How does the speed of sound change with altitude?

In a perfect gas the speed of sound depends on the square root of absolute temperature multiplied by a constant that includes the gas's composition. Pressure and density cancel out of the equation, which surprises many people: thin air at altitude carries sound just as fast as dense air at the same temperature. So the question of how sound speed changes with altitude is really the question of how temperature changes with altitude, and that is described by the International Standard Atmosphere used by ICAO and every aircraft manufacturer.

In the standard atmosphere the temperature at sea level is 15 °C and falls at 6.5 °C per kilometre through the troposphere. The speed of sound falls with it: 340.3 m/s (661 kn) at sea level, about 328 m/s at 3,000 m, 316 m/s at 6,000 m, and 295.1 m/s (574 kn) at the tropopause, set at 11,000 m (36,089 ft). Above that, in the lower stratosphere, the standard temperature is fixed at −56.5 °C up to 20 km, so the speed of sound is constant at 295 m/s. Higher still, ozone absorption warms the stratosphere and sound speed rises again, reaching about 330 m/s near 47 km before falling once more in the mesosphere. The real atmosphere departs from these numbers by day, season and latitude, and cold winter air over Siberia or hot air over the Arabian Gulf shifts the values by several percent.

Mach number is true airspeed divided by the local speed of sound, so the same true airspeed corresponds to a higher Mach number at altitude. An airliner cruising at Mach 0.82 at 11,000 m is doing about 242 m/s (470 kn) of true airspeed; at sea level the same Mach number would be 279 m/s (542 kn). This is why climb profiles are flown at a constant indicated airspeed until a crossover altitude and then at constant Mach: the critical Mach number of the wing, beyond which shock waves and drag rise, is reached at a lower true airspeed the higher the aircraft climbs. The same effect narrows the margin between low-speed stall and high-speed buffet at very high altitude, the region pilots call the coffin corner.

Rockets pass through all of these layers in minutes. The point of maximum aerodynamic pressure, max-Q, occurs low in the atmosphere at roughly Mach 1 to 1.5, and launch vehicles throttle down to limit loads there; higher up, the falling density matters far more than the small changes in sound speed.