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How high do supersonic planes fly?
Short answer
Supersonic aircraft cruise far higher than airliners. Concorde flew at 15,000 to 18,300 m (50,000 to 60,000 ft) and the SR-71 operated above 24,000 m (80,000 ft). Thin air at those altitudes keeps drag and structural loads manageable at Mach 2 and softens the sonic boom heard on the ground. Fighters go supersonic lower, typically 10 to 15 km.

Concorde and the civil ceiling
Concorde cruised between about 15,000 and 18,300 m (50,000 to 60,000 ft), typically climbing slowly as it burned fuel, with 60,000 ft as the certified maximum. That is 5 to 7 km above the busiest airline flight levels, so it needed no separation from subsonic traffic and could fly a direct track across the Atlantic. Its windows were small and its cabin pressurisation more demanding than an ordinary jet's, because at 18 km the outside pressure is under a tenth of sea level and a rapid decompression would leave little time of useful consciousness.
Why thin air suits Mach 2
Drag and structural loads depend on dynamic pressure, half the air density times speed squared. At Mach 2 near sea level dynamic pressure would be enormous, far beyond what any practical airframe can take, and skin heating would be extreme. Climbing to 17 km reduces air density to about a tenth of sea level, so the aircraft can fly twice the speed of an airliner while loading the structure no more than a jet at low altitude. Engines also benefit: cold air improves intake compression, and the low density keeps the airflow through the engine within limits. There is a floor as well as a ceiling, since a wing needs enough dynamic pressure to lift the aircraft, which is why Concorde accelerated through Mach 1 only above about 9 km.
Altitude also softens the sonic boom. The pressure signature attenuates as it spreads through many kilometres of atmosphere, and from 17 km the boom reaching the ground is around 2 pounds per square foot, loud but tolerable over the ocean.
Military aircraft
Reconnaissance aircraft went higher. The Lockheed SR-71 operated above 24,000 m (80,000 ft) at Mach 3.2, where the sky is dark blue and the crew wore full pressure suits. Fighters are different: they can exceed Mach 1 at almost any altitude above a few thousand metres but typically do so between 10 and 15 km, where they still have manoeuvring margin. At low altitude supersonic flight is possible for short dashes, as the Tornado and F-111 were designed to do, but fuel consumption and airframe loads rise so quickly that it is never sustained.
Future designs
Proposed supersonic airliners follow the same logic. Boom's Overture is planned to cruise around 18 km at Mach 1.7, and NASA's X-59 demonstrator flies at about 16.8 km at Mach 1.4, chosen to keep the boom faint on the ground. Higher cruise altitudes bring more scrutiny of emissions in the lower stratosphere, which is one reason environmental studies focus on this band.
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