Sound is a compression wave: the air is pushed, compresses, pushes the next air and decompresses. The speed of this wave depends on two properties of the air: how much it resists being compressed and how heavy it is.

Newton’s formula

In the Principia, of 1687, Newton calculated the speed of sound for the first time. Assuming that, when compressed, air obeys Boyle’s law, that is, that pressure is proportional to density, he obtained

$$v = \sqrt{\frac{P}{\rho}}.$$

At sea level, at 20 °C, the pressure is $P = 101\,325$ Pa and the density of air is $\rho = 1.204$ kg/m³. The formula gives

$$v = \sqrt{\frac{101\,325}{1.204}} \approx 290 \text{ m/s}.$$

Measurements, however, gave about 340 m/s, some 15% more. Newton tried to explain the difference with forced hypotheses, such as the size of the air particles and water vapour. For more than a century the discrepancy remained unexplained.

Air that heats up

Laplace found the mistake. When a gas is compressed, it heats up, as anyone who has pumped up a bicycle tyre with a hand pump knows. Boyle’s law holds when the temperature stays constant. But the compressions of a sound wave are so fast that the heat has no time to escape: the compressed air gets a little warmer and so pushes harder than Newton assumed. Today we say the compression is adiabatic, not isothermal.

In mathematical terms, the speed of a wave in a fluid is $v = \sqrt{\frac{dP}{d\rho}}$, how much the pressure rises when the density rises. For an isothermal compression, $P = k\rho$ and

$$\frac{dP}{d\rho} = \frac{P}{\rho}.$$

For an adiabatic compression, $P = k\rho^\gamma$, where $\gamma$ is the ratio between the specific heats of the gas at constant pressure and at constant volume, and

$$\frac{dP}{d\rho} = k\gamma\rho^{\gamma-1} = \gamma\,\frac{P}{\rho}.$$

The correct formula, which Laplace published in 1816, is therefore

$$v = \sqrt{\frac{\gamma P}{\rho}}.$$

For air, $\gamma \approx 1.40$. The correction multiplies Newton’s result by $\sqrt{1.40} \approx 1.183$:

$$v \approx 290 \times 1.183 \approx 343 \text{ m/s},$$

exactly the measured value. Laplace turned the argument around: he used the measured speed of sound to estimate $\gamma$, before it could be measured directly with precision.

Exercise: sound in the cold and in the heat

For an ideal gas, $\frac{P}{\rho} = \frac{RT}{M}$, so that $v = \sqrt{\frac{\gamma RT}{M}}$, with $R = 8.314$ J/(mol·K) and, for air, $M = 0.02896$ kg/mol. Compute the speed of sound at 0 °C and at 30 °C.

At 0 °C, $T = 273.15$ K: $v = \sqrt{\frac{1.40 \times 8.314 \times 273.15}{0.02896}} \approx \sqrt{109\,800} \approx 331$ m/s. At 30 °C, $T = 303.15$ K: $v \approx \sqrt{121\,800} \approx 349$ m/s. Sound does not depend on pressure, only on temperature: on a hot day it travels almost 20 m/s faster.

Exercise: how far away was the lightning?

You see a flash of lightning and hear the thunder 6 seconds later. How far away did the lightning strike?

The light arrives practically at the same instant. Sound, at about 343 m/s, covers in 6 s about $343 \times 6 \approx 2\,060$ m: the lightning struck about 2 km away. The rule of thumb is to divide the seconds by 3 to get the distance in kilometres.

Sources: Isaac Newton, Philosophiæ naturalis principia mathematica (1687), book II, propositions 48–50 · Pierre-Simon Laplace, “Sur la vitesse du son dans l’air et dans l’eau”, Annales de chimie et de physique 3 (1816) · Bernard S. Finn, “Laplace and the Speed of Sound”, Isis 55 (1964) · Robert Fox, “The Rise and Fall of Laplacian Physics”, Historical Studies in the Physical Sciences 4 (1974).

All topics on Pierre-Simon Laplace · ← The Laplace transform · Laplace’s demon →

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