Overview#

• When a source moves faster than sound (\(v_S > v\)), the Doppler formulas (section 17-7) no longer apply—they predict infinite or undefined frequencies when \(v_S \geq v\).

• Wavefronts bunch into a cone called the Mach cone. A shock wave travels along this surface—a sharp pressure change.

• The sonic boom is the audible burst when the shock wave passes an observer.

Why the Doppler formula breaks down#

For \(v_S = v\) (source at the speed of sound), the denominator \(v - v_S\) in the Doppler formula becomes zero → infinite detected frequency. The source keeps pace with its own wavefronts; they pile up at the source.

For \(v_S > v\), the source outruns its wavefronts. Spherical wavefronts emitted at different times overlap and form an envelope—the Mach cone.

The Mach cone#

When \(v_S > v\), wavefronts emitted at successive positions bunch along a V-shaped envelope (2D) or cone (3D). The half-angle \(\theta\) of the Mach cone satisfies

Derivation: In time \(t\), a wavefront expands to radius \(vt\); the source moves \(v_S t\). The cone is tangent to the wavefront—the tangent makes angle \(\theta\) with the direction of motion, so \(\sin\theta = (vt)/(v_S t) = v/v_S\).

Mach number#

The Mach number is

• Mach 1: \(v_S = v\) (speed of sound).

• Mach 2: \(v_S = 2v\) (twice the speed of sound).

• In terms of \(M\): \(\sin\theta = 1/M\) → faster source → narrower cone.

Shock wave and sonic boom#

A shock wave exists along the surface of the Mach cone—air pressure rises sharply, then drops below normal, then returns. When this surface passes an observer, the result is a sonic boom: a loud burst of sound.

Summary#

• \(v_S \geq v\): Doppler formula breaks down; wavefronts bunch into a cone.

• \(\sin\theta = v/v_S = 1/M\)—Mach cone half-angle.

• Mach number \(M = v_S/v\).

• Shock wave along the cone; sonic boom when it passes an observer.