Why does the speed of sound change with altitude?

Sound is a pressure disturbance handed from molecule to molecule, and the speed at which it travels is set by how fast those molecules are moving between collisions. In a gas, that molecular speed depends on temperature alone. Physically, the speed of sound is the square root of the gas constant times the temperature times a factor that depends on the gas, about 1.4 for air. Pressure and density cancel out of the relation: compressing air makes it denser but also stiffer by the same factor, so at a fixed temperature sound moves at the same speed at sea level and on a mountaintop. What changes with altitude is not the thinness of the air but its temperature, and that is what moves the speed of sound.

In the International Standard Atmosphere used for aircraft performance, sea-level air is 15 °C and sound travels at 340.3 m/s, or 1,225 km/h. The temperature then falls by 6.5 °C for every kilometre of climb through the troposphere, and the speed of sound falls with it: about 328 m/s at 3,000 m and 316 m/s at 6,000 m. At 11,000 m the standard temperature reaches −56.5 °C and sound moves at 295 m/s, 1,062 km/h or 573 knots. From there up to about 20,000 m the standard atmosphere is isothermal, so the speed of sound stays constant, and above that, in the stratosphere, temperature and the speed of sound rise again as ozone absorbs sunlight. Real air rarely matches the standard exactly, so aircraft measure outside air temperature and compute the local value continuously.

The practical consequence is that Mach number, the ratio of true airspeed to the local speed of sound, rises as an aircraft climbs even if its true airspeed does not change. A jet holding 900 km/h is at about Mach 0.73 near the ground but about Mach 0.85 at 11,000 m, close to the point where shock waves form and drag climbs steeply. At the same time the stall speed in true airspeed rises with altitude because the thin air makes less lift. High in the atmosphere those two limits converge, and pilots refer to the narrow band between them as the coffin corner. It is also why supersonic aircraft cross Mach 1 more easily at altitude: the target speed is lower in km/h, and the thin air produces less drag.