A column of air, ringing. Blow across the mouth of a tube and the air inside settles into a standing wave — and whether the far end is open or closed decides everything: an open pipe sounds all the harmonics fn = n·c/2L; a closed pipe sounds only the odd ones fn = (2n-1)·c/4L — and drops an octave lower for the same length. Down the center, the pipe goes in, the engine solves the boundary conditions, the spectrum comes out. The blue team builds it; the red team tries to break it.
source Standing waves in air columns — Bernoulli; Lord Rayleigh (J. W. Strutt), The Theory of Sound (1877). Cited, not stably canonical for this figure — archive.org/details/theorysound06raylgoog. Rendered, not quoted.
A standing wave is fixed by what the ends demand. An open end is a pressure release — a displacement antinode. A closed end is rigid — a displacement node.
OPEN–OPEN antinodes at both ends ⇒ L = n·λ/2 ⇒ fn = n·c/2L, the full series n = 1,2,3,… CLOSED–OPEN node at the stop, antinode at the mouth ⇒ L = (2n-1)·λ/4 ⇒ fn = (2n-1)·c/4L, only odd multiples of its fundamental.
Integer harmonics of the current pipe's own fundamental — present, or silent (the even ones a closed pipe cannot sound):
| k | present? | freq (Hz) |
|---|
The pipe is the-speed-of-sound trapped in a tube: every frequency here is c divided by a length, so warm the air and the whole organ sharpens. The open all-harmonic set n·c/2L versus the closed odd-only (2n-1)·c/4L an octave lower is exactly which members of the-overtone-series the air will carry.
Drop the even harmonics and you have not just a lower note but a different timbre — the hollow, clarinet-like voice of a stopped pipe. Each sphere is the next one's premise.
The blue team's live check: recompute the open series (all integer harmonics), the closed series (odd only), and the octave ratio between them. If red tampers, this badge is where it shows.
Three numbers make a pipe: its length L, the air temperature T (which sets the speed of sound c = √(γRT/M) ≈ 343 m/s at 20 °C), and the end condition — is the far end open or stopped?
That end condition is the whole game: it decides which wavelengths fit, and so which harmonics live. Feed L, T and the ends into the panel below.
Change any control — c, the fundamental and every harmonic are computed from the boundary conditions on the spot, never looked up.
What the pipe produces, proven: an open pipe carries all integer harmonics fn = n·c/2L; a closed pipe carries only the odd ones fn = (2n-1)·c/4L; and a closed pipe of the same length sounds an octave lower (c/4L vs c/2L, a factor of 2) — which is why a stopped organ pipe voices an octave below its length, in that hollow timbre.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
More: viscothermal losses at the wall damp and detune; wide pipes admit cross-modes above a cutoff; a real flue pipe is driven at an edge-tone jet, not a free resonator, so its spectrum is shaped by the mouth as much as by L. The clean series is the skeleton, not the sound.
"A stopped pipe sounds an octave higher." Cut. Backwards — closed fundamental c/4L is an octave lower than the open c/2L of the same length.
"A closed pipe has no overtones — just the fundamental." Cut. It has a full overtone set — the odd harmonics 3f, 5f, 7f…; it is the even ones that are missing.
"The resonant length equals the tube length exactly." Kept, corrected. Close, but add the end correction ≈ 0.6r per open end — the engine renders the ideal L; the wall notes the offset.
The red team's move: give the closed pipe all harmonics n·c/4L (evens included). The odd-only series is gone, and the octave-down and hollow-timbre facts break with it. The blue team's witness (window 7) is watching.
Add the even harmonics to the closed pipe and its 2nd mode becomes ×2 (not ×3) — the witness recomputes, the odd-only check fails, and it turns red. Nothing is faked; the attack is real and it is caught.