How does the speed of sound vary with temperature?

In a gas, sound travels at a speed set by how stiff the gas is and how much it weighs. For an ideal gas that reduces to a simple expression: the speed of sound equals the square root of the product of the adiabatic index, the specific gas constant and the absolute temperature. For dry air the index is 1.4 and the gas constant 287 J/(kg·K), which gives about 20.05 times the square root of the temperature in kelvin. Pressure and density appear in the derivation but cancel each other out, because compressing air raises both in the same proportion. Only temperature remains.

Using the International Standard Atmosphere, sea level at 15 °C gives 340.3 m/s, or about 661 kn (1,225 km/h). At 5,000 m the standard temperature is −17.5 °C and sound moves at 320.5 m/s. At the tropopause, 11,000 m and −56.5 °C, it drops to 295.1 m/s (574 kn). Above that the temperature is constant up to 20,000 m, so the speed of sound is constant too. Warm days push it up: at 40 °C on a desert runway sound travels at about 355 m/s, some 4 % faster than on a standard day. Humidity has a small effect because water vapour is lighter than the nitrogen and oxygen it displaces; fully saturated air at 30 °C carries sound about 0.5 % faster than dry air.

Aircraft speed limits in the transonic regime are expressed as Mach number, the ratio of true airspeed to the local speed of sound, because it is Mach number, not airspeed, that governs shock formation on the wing. An airliner cruising at Mach 0.85 in the stratosphere is doing about 250 m/s true airspeed (487 kn); the same Mach number at sea level would be 289 m/s (562 kn). This is why the Machmeter and the airspeed indicator disagree so much at altitude and why climb profiles switch from a constant indicated airspeed to a constant Mach number at a crossover altitude, typically around 8,000–9,000 m. It also explains why a hot day makes it slightly harder to reach a given Mach number and why the sonic boom of a supersonic aircraft reaches the ground with different timing in summer and winter.