Why does the speed of sound vary with temperature?

A sound wave is a small disturbance in pressure that travels through a gas by molecules bumping into their neighbours. A compressed region pushes the molecules next to it, which push the next ones, and so on. How fast the disturbance can move therefore depends on how fast the molecules themselves are moving between collisions. Kinetic theory links that directly to temperature: the average speed of a gas molecule is proportional to the square root of absolute temperature divided by molecular mass. At 20 °C a nitrogen molecule moves on average at roughly 500 m/s; the sound wave, which is carried by the ordered component of that random motion, travels at about 343 m/s, a fixed fraction of the molecular speed.

It is tempting to assume that thin air at altitude should carry sound differently, but the calculation shows it does not. The speed of sound equals the square root of stiffness divided by density. Lower pressure makes the air less stiff, but it also makes it less dense, and the two effects fall together exactly. What remains is the ratio of pressure to density, which for an ideal gas is simply the gas constant times temperature. The practical consequence is that a sound wave on a summit at 4,000 m travels at the same speed as one in a valley if both are at the same temperature; the summit is only slower because it is usually colder.

Temperature dominates, but the mass of the molecules also matters. Helium is light, so sound in helium travels at about 1,000 m/s at room temperature, which is why a voice sounds high after inhaling it: the vocal tract resonates at higher frequencies. Water vapour, with a molecular mass of 18 against air's average of 29, has a smaller effect: saturated air at 30 °C carries sound about 0.5 % faster than dry air. For aviation this is negligible against the temperature term, which is why Mach number computations in an air data computer use only static temperature. That is also why the speed of sound at the tropopause, −56.5 °C in the standard atmosphere, is 295 m/s, some 13 % below the sea-level value on a standard day, and why an aircraft's Mach limit corresponds to a lower true airspeed the higher and colder it flies.