The note too high for thin air

The speeds in a still room found that the molecules of the air move at hundreds of metres a second in every direction, and how far a molecule gets found that each gets about seventy nanometres before it hits another. The viscosity that does not care how much gas there is found Maxwell’s surprise, that a gas’s viscosity does not change when its pressure does, because fewer molecules each carry momentum further. The gas of sound that carries a diamond’s heat turned the kinetic picture round and treated the vibrations of a crystal as a gas of their own.

This essay puts those pieces together to ask how sound itself depends on collisions. Sound in a gas is the one wave whose medium is literally a crowd of free particles, and the crowd can pass a compression along only if its members bump into each other often enough. That sets a highest frequency any gas can carry. At sea level it is so high that nobody notices; with height it falls, steadily and enormously, until above about a hundred kilometres the air cannot carry anything audible at all.

A path that lengthens with height #

The mean free path of a molecule is inversely proportional to the number of molecules per unit volume, which why the air thins with height found falling roughly exponentially, by a factor of e every seven or eight kilometres. At sea level a molecule of air goes sixty-five nanometres between collisions and collides about seven thousand million times a second. At fifty kilometres, near the top of the stratosphere, the path has grown a thousandfold, to tens of micrometres. At a hundred kilometres, the conventional edge of space, it is fourteen centimetres, and at a hundred and fifty kilometres tens of metres, longer than the wavelength of most of the sounds a person can hear.

A sound wave of wavelength λ is a pattern of compressions repeating every λ. If the molecules travel further than that between collisions, a molecule in a compression at one moment is in a rarefaction, or several wavelengths away, at its next collision, and the compression has no way of being handed on intact. The condition for sound is that the mean free path be much smaller than the wavelength, or equivalently that the molecules collide many times in each period of the wave.

How many collisions a sound wave has to spare at sea level is worth counting once. A tone of one kilohertz has a period of a millisecond, and in that time each molecule collides about seven million times. Its random walk over the period takes it, on average, the square root of that many free paths, about a fifth of a millimetre, while the wavelength is thirty-four centimetres. The molecules are thoroughly mixed with their neighbours on every scale smaller than a millimetre and know nothing of what is happening a wavelength away, except through the pressure passed from collision to collision. That is the separation of scales that lets air be treated as a continuous fluid for sound, and the separation shrinks as the pressure falls.

The loss in one wavelength #

Below that limit the gas carries sound, with a small loss. The distance that takes the treble out computed it from the air’s viscosity and thermal conductivity: in each compression some of the wave’s ordered motion is converted into heat, because the molecules in the compressed region are not quite in equilibrium with each other, and that conversion depends on how long they take to re-equilibrate compared with the period of the wave. Written per wavelength, the classical loss has a simple form,

$$\alpha\lambda = \frac{\pi}{\gamma}\,\omega\tau\left[\frac43 + \frac{\gamma-1}{\Pr}\right], \qquad \tau = \frac{\eta}{p},$$

where τ, the viscosity divided by the pressure, is close to the mean time between collisions, and Pr is the Prandtl number, the ratio of viscous to thermal diffusion. For air the bracket and its factor make 4.26ωτ. An audible tone at sea level has ωτ of about a millionth and loses a few millionths of its amplitude in each wavelength — which is why sound carries across a valley at all.

The formula has a domain, and its own result marks the edge. When ωτ reaches about a quarter, the wave loses a factor e of its amplitude within one wavelength, which is to say it is no longer a travelling wave in any useful sense. Beyond that the description underneath the formula, a continuous fluid with a viscosity and a conductivity, fails as well: it assumes the gas is nearly in equilibrium everywhere, which requires many collisions per period. Shaken faster than its molecules collide, a gas responds not as a fluid but as a crowd of independent particles streaming past one another, and what a loudspeaker would set moving there is not a sound wave but a spreading cloud of faster and slower molecules, a free-molecular disturbance that thins out and smears as it goes.

A ceiling that falls with the pressure #

The essential step is Maxwell’s. The viscosity of a gas does not depend on its pressure: halve the number of molecules and each one carries momentum twice as far, and the two changes cancel. So τ = η/p grows in exact inverse proportion to the pressure, at fixed temperature, and the frequency at which the loss reaches a factor e per wavelength falls in exact proportion to it.

At sea level that frequency is about two hundred megahertz, far into the ultrasound used to inspect materials and beyond anything audible by a factor of ten thousand. The pressure falls by a factor of about ten every sixteen kilometres, and the ceiling with it. It passes the top of human hearing, twenty kilohertz, near sixty-eight kilometres, in the mesosphere; at eighty kilometres it is about three kilohertz; and somewhere above a hundred and ten kilometres it passes the bottom of hearing, twenty hertz. Above that height there is no frequency a person could hear that the air can carry over even a single wavelength, at any loudness.

That is a sharper statement than the usual one, that space is silent because there is nothing in it. The thermosphere at a hundred and twenty kilometres is far from empty; a cubic centimetre of it holds about five hundred thousand million molecules. What it lacks is not matter but collisions frequent enough to carry anything faster than a slow rumble.

How far a note carries #

Below the ceiling, the distance a tone carries falls as the square of its frequency and in proportion to the pressure. At sea level the classical loss alone would let a two kilohertz tone carry eighteen kilometres before falling by a factor e; the real air absorbs more strongly than that, because the oxygen and nitrogen molecules’ vibrations take energy from the wave and give it back late, a relaxation the earlier essay on the treble measured at thirty times the classical value. With height the classical loss grows so fast that it soon dominates everything: at sixty kilometres the same tone fades within a few metres, and at seventy-five within half a metre, a handful of wavelengths.

The low notes survive. A twenty hertz rumble, fifteen metres long, still travels several wavelengths at a hundred kilometres, and infrasound at a fraction of a hertz, with wavelengths of kilometres, travels through the whole upper atmosphere. That is why the sounds that reach the ground from the thermosphere — from meteors burning up at eighty to a hundred kilometres, from rockets and re-entering spacecraft, from the aurora — arrive as infrasound, recorded by the microbarometer arrays built to detect nuclear explosions. Whatever higher-pitched noise a meteor makes is absorbed within metres of it.

Mars, where the speed of sound splits #

The thin carbon dioxide atmosphere of Mars, at about six hundred pascals, a hundredth of the Earth’s surface pressure, shows the same physics at a frequency people can hear. Its collision ceiling is still far above the audible range, but a carbon dioxide molecule’s vibrations exchange energy with its motion slowly, after many collisions, and the frequency at which that exchange can keep up with a sound wave falls with pressure just as the collision ceiling does. On Mars it falls to a few hundred hertz. When the Perseverance rover recorded sounds with its microphones in 2022, its team found two speeds of sound: notes below about two hundred and forty hertz travelled at about 240 metres a second, higher notes about ten metres a second faster, because the vibrations could not take part in the faster compressions. High notes were also absorbed within a few metres, so the planet’s soundscape is a muffled one — a laser’s crack on a rock heard from a few metres away, the low drone of the rover’s helicopter heard from much further.

The relaxation that splits Mars’s sound speed is the same kind that the distance that takes the treble out found making the Earth’s air absorb thirty times more than the classical loss. In both, the gas carries a wave faithfully only below a rate set by how often its molecules meet, whether to share their motion or their vibration, and both rates fall with the pressure. Titan, with a surface pressure half again the Earth’s and a cold, dense nitrogen atmosphere, would carry high notes better than the Earth does; the thin air of Mars carries them worse, and the thermosphere of every planet carries none.

Sound slower than the molecules that carry it #

The speed of sound itself is fixed by the molecules’ speeds and does not depend on the pressure, which is why a thin gas carries sound just as fast as a dense one, provided it carries it at all. The ratio of the sound speed to the mean molecular speed is $\sqrt{\gamma\pi/8}$, three-quarters for air and four-fifths for helium, set by how the molecules store energy and by nothing else. A compression cannot travel faster than the molecules that carry it, and it travels somewhat slower, because each molecule hands it on only after a random series of collisions in all directions.

The figure shows how directly sound belongs to the molecular speeds. Helium’s molecules are seven times lighter than air’s and move more than two and a half times as fast, and sound in helium is correspondingly faster; that is why a voice in helium sounds higher, as the resonances of the throat move up with the speed of sound while the vocal cords vibrate at much the same rate. The same ratio holds in the thin upper atmosphere. What changes with height is only how often the carriers meet.

A ceiling met on a chip #

The same crossover is met on a laboratory bench, not by going up but by going small. The vibrating structures inside the gyroscopes and accelerometers of phones are silicon beams and plates a few micrometres apart, oscillating at tens of kilohertz, and at atmospheric pressure the gap between a moving plate and its neighbour is only a few dozen mean free paths. The air in the gap is squeezed out and sucked back each cycle, and it damps the motion strongly, by the same viscous loss that absorbs sound. Sealing the device in a package pumped down to a fraction of a pascal lengthens the mean free path past the size of the gap, the gas stops behaving as a fluid, and its damping falls in proportion to the pressure, as individual molecules strike the moving plate and carry off their share of its momentum one at a time.

The quality factor of such a resonator, the number of cycles it rings for, rises from tens in air to tens of thousands in vacuum, and the noise a high Q moves out of the way explains why that is worth the cost of the package: the resonator’s own thermal noise is concentrated into a narrower band and its sensitivity improves. Designing the package means choosing a pressure on the curve this essay has drawn, between a gas that damps as a fluid and one that barely damps at all.

The same ratio, read as a frequency #

The collisions set other lengths too, which the essays on sound met without naming. A loud wave that steepens until it cannot is held at a finite thickness by the same viscosity and conduction that absorb ordinary sound, and that thickness is a few mean free paths. Around a spacecraft re-entering the atmosphere the gas is first a fluid, then a fluid with slip at the walls, and finally, highest up, molecules striking the hull one at a time — the regime of the gas that leaves, where no fluid equation applies.

What this essay adds is the version of that ratio that a wave carries with it. A size compared with a mean free path says whether a gas behaves as a fluid around an object; a period compared with a collision time says whether it behaves as a fluid for a disturbance passing through it. Sound turns the second comparison into a frequency, and the atmosphere, whose collision time grows by a factor of ten every sixteen kilometres, lays the whole range of that frequency out in height.

Where the model stops #

The ceiling and reach figures use the classical, Navier–Stokes loss throughout, with a single atmospheric composition and an approximate standard atmosphere interpolated in pressure. That is a simplification in three ways. Above about ninety kilometres the air’s composition changes, oxygen dissociating into atoms and the gases separating by mass, which changes its viscosity, its ratio of heat capacities and its molecular weight. The molecular relaxations that dominate absorption near the ground are left out; they matter little in the upper atmosphere, where the classical loss is overwhelming. And the ceiling is defined at ωτ near a quarter, where the continuum formula is already at the edge of its validity; kinetic theory beyond the Navier–Stokes equations, solved by Boltzmann’s equation or by simulating molecules directly, predicts the behaviour there and agrees with the measurements of sound in low-pressure gases made in the 1950s, which found the wave’s speed rising and its absorption saturating rather than following the classical formula past the limit.

The domain of the essay’s claim — that the highest audible frequency falls in proportion to the pressure, and passes out of the audible band at about a hundred kilometres — rests on the ratio of the wave’s frequency to the collision rate, which the figures compute correctly to within the uncertainties of the atmosphere’s structure.

Still open: how sound behaves at the edge of the continuum #

Between the fluid limit and the free-molecular limit, when a wave’s period is comparable with the collision time, the gas behaves in ways that neither description captures, and extending fluid equations into that regime has been an open problem since Boltzmann. The question is practical for the design of microscopic acoustic and mechanical devices, whose gaps are comparable with the mean free path at ordinary pressure, for the aerodynamics of vehicles in the upper atmosphere, and for interpreting the infrasound from the thermosphere that monitors use to detect explosions and meteors. Generalised hydrodynamic equations, moment methods and direct simulation each work in part of the range, and a single description valid across it remains unfound.

The habit worth carrying away is to find the time a medium takes to respond, before asking what it can carry. Air passes a compression on by collisions, losing 4.26ωτ of its amplitude per wavelength with τ = η/p, and since viscosity does not depend on pressure, the highest frequency that survives a wavelength falls in proportion to p — 212 MHz at the ground, 3 kHz at 80 km, 92 Hz at 100 km. The upper atmosphere is not too quiet to hear. Its molecules are too far apart to speak.