The Physics
Opus in profectus

Shock Waves

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When the speed of a source equals the speed of sound (v = c) the wave fronts cannot escape the source. The resulting pile of waves forms a large amplitude "sound barrier" that makes sustained flight at this speed difficult and risky.

Cartoon animation of sound waves from a source moving near the speed of soundAnimate

The term "sound barrier" or "sonic barrier" first came into use during World War Two. Fighter pilots engaged in high speed dives noticed several irregularities as flying speeds approached the speed of sound: aerodynamic drag increased markedly, much more than normally associated with increased speed, while lift and maneuverability decreased in a similarly unusual manner. Pilots at the time mistakenly thought that these effects meant that supersonic flight was impossible; that somehow airplanes would never travel faster than the speed of sound. They were wrong.

Cartoon animation of sound waves from a source traveling faster than the speed of soundAnimate

When the speed of a source exceeds the speed of sound (v > c) the wave fronts lag behind the source in a cone-shaped region with the source at the vertex. The edge of the cone forms a supersonic wave front with an unusually large amplitude called a "shock wave". When a shock wave reaches an observer a "sonic boom" is heard.

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Unlike ordinary sound waves, the speed of a shock wave varies with its amplitude. The speed of a shock wave is always greater than the speed of sound in the fluid and decreases as the amplitude of the wave decreases. When the shock wave speed equals the normal speed, the shock wave dies and is reduced to an ordinary sound wave.

The ratio of the speed of a moving object (v) to the speed of sound (c) in a fluid is known as the Mach number (Ma) in honor of Ernst Mach (1838–1916), the Moravian physicist, psychologist, and philosopher who studied sound and ballistics.

Ma =  v

The Mach number is a dimensionless measure of speed common in aerodynamics. Mach 0.5 is half the speed of sound, Mach 2 is twice the speed of sound, and so on. Speeds less than the speed of sound have a Mach number between zero and one and are described as subsonic. Those greater than the speed of sound have Mach numbers greater than one are a described as supersonic. Speeds approximately equal to the speed of sound have Mach numbers approximately equal to one and are described as transonic.

The shock wave from a supersonic object is a cone composed of overlapping spherical wavefronts. As any one of these wavefronts forms, it propagates radially outward at speed c and acquires a radius ct. At the same time the source, traveling at speed v moves forward vt. These two displacements form the leg and hypotenuse, respectively, of a right triangle and can be used to determine the Mach angle μ at the vertex of the shock cone.

sin μ =  c

When an object travels slower than sound, the ratio in this equation is greater than one, and the equation does not have a real solution. This makes absolute sense as there is no shock wave to speak of at subsonic speeds. Traveling at the speed of sound makes the ratio equal one and results in a Mach angle of ninety degrees. At transonic speeds the shock wave is a wall of high pressure moving with the object, perpendicular to its velocity. Above the speed of sound, the ratio is less than one and the Mach angle is less than ninety degrees. The faster the object moves, the narrower the cone of high pressure behind it becomes. Measuring the vertex angle is thus a simple way to determine the speed of a supersonic object.

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