The Speed of Sound

Put your ear to a long steel railing while a friend taps the far end: you hear the tap twice, first through the metal and then through the air. A stiffer medium carries sound faster.

Speed in a fluid#

Take a slab of fluid of thickness Δx and cross-section A in a sound wave. The pressures on its two faces differ slightly, and this difference accelerates the slab. Newton’s second law, together with $p = -B\,\partial s/\partial x$ from the last lesson, gives

This is the wave equation, and the number in front is v². So:

In a solid rod

A thin rod can bulge sideways as it is squeezed along its length, so the relevant stiffness is Young’s modulus Y:

Speed of sound in a gas: Newton’s attempt#

For a gas, what is B? Newton assumed the temperature of each bit of air stays constant as it is squeezed and stretched. For an isothermal process $PV$ is constant, which gives $B = P$, so

The measured value at STP is about 332 m/s. Newton’s answer is about 15% too low.

Laplace’s correction#

Laplace pointed out that the compressions and rarefactions come and go far too quickly for heat to flow in or out, and air is a poor conductor anyway. A compressed region warms up; a rarefied region cools. The process is adiabatic, for which $PV^\gamma$ is constant and $B = \gamma P$, where γ = C_p/C_v ≈ 1.4 for air.

What changes the speed of sound in air?#

Pressure: no effect (at fixed temperature)

For an ideal gas $P/\rho = RT/M$. Doubling the pressure at the same temperature doubles the density too, so P/ρ and hence v are unchanged.

Temperature: v ∝ √T

Using $P/\rho = RT/M$, $v = \sqrt{\gamma RT/M}$, so v grows as the square root of the absolute temperature. For ordinary temperatures, with v₀ the speed at 0 °C and t in °C,

Humidity: moist air is faster

Water vapour (M = 18 g/mol) is lighter than dry air (about 29 g/mol). Humid air at the same pressure and temperature is therefore slightly less dense, and sound travels slightly faster in it.