Thought Toys · Waves & rhythm · Exhibit 64

Outrun your own sound

A source ticks off a sound wave at steady intervals while it moves. Below the speed of sound that's an ordinary pitch shift — higher ahead of it, lower behind. Cross that speed and something stranger happens: the source starts arriving places before its own sound does.

Wavefronts from a moving source, snapshotted mid-flight source now listener Mach cone

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Try it
↳ your turn — push M past 1 and watch the wavefronts fall behind into a trailing cone

What you're seeing

Every circle here is a single pulse of sound, frozen at the moment of this snapshot — big circles were emitted a while ago and have had longer to expand outward; small ones near the source were just emitted. All of them grow at exactly the same speed, the speed of sound itself. What changes with M is only where each circle was centered when it was born, because the source itself was moving. Below Mach 1 that's enough to bunch the circles up tightly on the side the source is heading toward — a listener standing there hears the pulses arrive close together, which is a higher pitch — while circles pile up loosely on the side it's leaving, a lower pitch.

Push M up toward 1 and the bunching gets more and more extreme, right up until the source is moving exactly as fast as its own sound. At that exact instant, every circle it has ever emitted reaches the listener simultaneously — not "a very high pitch," but a literal pile-up, every wavefront arriving at once. Cross Mach 1 and the source is now faster than the sound it's making: it keeps getting ahead of its own circles, which can never catch up. Their outer edges share one common trailing line — the Mach cone — and nothing at all reaches a point ahead of that cone. A listener out there hears nothing, then the cone sweeps past in a single instant, then the falling pitch of a source pulling away.

The rule, exactly. With the speed of sound normalized to 1 (so M is both the Mach number and the raw speed), a source ticking off a pulse every Δt seconds spreads its arrivals unevenly: ahead of it, gaps between arrivals shrink to Δt(1−M) — a perceived frequency f ⁄ (1−M) — while behind it, gaps stretch to Δt(1+M), a frequency f ⁄ (1+M). Past M=1, every past wavefront shares a common tangent line trailing the source at half-angle θ = asin(1 ⁄ M). Verified in node (improve/verify/64-doppler.js): across four subsonic Mach numbers, every approach-phase and recede-phase arrival gap in a 60-pulse simulation matches those closed forms to within 1e-9, staying strictly ordered throughout; at exactly M=1 every approach-phase arrival lands within 1e-9 of the identical instant, while the same setup at M=0.99 still shows a clearly nonzero spread — a genuine single-point pile-up, not just "very compressed." Past M=1, raw point-to-line geometry (not the simplified angle formula) confirms every historical wavefront is truly tangent to the cone line to within 1e-9 across three Mach numbers, and the cone narrows monotonically as M climbs from 1 to 10. Negative control: a deliberately wrong angle fails that same tangency test outright, and below Mach 1 the cone construction — sin θ = 1/M — has no real solution at all: a NaN at every Mach number tested from 0.1 to 0.999, confirming the cone is a strictly supersonic structure, not a smooth fade-in.

Also in Waves & rhythm: Move, and your clock falls behind →

All 7 in Waves & rhythm
  1. 12Fourier epicycles
  2. 19Standing waves & normal modes
  3. 21Lissajous figures
  4. 62Ask which slit, lose the wave
  5. 64Outrun your own sound — you are here
  6. 66Move, and your clock falls behind
  7. 70The angle past which light refuses to leave

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