17.16:
Shock Waves
Consider a sound source, for instance an airplane, moving towards a stationary observer with a speed vs. If the speed of sound in air is v, the Mach number is defined as the ratio of the two speeds.
If vs is greater than v, the Mach number becomes greater than unity, and the speed is called supersonic. Then, according to the Doppler formula, the sound's observed frequency is negative, which is practically impossible.
Instead, the observer perceives no sound until the source has passed them.
The sound waves interfere constructively along the lines of a three-dimensional cone. Within the cone, they interfere destructively. Thus, the cone's edge is a sharp divider of regions of no sound and destructive interference. The resultant pattern is called a shock wave.
Applying geometric concepts, the sine of the cone angle equals the inverse of the Mach number.
The sudden arrival of the shock wave is known as a sonic boom. Because of the high pressure, sonic booms created by moving airplanes can shatter windows and rattle our nerves.
17.16:
Shock Waves
While deriving the Doppler formula for the observed frequency of a sound wave, it is assumed that the speed of sound in the medium is greater than the source's speed through it. When this condition is breached, a shock wave occurs.
When the source's speed approaches the speed of sound, constructive interference between successive wavefronts emitted by the source occurs immediately behind it. Initially, scientists believed that this constructive interference would result in such high pressures that an airplane, for instance, would be unable to breach the speed of sound and would eventually get destroyed. However, airplanes can now fly at speeds greater than the speed of sound, also called supersonic speeds. They create a familiar conical structure of waves behind them.
The phenomenon of shock waves is three-dimensional, and gets its name from the sharp difference of pressure along the edges of the cone. This cone moves along with the sound's source, and is narrower if the Mach number (the speed of the source divided by the speed of sound) is higher. An observer can only experience the shock wave only after the source has passed them. When they do, they experience a sudden change of pressure, known as a sonic boom.
Shock waves are a specific case of a general wave phenomenon called bow wakes. The same principle of shock waves applies to other waves, such as surface water waves. When a duck or a steamer moves over the water's surface at high speed, it leaves behind a bow wake over the water's surface.
This text is adapted from Openstax, University Physics Volume 1, Section 17.8: Shock Waves.