The Sound Inside A Bottle: The Shape Of Empty Space

Reading A Bottle From Within

The air-filled space opening to the outside through a bottle’s neck is called a cavity. Under suitable conditions, the air in the neck and this cavity can behave as a Helmholtz resonator. An audible response requires excitation; closing the cap also changes the conditions assumed for an open neck.

Body, shoulder and neck bound the space above the sauce. That inner space is small in a full bottle and grows as the contents diminish. We continue to recognise the same container while one variable of the acoustic model changes inside. The object followed by acoustics includes the air enclosed by the wall as well as the glass itself.

The Invisible Spring Of Air

UNSW explains this air through a mass–spring system. Air in and near the neck provides the moving mass; the compressibility of the body’s air provides the spring. Inward movement raises pressure, which pushes the air back; inertia carries it past equilibrium. Without an external energy supply, losses in the real system gradually damp that back-and-forth movement.

When the sound wavelength is much larger than the cavity, pressure throughout the cavity is assumed to vary approximately in phase: it rises and falls at nearly the same time at different points. This relationship between size and frequency supports treating the cavity’s air as a single spring.

How Geometry Sets The Pitch

In the ideal model, the size of the air space is only one influence on pitch. Sound speed, opening area and effective neck length also enter the frequency. Halving the air volume with other conditions fixed raises the frequency by about 1.41 times. Effective length includes more than a ruler measurement of the neck: nearby moving air at both ends also contributes.

This relationship can be written as:

$$f_{\mathrm{H}}=\frac{c}{2\pi }\times \sqrt{\frac{A}{V\times L_{\text{eff}}}}$$

Here c is the speed of sound in air, A is the opening’s cross-sectional area, V is the air volume in the cavity and L_eff is the effective neck length. Volume appears in the denominator inside the square root, explaining why a larger air space gives a lower frequency when the other conditions remain unchanged.

The geometry seen at a sauce bottle’s neck acquires a more detailed counterpart in the model. Using up contents changes the air volume; imagining a different opening may change area and effective length too. The relation lets us specify which quantity a predicted change in sound is being linked to.

Monteiro and colleagues investigated the relation in 2018 using an approximately cylindrical tea bottle. Water was added at different levels, reducing the air volume, and the frequency produced by blowing across the mouth was tracked with a smartphone. Water level was the experimental variable. Considering the same relation for a sauce bottle likewise requires an open neck and suitable dimensional and frequency conditions.

The Tone Of Air And The Ring Of Glass

Striking a bottle can excite elastic motion in its glass wall. The participation of liquid in the moving system then becomes important. The model built from neck air and cavity air does not by itself describe every vibration of a sounding glass object.

In 2006, Jundt and colleagues used holographic methods to examine vibrations in partly filled glasses. Adding liquid lowered the glass resonance frequencies in their experiment. This result and the rising frequency of an ideal Helmholtz model with decreasing air volume describe two different mechanisms: one examines glass and liquid moving together, the other the response of an air cavity.

In the wine glass, liquid adds to the effective vibrating mass. With the same stiffness, increased mass lowers frequency. In an air cavity, shrinking air volume instead raises the ideal model’s frequency. Predicting the direction of a sound change as a vessel fills first requires identifying which vibration is being followed.

A Way Of Listening In An 1865 Catalogue

Returning to 1865 brings us to an example of the volume–opening relationship shaped specifically for listening. Page ten of Rudolph Koenig’s catalogue shows a spherical Helmholtz resonator and separately lists sets of nineteen and ten. This vessel houses an air volume intended to make a particular sound component more apparent.

The catalogue describes the sphere as a hollow copper body. One of its two openings communicates with the surrounding air; the other ends in a small tube placed in the ear. The openings therefore serve different purposes: one receives the surrounding sound while the other connects the resonator to the listener.

The catalogue describes hearing the component to which the resonator is tuned more clearly within a complex sound. As one opening reaches the surrounding air and another the ear, the space inside becomes a working part of an instrument. The volume left behind by contents in an everyday bottle is deliberately formed here for listening.

Accounting For The Cavity In A Laboratory

In 2010, Webster and Davies investigated the volumes of water, spheres and cubes using a purpose-built Helmholtz apparatus. Their system had nominal one-, two- and three-litre cavities, used microphone measurements and accounted for temperature. Calibration beyond the ideal relation was needed: water-filling experiments and regular solid samples were evaluated with different correction curves.

In this experiment, frequency leads to a volume estimate through the microphone and suitable calibration. Comparisons involving water, spheres and cubes make the relation between occupied space and remaining air measurable. The relation is also visible in a sauce bottle at the table; measuring quantity by sound requires a defined apparatus and appropriate calibration, as used in this study.

The Unseen Half Of A Familiar Form

Looking at a hot sauce’s colour can also reveal the volume above its fill line. The familiar shoulder, neck and opening become the boundaries of the air inside. As the sauce is used, what changes is the share of that bounded space occupied by air.

Acoustics adds a further question to the bottle’s appearance: what does the surface in our hand enclose? Behind label and glass is an interior space that, under suitable conditions, can contribute to a sound response.

• The design language of bottles
• Sauce structure: emulsions and flow
• Ingredients and the senses at the table