9.3 Supersonic motion and the Mach cone

For a source moving at velocity $v_s$ through still air, the wavefronts it emits are centered on the positions where it was when each wavefront was emitted, expanding at speed $c$. The geometry depends on whether $v_s < c$, $v_s = c$, or $v_s > c$.

Subsonic ($v_s < c$): asymmetric, but unbroken

Wavefronts emitted at different times are concentric in their own centers but the centers are at different positions. The Doppler shift (lesson 9.1) is the kinematic consequence: wavefronts ahead are compressed, wavefronts behind are stretched.

Transonic ($v_s = c$): wavefronts pile up at the source

All the wavefronts emitted by the source pile up exactly at the source’s current position — a kind of acoustic singularity. In real fluids this is mitigated by nonlinear effects (chapter 10) and by viscosity, but in the idealised linear theory it is a degenerate case.

Supersonic ($v_s > c$): the Mach cone

When the source outruns its own sound, the wavefronts it emits all lie behind the source, inside a cone with half-angle

where $M = v_s / c$ is the Mach number. The cone opens behind the source; outside the cone there is no acoustic disturbance at all.

The locus of all the wavefronts is the cone itself: the Mach cone. It is an envelope of wavefronts, all arriving at a given downstream point simultaneously. This is a strong wave — much stronger than any individual wavefront would be, because many wavefronts have piled up. In real fluids this superposed wave is a shock wave (the linear theory’s approximation to a true discontinuity in pressure and density).

What this sounds like

A supersonic aircraft is silent ahead of itself: no sound has reached the ground because the aircraft outruns it. Then the Mach cone passes overhead — a single sharp pressure pulse, the sonic boom. After the boom, the engine and aerodynamic noise of the aircraft arrives in normal subsonic fashion, but the boom itself is the signature of the supersonic motion.

The intensity of a sonic boom is set by the aircraft’s size and Mach number; it is typically 1–5 psf (pounds per square foot, $\sim$ 50–250 Pa) at ground level for a fighter jet at $M = 2$. This is loud enough to break windows and is the reason supersonic flight is banned over populated land in most countries.

Cherenkov radiation — the same phenomenon for light

In a dispersive medium, light travels slower than $c_\text{vacuum}$. A charged particle moving through such a medium faster than the speed of light in the medium produces a Cherenkov cone — the optical analogue of a Mach cone. It glows blue. The Cherenkov detector at Super-Kamiokande uses exactly this principle to detect high-energy neutrinos.

The same mathematics — same equation form — appears in many fields. A boat moving faster than the wave speed on water produces a Kelvin wake. A meteor moving faster than ionospheric Alfvén speed produces a magnetosonic shock. The conical envelope is a universal feature of supersonic point sources.

The supersonic regime in this book

For the rest of the book we will mostly stay subsonic. The supersonic regime is well-studied in aerodynamics and gas dynamics; we touch on it here because it provides a clean illustration of what happens when a source outruns its own waves. The phenomenon recurs in the next-to-last lesson when we discuss shocks more carefully.

Next lesson: when more than one of source, observer, and medium are moving, the Doppler formulas need careful bookkeeping.