Parameters
f₀ = (c / 2π) √(A / (V Leff))
Leff = L + 1.7r (assumed total end correction; termination-dependent)
c = 331.3 √(T/273.15) m/s
H(f) = 1 / √((1−(f/f₀)²)² + (2ζf/f₀)²)
Q = 1/(2ζ) Δf ≈ f₀/Q (light damping)
fpeak = f₀√(1−2ζ²) for ζ<1/√2
pipe, open–open: fn = nc/2L closed–open: fn = (2n−1)c/4L
Live readout
What to watch for
Volume and neck pull opposite ways
f₀ goes as 1/√V but as √A, so a bigger body drops the note while a wider neck raises it. Halving the cavity volume raises frequency by √2, about six semitones (an equal-tempered tritone), not a perfect fifth.
The end correction is not a detail
On the neck tab, compare the geometric and effective lengths. With this page's assumed total correction of 1.7r, a 3 cm-diameter neck adds 2.55 cm to the effective moving-air length. Real corrections vary with termination geometry.
One note, not a harmonic series
The pipe tab shows the difference plainly: the ideal lumped Helmholtz model has one dominant low-frequency resonance, while an ideal pipe has a fundamental and a ladder of standing-wave modes. Real bottles can also support higher acoustic modes.
Temperature moves the tuning
c goes as √T, so a resonator tuned in a cold workshop drifts sharp when the duct warms up. Over a 40 K swing the tuning moves about 7 % — enough to walk a narrow absorber off the noise it was installed to kill.
Narrow is the point, and the problem
On the absorber tab, raising damping lowers the peak but widens the band. A lightly damped, high-Q resonator is highly selective around its tuning. Practical absorption also depends on coupling and losses; broadband control generally needs broader damping or multiple treatments.
Scale it and it still works
The same equation covers a beer bottle at 100 Hz and a car exhaust chamber at 40 Hz. Only the ratio A/(V·Leff) matters, so tuning is a geometry problem rather than a size problem.
Sources and technical basis
Lumped-element Helmholtz theory with an explicitly assumed 1.7r total end correction, ideal pipe modes and an ideal-gas speed-of-sound approximation.