We finished a bottle of wine this evening, and I blew across the top as I often do. (Don’t worry: I only do this at home. If we’re ever in a restaurant together, I won’t embarrass you by blowing across the neck of an empty bottle.)

The pitch sounded lower than I expected, so I revisited some calculations I did last year.

As I wrote about here, a wine bottle is approximately a Helmhotz resonator. The geometric approximation is not very good, but the pitch prediction usually is. An ideal Helmholtz resonator is a cylinder attached to a sphere, and a typical wine bottle is more like a cylinder attached to a larger cylinder. But the formula predicting pitch is robust to departures from ideal assumptions.

As noted before, the formula for the fundamental frequency of a Helmholtz resonator is

where the variables are as follows:

*f*, frequency in Hz*v*, velocity of sound*A*, area of the opening*L*, length of the neck*V*, volume

The opening diameter was 2 cm, the neck length 9 cm, and the volume 750 cm³. All these are typical. The predicted frequency is *f* = 118 Hz. The measured frequency was 106 Hz, measured by the Sonic Tools phone app.

The actual frequency was about 10% lower than predicted. This is about a whole step lower in musical terms. I could certainly hear an interval that large if I heard the two pitches sequentially. But I don’t have perfect pitch, and so I’m skeptical whether I could actually notice a pitch difference of that size from memory.

German Wikipedia

https://de.wikipedia.org/wiki/Helmholtz-Resonator

gives a resonance frequency formula with end correction

(End-Korrektur der Resonatorlänge)

which results in a lower frequency:

L is replaced by (L + diameter*pi/4) i.e.

9 is replaced by (9+2*pi/4) = 10.57 and sqrt(9/10.57) = 0.92

which is quite close to the observed 106/118 = 0.90

Typo: 2nd para, 3rd sentence: An idea Helmholtz … -> An ideal … [missing l]