The correction that took a century

Newton’s Principia contains a derivation of the speed of sound, which is a remarkable thing to find in a book of 1687 — nobody had previously suggested that the speed of a wave could be computed from properties of the substance it travels in. His answer for air was about 290 metres a second. The measured value, which he knew, was around 340. He spent the second edition adding corrections for the size of air particles and for water vapour, and got no nearer.

What Newton computed #

A longitudinal wave in a fluid travels at

$$c = \sqrt{\frac{K}{\rho}},$$

with $K$ the bulk modulus — how much pressure a fractional squeeze costs — and $\rho$ the density. That is the general result and it is not in dispute; it is the same form as the speed on a string, with a restoring stiffness over an inertia.

The question is which $K$. Newton took the compressions to happen at constant temperature, so he used Boyle’s law: $PV$ constant gives $K_T = P$, and

$$c_{\text{Newton}} = \sqrt{\frac{P}{\rho}} = \sqrt{\frac{RT}{M}}.$$

For air at 0 °C that is 280.0 m/s. It is an entirely respectable derivation and it is 15% low. The density it needs comes from the ideal gas law, which is the one part of the calculation nobody has ever had to revise.

Which stiffness a sound wave actually uses #

A gas has two bulk moduli, not one. Squeeze it slowly, letting heat leak away to keep the temperature fixed, and it resists with $K_T = P$. Squeeze it quickly, so that no heat has time to move, and the temperature rises with the compression, which makes it resist more: $K_S = \gamma P$, where $\gamma = C_P/C_V$.

Which applies is a question about timescales, and it is the same question that decides whether a fluid can be treated as a continuum at all — a ratio of a molecular length or time to a length or time belonging to the flow. Heat diffuses; in a time $t$ it reaches a distance of order $\sqrt{Dt}$ with $D$ the thermal diffusivity. In one period of a sound wave, $t = 1/f$, so heat gets $\sqrt{D/f}$. The distance it would need to travel to equalise a compression with the rarefaction beside it is of order the wavelength, $c/f$. The ratio is

$$\frac{\sqrt{D/f}}{c/f} = \frac{\sqrt{Df}}{c},$$

which for audible sound in air is around $10^{-3}$ — heat gets a thousandth of the way. So the compressions are adiabatic, overwhelmingly, and

$$c = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}} = 331.3\ \text{m/s},$$

which is the measured value to a tenth of a per cent.

There is a second way to see that the adiabatic modulus is the larger one, and it is worth having because it needs no thermodynamics at all. Compressing a gas quickly does work on it, and the work has nowhere to go, so the gas gets hotter; a hotter gas at the same density pushes back harder. So the quick squeeze always resists more than the slow one, in any substance whose temperature rises on compression — which is every gas. The factor by which it resists more is exactly the ratio of the two heat capacities, and that identity is the content of the thermodynamic derivation.

Laplace supplied this in 1816, a hundred and twenty-nine years after the Principia. The delay is not surprising in retrospect: the distinction between isothermal and adiabatic requires the concept of heat capacity at constant volume and at constant pressure, which required a theory of heat as something other than a fluid, and that arrived in the 1820s and 1840s. Newton could not have made the correction, because the vocabulary in which it is stated did not exist.

The speed that does not depend on the pressure #

Written as $\sqrt{\gamma P/\rho}$ the formula looks as though squeezing a gas should make sound travel faster in it. Written as $\sqrt{\gamma RT/M}$ it plainly does not depend on the pressure at all, and both expressions are the same expression.

The reconciliation is that raising the pressure at fixed temperature raises the density in exact proportion, so the ratio is unchanged. A gas made twice as stiff has been made twice as heavy, and a wave speed is a stiffness over an inertia. Sound travels at the same speed at the bottom of a mine and at the top of a mountain, provided the two are at the same temperature.

That has a consequence worth stating because it is routinely got wrong. An aircraft’s Mach number is its speed divided by the local speed of sound, and the local speed of sound is a function of the air temperature and of nothing else — not the altitude, not the pressure, not the density. The reason the sound speed falls with height in the atmosphere is entirely that the temperature falls with height; it stops falling at the tropopause because the temperature stops falling there, and it is constant through the lower stratosphere for the same reason. An instrument that reads Mach number is reading a ratio in which the denominator is a thermometer.

It also explains why the composition matters as much as it does. $M$ is in the denominator under a square root, so what a gas is made of enters through its molar mass and its degree-of-freedom count, and nothing else about it survives. Two gases of the same $\gamma$ and the same $M$ carry sound at identical speeds however different their chemistry.

The factor as a count #

The repaired formula contains $\gamma$, and $\gamma$ is not a fitted number. Equipartition gives each accessible quadratic degree of freedom an average energy of $\tfrac12 k_BT$, so a molecule with $f$ of them has $C_V = \tfrac{f}{2}R$ per mole, $C_P = C_V + R$, and

$$\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}.$$

A monatomic gas has three translational degrees and nothing else, so $\gamma = 5/3$. A diatomic gas at room temperature has three translations and two rotations — rotation about the bond axis is not excited — so $f = 5$ and $\gamma = 7/5$. Carbon dioxide is linear and triatomic, with three translations, two rotations and a pair of low-frequency bending modes partly active at 0 °C, giving something near seven.

Turn that round, and the sound speed becomes a measurement. Divide the measured speed by Newton’s, square it, and $\gamma$ falls out; invert $\gamma = 1 + 2/f$ and $f$ falls out. The four gases in the opening figure give 5.0, 3.0, 3.0 and 6.9 — the count of the ways each molecule can hold energy, obtained by timing a pulse down a tube.

Counted as degrees of freedom, $\gamma$ stops being a measured constant and becomes arithmetic. A monatomic gas has three ways of moving and $\gamma = 5/3$; a diatomic one has three plus two rotations and $\gamma = 7/5$; and the ratio is $(f+2)/f$ for $f$ active degrees of freedom. Air is mostly diatomic, so 1.4 is not fitted to the sound speed — it is counted, and then the sound speed is predicted.

That last figure is the point at which the argument stops being classical. Equipartition says every degree of freedom takes its half a $k_BT$; the plateaus say some of them do not, until the temperature is high enough. The temperature at which each switches on is $\hbar^2/2Ik_B$ for rotation and $\hbar\omega/k_B$ for vibration, so the shape of the heat capacity curve — and therefore the temperature dependence of the speed of sound — contains Planck’s constant. A careful measurement of how sound speed varies with temperature in hydrogen would recover $\hbar$ from a thermometer and a stopwatch.

Why the collision rate appears twice #

The most surprising thing in the crossover figure is that two apparently unrelated frequencies land almost on top of each other.

The isothermal crossover is at $f_{\text{iso}} = c^2/D$. The continuum limit is at $f_{\text{mfp}} = c/\ell$, where $\ell$ is the mean free path. These look like different quantities and they are not, because kinetic theory relates both to the same molecular scale: $D \approx \tfrac13 \ell\bar v$ and $c \approx \sqrt{\gamma/3}\,\bar v$, with $\bar v$ the mean molecular speed. Substituting,

$$f_{\text{iso}} = \frac{c^2}{D} \approx \frac{\gamma \bar v^2/3}{\ell \bar v /3} = \frac{\gamma \bar v}{\ell}, \qquad f_{\text{mfp}} = \frac{c}{\ell} \approx \frac{\bar v}{\ell}\sqrt{\gamma/3},$$

and both are the collision rate $\bar v/\ell$ to within a factor of two. They cannot be separated by changing the pressure, because both scale as $1/\ell$ and the ratio between them contains only $\gamma$.

So Newton’s assumption is not merely wrong for the sound anybody has ever heard. It is unreachable: the frequency at which a gas would carry an isothermal sound wave is the frequency at which the gas stops carrying sound waves at all.

The same length appears in both frequencies, which is why the collision rate turns up twice. A molecule’s free flight between collisions is 68 nm in air at atmospheric pressure, and it sets the thermal diffusivity and the viscosity together. That is the physical content of the assumption being made: heat has to fail to leave a compression, and whether it fails depends on how far a molecule gets before it is stopped.

Sound travels at one definite speed while the molecules carrying it move at a spread of speeds averaging rather more. The sound speed comes out at $\sqrt{\gamma/3} \approx 0.68$ of the mean molecular speed — slower than the molecules, because a wave is a coordinated disturbance passed along rather than anything being carried. That ratio is a number rather than a coincidence, and it is where $\gamma$ enters from the kinetic side.

What the correction is worth, in practice #

The 15% Newton was out by is not a small quantity in any application that uses the number.

A concert pitch of A = 440 Hz in air at 20 °C corresponds to a wavelength of 780 mm. Using Newton’s speed instead would put it at 663 mm — a difference that would misplace the length of every organ pipe by a fifth of its length, which is about four semitones. The tuning of wind instruments is a measurement of $\sqrt{\gamma R T/M}$ made with an ear, and it is sensitive enough that players hear the temperature: the speed rises 0.6 m/s per kelvin, so an instrument that has warmed by ten degrees has gone sharp by about 30 cents.

The displacement in a sound wave is along the direction of travel rather than across it, so the crests are compressions and the troughs are rarefactions. That is worth picturing because it is where the stiffness enters: what resists is the gas’s reluctance to be compressed, and the question this essay turns on is which reluctance — the one measured slowly, or the one measured too fast for heat to escape.

Helium’s higher speed — 972 m/s, nearly three times air’s — is the same formula with a smaller $M$ and a larger $\gamma$, and it is why a voice in helium changes. It changes in a specific way: the vocal cords vibrate at the same frequency, since their tension and mass are unchanged, but the resonances of the vocal tract all move up by the speed ratio, so the formants shift and the pitch does not. What is heard is a change of timbre, and describing it as a raised pitch is a misreading of what a resonant cavity does.

The measurement that fixed the kelvin #

The formula runs in the direction Laplace used it — known gas, computed speed — and it is far more valuable run backwards. For a monatomic gas, $\gamma$ is not measured but known: three translational degrees of freedom and nothing else gives exactly $5/3$, with no approximation and nothing to fit. So

$$c^2 = \frac{5RT}{3M}$$

contains one unknown if the temperature is known, and one temperature if the gas constant is.

That is the basis of acoustic gas thermometry, and it is not a demonstration but the most accurate route to the quantity there has ever been. Fill a resonator of accurately known shape with argon, measure the frequencies of its acoustic modes, and the speed of sound follows from the frequencies and the geometry. Hold the gas at the triple point of water, whose temperature was fixed by definition, and the gas constant falls out — and with Avogadro’s number, so does Boltzmann’s.

The precision reached is a few parts in a million, which is why the value of $k$ carried in the tables for years came from this measurement rather than from anything more obviously fundamental. When the kelvin was redefined in 2019 by fixing $k$ at an exact number, the number chosen was determined largely by acoustic thermometry in argon.

There is a pleasing closure in that. The quantity Newton’s derivation was missing is a count of degrees of freedom; the case where that count is certain is a monatomic gas; and the measurement of sound speed in a monatomic gas is what pinned down the constant that converts a temperature into an energy per degree of freedom. The correction and the instrument are the same fact, used in opposite directions, two hundred years apart.

What Newton did with his fifteen per cent #

Newton knew the answer was wrong. The Principia’s later editions contain his attempt to close the gap, and it is worth recording because of how it fails.

He reasoned that air is not a continuum of nothing but that its particles occupy some fraction of the volume, so a pulse crosses the solid parts instantaneously and only has to travel through the gaps — which would raise the effective speed. He assigned the particles a size, obtained a correction of about a tenth, and added a further allowance for water vapour in the air. The combination brought his figure up to something close to the measurement.

Every piece of that is a real effect and every one is the wrong size. Particle volume does correct the gas law, by parts in ten thousand at atmospheric pressure rather than by ten per cent. Water vapour does raise the speed, by a few tenths of a per cent rather than by several. Two corrections of the right sign and the wrong magnitude, tuned until they summed to the discrepancy.

The instructive part is not the error but that the discrepancy varies from gas to gas. Air is 15.5 per cent low, helium 22.5 and carbon dioxide 12.0 — and no account built on the size of particles or on humidity can produce that pattern, because it is a pattern in the number of ways a molecule can move. A single fudge factor tuned on air would have been invisible for another century; four gases disagreeing by four different amounts is what makes the missing quantity findable.

Where the model stops #

The gas is ideal. Real gases have interactions, so $K$ is not exactly $\gamma P$. The correction is small at atmospheric pressure — a few tenths of a per cent for air — and it is not small near a condensation point, where the sound speed dips sharply.

The wave is small-amplitude. A finite-amplitude compression travels faster than the rarefaction behind it, because it is both hotter and moving forward, so a wave steepens as it goes. That is a different subject and belongs elsewhere; the linear speed derived here is the limit as the amplitude goes to zero.

The gas is not a mixture with slow components. Humid air is a mixture, and water’s molar mass is lower than nitrogen’s, so damp air is less dense than dry air at the same pressure and sound travels slightly faster in it — about 0.3% at 30 °C and complete saturation. Newton, hunting his 15%, tried to attribute it to water vapour and to the finite size of air particles, and both were real effects of the wrong order.

The energy exchange is instantaneous. Rotational and vibrational modes do not equilibrate with translation immediately: they take a characteristic number of collisions, and for vibration in carbon dioxide that number is enormous. If a mode cannot keep up with the compression cycle it does not contribute to $\gamma$, so the effective $\gamma$ depends on frequency — which makes the sound speed dispersive and absorbs energy. That relaxation absorption is why carbon dioxide is far more absorbing of sound than nitrogen, and it is a case where the adiabatic assumption fails in a way that has nothing to do with heat conduction.

An isotherm follows $PV =$ constant and an adiabat follows $PV^\gamma =$ constant, so the adiabat is steeper by exactly the factor $\gamma$ at every point. That is the whole correction drawn as two curves: Newton computed the slope of the first and the wave obeys the second, and the ratio of the two slopes is the ratio of the two answers.

What the pictures cannot show #

The bar chart shows three numbers per gas and cannot show why the measured value belongs beside the adiabatic one rather than the isothermal one. That is an argument about timescales, and it appears in the second figure as a ratio far below one over the whole audible range — but a ratio of $10^{-3}$ is not visible on any axis that also has to reach one.

The crossover figure draws two frequencies as though the transition were at a point. Both are order-of-magnitude statements: the wave becomes partly non-adiabatic well before the ratio reaches one, and the continuum degrades gradually rather than at a threshold. The figure marks scales, not edges.

And nothing here shows absorption. A sound wave in a real gas loses energy at a rate that rises as $f^2$, from viscosity and conduction together, so the high-frequency end of every figure describes a wave that would not survive a centimetre. The arithmetic is about a wave’s speed and is silent on whether there is a wave left.

Where the ladder goes next #

This rung took a wave speed made of a stiffness and an inertia and asked which stiffness — and the answer turned out to depend on a comparison of timescales, then on a count of molecular degrees of freedom, and finally on which of those degrees of freedom are affordable at the temperature in question.

The rungs above are the ones where the two timescales become comparable rather than remote: relaxation absorption, where a single internal mode lags and the effective $\gamma$ becomes frequency-dependent; and the transition to free-molecular flow, where the continuum gives out and the disturbance stops being a wave. Both are what the Knudsen number governs in a different guise.

The habit is the one Newton’s failure teaches. When a derivation is right and the answer is wrong, the error is in an assumption made so early that it was never written down. Newton wrote every step of the algebra and did not write the words “at constant temperature”, because there was as yet no alternative to contrast it with.