THE MODEL
A Helmholtz resonator is a cavity of volume V connected to the outside by a short neck of area A and effective length L. The neck air is a lumped mass m = ρ·A·L; the cavity air is a spring of stiffness k = ρc²A²/V.
Since ω = √(k/m), the A, ρ, L pieces collapse and out drops the singing frequency. It depends on 1/√V — not on any length being a wavelength fraction.
THE LINEAGE
Neighbour: the-organ-pipe. A pipe sings because a wave fits its length (fn = n·c/2L, all harmonics; closed = odd only). The bottle is the opposite kind of resonator: no standing wave, one lumped note, f ∝ 1/√V not 1/L. Same air, same c — different physics. Helmholtz drew the line in 1863.
THE WITNESS
Live re-check of the load-bearing law: a bigger cavity must sing lower (1/√V). Re-runs on every tamper.
DATA IN in ↓
c = 343 m/s (air, 20 °C)
THE PANEL lit
DATA OUT out ↓
The bottle sings . Double the cavity → pitch drops by √2 (a tritone-ish fall), not by an octave. Halve the neck area → pitch drops by √2. Proven live from f = (c/2π)√(A/VL).
THE ADVERSARY wall
"It's just a short pipe — treat the bottle as a tube of length L and use f = c/4L." Wall: that ignores the cavity entirely. The restoring spring is the compressed cavity air, so f depends on V. A pipe formula predicts the same note for a thimble and a carboy with equal necks — the real bottles differ by hundreds of Hz.
THE GRAVEYARD
"Bigger bottle → higher note."
Correction: f ∝ 1/√V — a larger cavity is a softer spring,
so it sings lower. That is why a near-empty bottle is deep and fills up bright.
"The neck length doesn't matter."
Correction: f ∝ 1/√L. And the real acoustic length is L + an
amber end correction (≈0.6–0.85·r per open
end), which lengthens Leff and lowers f slightly.
THE TAMPER
Planted void: swap the mass–spring ratio A/(V·L) → A·V/L, making f ∝ √V — a bigger bottle would sing higher. The Witness (7) catches it live.