Definition: The ratio between the speed of an object in a medium to the speed of sound in the same medium.
Significance: The speed of sound is the speed at which the weakest disturbances in pressure propagate through a medium. In a gas, the speed of sound is close to the average speed of random thermal motion of molecules, because disturbances propagate by the collisions between molecules. Thus, the Mach number is also viewed as the ratio between the speed of organized motion of the gas, or a body relative to the gas, and the average speed of random thermal motion in the gas.
Calculation
The Mach number is named in honor of Ernst Mach, a nineteenth century Austrian scientist who conducted experimental research on the aerodynamics of artillery shells. Speed ranges in the field of aerodynamics are classified according to Mach number. A Mach number of 1 corresponds to motion at the speed of sound. The speed of sound thus depends on the temperature of the gas. The speed of sound in air at a temperature of 0 degrees Celsius (273.15 Kelvins) is 331 meters per second. To find the speed of sound in meters per second at other air temperatures in the range 200 to 700 Kelvins (from minus 73.15 to 426.85 degress Celsius), divide 5,471 (obtained as product of 331 and the square root of 273.15) by the square root of the air temperature in Kelvins. The Mach number is then the ratio of an object’s speed to this speed of sound.
Mach Ranges
The low-speed regime is generally identified with Mach numbers of less than0.3. In this range, the maximum change in density that can occur when the flow is stopped is less than about 5 percent of the actual density. Hence, such flows are also described as being incompressible. Small propeller-driven airplanes and helicopters fly in this speed range, though their propeller or rotor tips move quickly enough relative to the air to encounter Mach numbers close to 1.
The subsonic range is generally the range from Mach 0.3 to 1.0. In this range, the changes in density associated with changes in Mach number become significant. The lift coefficient and drag coefficient rise more steeply with increased angle of attack as Mach number increases. According to the Prandtl-Glauert rule, the lift coefficient associated with a given angle of attack scales with the number resulting from 1 divided by the square root of 1 minus the square of the Mach number. The drag coefficient also increases with Mach number and with thickness. Thus, aircraft flying in this range use thinner airfoils and smaller angles of attack. Turboprop aircraft reach speeds well into the subsonic regime. Much of the close air combat between fighter planes that involves sharp maneuvers occurs in this speed range.
The range that includes Mach 1 and values slightly greater and less than Mach 1 is called the transonic regime, roughly taken as Mach 0.8 to 1.2. Transonic flows include both supersonic and subsonic regions. Most modern airliners cruise in the transonic regime. Although the actual flight Mach number is below 1.0, there is some supersonic flow over the wings. The critical Mach number is the lowest flight Mach number where sonic conditions, in which the Mach number equals 1, are first encountered on the airfoil.
When the flow speed, or the speed of an object relative to the medium, is clearly greater than the speed of sound, the speed is said to be supersonic. Bullets, artillery shells, surface-to-air missiles, and air-to-air missiles all typically operate in the supersonic regime, with Mach numbers up to about 3.5. Most modern fighter aircraft can fly at supersonic speeds for short durations, with their engines operating on afterburners. The Lockheed Martin F-22 is capable of cruising at supersonic speeds without afterburners. Large aircraft capable of cruising at supersonic speeds are the North American B-70 Valkyrie bomber, the British Aerospace/Aerospatiale Concorde, the Tupolev Tu-144, and the Tupolev Tu-44 Backfire Bomber.
The shock waves caused by supersonic flight can be seen emanating from a model of the X-15 as it flies through a supersonic pressure tunnel. (NASA)
Speeds greater than five times the speed of sound are described as hypersonic speeds. Spacecraft and missile warheads reentering Earth’s atmosphere fly at hypersonic speeds, with Mach numbers as high as 36 for the Apollo capsule and about 25 for the space shuttle.
The importance of Mach number to flight can be seen from the Mach cone. An object moving at a Mach number of 2 through air generates pressure disturbances that propagate in all directions at the speed of sound in the medium of air. If the speed of sound were 300 meters per second, when the object reached a given point, the disturbances it had generated a second before would have spread within a sphere whose radius was only 300 meters, yet the object itself would have traveled 600 meters in the same second. Disturbances generated 0.1 seconds before would have reached a radius of only 30 meters, yet the object would have traveled 60 meters. The disturbances generated at each intermediate point propagate out to smaller and smaller distances, all lying within the Mach cone.
The Mach cone has its apex at the location of the object, and its axis is the path taken by the object to reach that point. The angle made by the axis of the Mach cone surface is called the Mach angle. This is the inclination of the weakest wave front generated by an object moving at supersonic speeds through a medium. Such a weak wave is called a Mach wave.
The region ahead of the Mach cone is called the zone of silence. In this region, the sound from the approaching object cannot have reached at the instant in question. The Mach angle is thus given by the inverse sine of the reciprocal of Mach number. For example, if the Mach number is 2, the Mach angle is 30 degrees, whereas for a Mach number of 3, the Mach angle is 19.47 degrees.
Most disturbances created by moving objects involve large pressure differences. These disturbances raise the temperature of the air, and hence move faster, accumulating along a shock front. Shocks formed by blunt objects can reach a shock angle of 90 degrees: Such shocks are called normal shocks, and they result in subsonic flow on the downwind side of the shock. As the flight Mach number of a wing exceeds the critical Mach number, the drag caused by the occurrence of shocks rises sharply. Early theoretical analyses predicted that the drag would rise to extremely high values, preventing an aircraft from accelerating through the speed of sound. This was called the sound barrier, the existence of which was conclusively disproved when Air Force Captain Charles E. “Chuck” Yeager flew the Bell X-1 rocket plane faster than the speed of sound in 1947 and became the first person to fly faster than Mach 1.
