lost in one cycle is about three parts in ten thousand and $Q$ is around twenty thousand. That is better than most mechanical resonators anybody builds. It sounds wrong only because a wave in open air gets through so many cycles: twenty thousand of them is seven kilometres, and by then the spreading loss has finished the job several times over.

The same arithmetic explains why the effect is so much more visible at the top of the spectrum without the exponent having to be quoted. At sixteen kilohertz the wavelength is sixteen times shorter, so a metre holds sixteen times as many cycles, and the loss per cycle has itself grown — which is the $f^{1.42}$ read as a statement about cycles rather than about metres. Absorption per wavelength is the quantity with the mechanism in it, and absorption per metre is the quantity a listener meets.

Where this ladder goes next #

This rung has treated absorption as a number that removes energy. What it has not asked is where the energy goes, and the answer has a consequence that can be measured: it becomes heat, in the gas, in step with the wave — so a standing sound wave heats the medium at its antinodes and not at its nodes, and a strong enough one drives a steady flow. The next rung is that: acoustic streaming, the pressure a sound wave exerts, and why an absorption coefficient and a radiation force are two readings of the same quantity.