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Shock Waves & the Sonic Boom

Source lecture(s): SC133 Lec 25

Intuition

A moving sound source pushes its wavefronts ahead of it — until it moves as fast as the sound itself. Then the fronts can't escape: they pile up into a single, knife-edged wall of pressure. Faster still, and the wall folds back into a cone trailing the source. Crossing that cone is the sonic boom — all the sound of the passage, delivered at once.

The Mach cone

For source speed \(v_s\) greater than sound speed \(v\), the piled-up wavefronts form a cone of half-angle \(\theta\):

\[\boxed{\,\sin\theta = \frac{v}{v_s} = \frac{1}{M}\,} \qquad M \equiv \frac{v_s}{v}\ \text{(Mach number)}\]
  • \(M < 1\) (subsonic): fronts bunched ahead (Doppler), no shock.
  • \(M = 1\): fronts coincide — the "sound barrier," a wall of pressure at the nose.
  • \(M > 1\): the cone; larger \(M\), narrower cone.

The boom isn't a one-time bang at Mach 1 — the cone drags along the whole supersonic flight, sweeping a "boom carpet" tens of kilometres wide.

Everyday supersonics

  • Bullwhip crack: the tapering whip accelerates its tip past ~340 m/s — the crack is a miniature sonic boom (humanity's first supersonic machine).
  • Thunder: lightning heats a channel so fast the expansion is a shock that decays into the rumble.
  • Boat wakes: the V-wake is the same geometry with water-surface waves.
  • Cherenkov radiation: a particle exceeding light's speed in a medium makes the optical analogue — the blue glow of reactor pools.

Inside the shock

Across the thin front, pressure, density, and temperature jump almost discontinuously — ordinary sound is a gentle (~Pa) wiggle, a shock is a finite (~kPa–MPa) step that heats the gas irreversibly. The jump obeys conservation laws (mass, momentum, energy) written across the front — the Rankine–Hugoniot conditions, treated fully in PC316's shock chapter, and the reason re-entry capsules need heat shields (the shock, not friction, does the heating).

Worked example: how high is the jet?

You see a fighter pass directly overhead; the boom arrives 6 s later. The jet flies at Mach 1.5 (\(v = 340\) m/s). Altitude?

Cone half-angle: \(\sin\theta = 1/M = 1/1.5 \Rightarrow \theta \approx 41.8°\).

You hear the boom at the instant the cone surface sweeps over you. By then the jet has moved past overhead by \(d = v_s t = 1.5(340)(6) = 3060\,\text{m}\), and the cone geometry relates your position to the jet: \(\tan\theta = h/d\), so

\[h = d\tan\theta = 3060\times\tan 41.8° \approx 2.7\,\text{km}\]

Common mistakes

  • "The boom happens only when breaking the barrier." The cone persists throughout supersonic flight; everyone under the carpet hears it, sequentially.
  • Thinking the pilot hears the boom. The cone trails behind; the aircraft outruns its own noise.
  • Confusing shock heating with friction. Compression across the shock does the heating — that's why blunt re-entry shapes (stronger, detached shock) protect better than needles.
  • Using small-signal sound formulas across a shock — shocks are nonlinear; ordinary acoustics breaks down (the full story).

Knowledge graph position

Prerequisites: Sound waves. Leads to: compressible flow (PC316), aerodynamics, astrophysical shocks.

Quiz

Q1 (computational). A jet at Mach 2: what is the Mach-cone half-angle?

Answer

\(\sin\theta = 1/2 \Rightarrow \theta = 30°\) — double the speed of sound, a 30° cone.

Q2 (conceptual). Why does a bullwhip crack while no part of your arm approaches the speed of sound?

Answer

Momentum funnels into ever-less mass: as the wave runs down the tapering whip, conservation drives the thin tip to enormous speed — past Mach 1 — with only a modest hand motion at the heavy end.

Q3 (multiple choice). As a supersonic jet flies faster, its Mach cone becomes: (a) wider (b) narrower (c) unchanged

Answer

(b). \(\sin\theta = 1/M\) — at very high Mach the boom concentrates into a slender, intense cone.