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.
