1. The Air in the Neck Is a Weight and the Air in the Bottle Is a Spring
When you blow across the opening, a small plug of air in the neck gets pushed in and out. As it moves in, it squeezes the air inside the bottle. Squeezed air pushes back, like a compressed spring, and shoves the plug out again. The plug overshoots, the air inside is stretched thin, and it pulls the plug back in.[2]
A weight on a spring has a natural rate of bouncing, set by the spring and the weight. The air in a bottle's neck and body has a natural rate too, and that is why the bottle has "its" note. The same system is called a Helmholtz resonator, and it needs just three parts: an enclosed volume of air, a narrow neck, and the plug of air that slides in the neck.[2]
The picture also tells us what to expect when the bottle changes. Let us turn it into a rule you can test.
2. A Bigger Bottle or a Longer Neck Gives a Lower Note, and a Wider Opening Gives a Higher One
For a simple resonator, the pitch (frequency) can be estimated with f = (c / 2π) × √(A / (L × V)). Here c is the speed of sound, A is the area of the opening, L is the length of the neck, and V is the volume of the bottle.[3]
Read the formula like a recipe. A bigger volume V or a longer neck L makes the note lower. A wider opening A makes it higher, which surprises many people who expect a wide mouth to sound deep.
How good is the formula for a real bottle? The engineer John D. Cook tried it on a wine bottle with an opening about 2 cm across, a neck about 9 cm long, and a volume of 750 cm³. The formula predicted about 118 Hz. A phone app measured about 106 Hz, roughly 10% lower. A bottle is not a neat sphere with a neat tube, so the geometry is only a rough fit. This one example does not tell us how accurate the formula is for bottles in general.[3] A common explanation for the gap is that air just outside the neck also moves, which makes the neck act a little longer than it looks. That explanation is a widely used correction, but we did not confirm its exact numbers in the sources we could read.
So a hollow shape with a neck can single out one pitch. Why would a scientist want that?
3. Helmholtz Used the Shape to Pull One Pitch Out of a Messy Sound
Helmholtz wanted to hear the separate pitches inside a sound. In his 1862 book On the Sensations of Tone, Helmholtz described a device that could pick one frequency out of a sound.[1]
The device was a hollow container with a small neck and a hole on one side, plus a narrow tube-shaped tip on the other side that fits into the ear. If the sound around you contains the resonator's own pitch, the air inside responds strongly and that one pitch rings clearly in your ear.[1] Helmholtz wrote that the resonator's own tone can sometimes be heard even in the whistling of the wind, the rattling of carriage wheels, and the splashing of water.[1]
The idea was simple and clever: instead of trying to hear every pitch at once, listen through a tool that favors only one. But a single fixed bottle only catches one pitch. How could he catch others?
4. A Paris Instrument Maker Built Metal Resonators You Could Tune
Kenyon College's collection notes say that, until about 1860, Helmholtz used whatever glass cylinders and tubes were at hand. Around that year he asked Rudolph Koenig, an instrument maker, to build metal resonators to exact measurements for more careful work. Koenig made both cylinder-shaped and ball-shaped ones.[2]
In the cylinder type, tubes slide in and out to change the volume of air inside, and the note each one favors is engraved on its side.[2] That fits the formula: a changeable V means a changeable pitch.
The formula gives a useful estimate. If the air space is cut in half, the pitch rises by a factor of √2, about 1.41, the same ratio as six piano keys (a tritone). Shrink the air space by just a tenth and the pitch rises by about 5%.
Resonators like these were not only lab curiosities. They were sold to universities: the Kenyon notes mention sets at Toronto and Vanderbilt.[2] Now to the question in the title's second half: where else does this shape turn up?
5. The Same Shape Boosts Bass in Speakers and Quiets Jet Engines
Many loudspeaker boxes have a tube-shaped hole called a port. In this design, called "bass reflex," the air in the port acts like a weight and the air in the box acts like a spring, and together they form a resonant system that helps the speaker's low-frequency (bass) output.[5] Same kind of parts as the bottle.
Jet engines use the shape to remove sound instead. A research paper describes "acoustic liners" for aircraft engines that have a honeycomb core of tiny cells and work as Helmholtz resonators to absorb noise.[4] The paper notes that such liners often absorb sound only in a narrow range of pitches, which is why engineers keep looking for ways to widen it.[4]
So the same neck-and-cavity shape is used to add sound in one place and take it away in another. The common thread is that it responds strongly to one pitch.
6. You Can Test the "Smaller Volume Means Higher Note" Rule With Two Bottles
Take two empty bottles of different sizes. Rest your lower lip on the rim and blow gently across the top, like playing a flute. Which one sounds lower? The formula says the larger one.
Next, pour a little water into one bottle and blow again. Water takes up room, so the air volume V shrinks and the note should rise. Try adding a bit more each time and listen for the steps. Blow gently across the opening; do not put your mouth over it or suck air out of the bottle. If you use a glass bottle, ask an adult to help, since glass can break.
If you want to read the originals, the Case Western Reserve University and Kenyon College pages listed below both have photos and short descriptions of Koenig's resonators.