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The Mach number is the ratio of a sound source's speed to the speed of sound in the medium. When the Mach number exceeds one, the source travels faster than sound, creating supersonic speeds. At these speeds, sound waves interfere constructively along a three-dimensional cone, forming a shock wave. The cone's angle depends on the Mach number: higher Mach numbers produce narrower cones.

The Doppler formula assumes the speed of sound exceeds the source's speed. When this condition breaks, the formula predicts a negative observed frequency, which is physically impossible. Instead of hearing sound, an observer perceives nothing until the source passes. This breakdown occurs because sound waves cannot propagate ahead of a supersonic source, creating a shock wave phenomenon.

Sound waves from a supersonic source interfere constructively along the cone's edges and destructively within the cone's interior. This creates a sharp boundary between regions of no sound outside the cone and destructive interference inside. The cone's edge acts as a divider, producing the characteristic shock wave pattern that moves with the source.

A sonic boom is the sudden arrival of a shock wave when it reaches an observer after the source has passed. The shock wave carries extremely high pressure differences across its sharp edge. This sudden pressure change can shatter windows and cause physical disturbance. The intensity and pressure of sound waves at shock wave boundaries far exceed normal sound levels.

The sine of the shock wave cone's angle equals the inverse of the Mach number. As the Mach number increases, the cone becomes narrower because the sine of the angle decreases. This geometric relationship means supersonic aircraft traveling at higher speeds produce more acute cone angles, concentrating the shock wave's effects more sharply behind the aircraft.

Shock waves are a specific case of a general wave phenomenon called bow wakes. The same principle applies to other wave types, such as surface water waves. When a duck or steamer moves over water at high speed, it creates a bow wake. This demonstrates that shock wave formation is not unique to sound but occurs whenever any wave source exceeds the wave's propagation speed in its medium.

Early scientists believed that constructive interference between successive wavefronts behind a supersonic source would create such extreme pressures that an aircraft would be destroyed before breaching the sound barrier. However, aircraft now routinely fly at supersonic speeds, disproving this theory. Modern engineering successfully manages the intense pressure conditions created by shock waves during supersonic flight.