The note a bottle sings whatever its shape

The frequency that gets an answer drove a mass on a spring and found it answering most strongly at its own frequency, a quarter cycle behind the push. The width that is a lifetime found that the sharpness of the answer and the time the oscillator rings are the same number. Later essays found resonances pumped rather than pushed, masses that hold machines still, resonances with zeros in them, resonances that refuse to leak, and clocks pulled into step. Every one began with an oscillator whose mass and spring were obvious: a weight on a coil, a pendulum, an electron bound to an atom.

Sound has resonators too, and the familiar ones are not like that. A flute or an organ pipe resonates because a standing wave fits in it: a pressure pattern with nodes and antinodes, whose wavelength is set by the pipe’s length, as only some notes fit found for a string and how many ways there are to vibrate counted for a box. But there is a second kind of acoustic resonator, older than any theory of it, in which nothing fits at all. Blow across the mouth of a bottle and the note it makes has a wavelength many times the bottle’s size. No standing wave is involved. Helmholtz worked out what is going on in the 1850s, and his answer turns the bottle into exactly the mass on a spring of that first essay, made of nothing but air.

A mass of air on a spring of air #

The air in a bottle’s neck is a short plug, free to slide in and out. The air in the body is a closed volume: push the plug in a little and the body’s air is compressed, its pressure rises, and it pushes the plug back out. Overshoot, and the body’s air is rarefied and sucks the plug back in. The plug is a mass, the body’s air is a spring, and the pair oscillates.

Both quantities can be computed. The plug’s mass is the density of air times the neck’s area $A$ times its effective length $L'$, which is the neck’s actual length plus a little air beyond each end that moves with it — about one and a half neck radii in all. For a wine bottle that is thirty-two milligrams. The spring’s stiffness comes from how the body’s pressure responds to a small change of volume: compressed quickly, without time to exchange heat with the glass, air’s pressure rises by $\gamma p$ times the fractional change in its volume, which for a piston of area $A$ on a volume $V$ gives a stiffness of $\rho c^2 A^2/V$, fifteen newtons per metre for the bottle. The frequency of a mass on a spring follows:

$$f = \frac{c}{2\pi}\sqrt{\frac{A}{V L'}}.$$

For the bottle that is 109 hertz, close to what one hears. The speed of sound enters only because it packages the air’s density and stiffness; no wave has to travel anywhere.

A spring that has no time to warm #

The stiffness of the air spring hides a classic mistake. Compress a gas slowly, letting it stay at the temperature of its surroundings, and its pressure rises in proportion to the squeeze. Compress it quickly and the work done on it heats it, and a warmer gas pushes back harder: its pressure rises by $\gamma$ times as much, where $\gamma$, the ratio of the gas’s two heat capacities, is 1.4 for air because its molecules have two ways of turning as well as three of moving. Newton calculated the speed of sound in 1687 assuming the slow, isothermal stiffness and came out fifteen per cent too low, a discrepancy that stood for over a century until Laplace saw that sound is too fast for heat to flow.

The bottle is in the same position. At 109 hertz, heat can diffuse through air only about a quarter of a millimetre in one cycle, so all but a thin skin of the body’s air next to the glass is compressed without exchanging heat, and the spring’s stiffness is the adiabatic one. Had it been isothermal, the bottle would sing at $109/\sqrt{1.4}$, about 92 hertz. The thin skin where heat does flow is one of the places the oscillation loses energy, alongside the friction in the neck, and in very small resonators, where the skin is a large fraction of the volume, it matters.

Why the shape does not matter #

The formula contains the volume of the body and nothing else about it, and that is the remarkable claim. It says that every vessel of the same volume, given the same neck, sings the same note, whether it is a sphere or a cube, a tall narrow cylinder or a flat box.

The reason is that the body’s air is squeezed as a whole. The wavelength of the note, three metres, is so much larger than any dimension of the body that a pressure change made at the bottom of the neck spreads through the whole body long before the plug has moved appreciably: the pressure in the body is uniform, rising and falling together everywhere. A uniform squeeze depends only on how much air there is. The shape would matter only if the pressure had time to vary from one part of the body to another, which is what the body’s own standing waves are, and they lie far higher: the lowest mode of a sphere of this volume is at two kilohertz, of a cube nineteen hundred hertz, of a tall cylinder six hundred and fifty. The Helmholtz note sits well below all of them, in a range where the body cannot slosh and can only breathe.

That is what it means for a resonator to be lumped. Its mass sits in one place and its springiness in another, and each can be described by a single number. The limit of the description is visible in the figure: the tall cylinder’s own mode is closest to the Helmholtz note, and a vessel long and thin enough would bring them together, at which point the body’s air would start to slosh as well as breathe and the formula would fail.

Filling the bottle #

Pour water into the bottle and the note rises. The water does not take part in the oscillation; it only takes up room, leaving a smaller volume of air to act as the spring, and a smaller spring is a stiffer one, since the same movement of the plug compresses a smaller volume by a larger fraction. The frequency goes as one over the square root of the air’s volume. Half full, the bottle sounds 155 hertz, a factor of $\sqrt{2}$ higher, the musical interval called a tritone; three-quarters full, 219 hertz, an octave above empty. Each further octave needs the air reduced to a quarter again, so the top notes of a row of bottles tuned with water are bottles nearly full.

It is tempting to explain the rising note by saying the water shortens the column of air above it, as if the bottle were a pipe. For a bottle with a narrow neck that explanation is wrong; the column’s length enters only through the volume. For a wide-mouthed jar, whose opening is as wide as its body, there is no neck and no plug, and the jar does behave more like a pipe — which is why a jar and a bottle being filled change their notes in different ways.

What keeps the note going #

A bottle struck or tapped rings briefly and dies away, a damped oscillator answering a kick. A bottle blown across sings for as long as the breath lasts, and that needs something more than a resonance: a source of energy that the resonance itself controls. The source is the jet of air from the lips. Directed across the bottle’s mouth, a jet is unstable — it flaps from side to side — and the flapping is steered by the air moving in and out of the neck. When the plug of air moves out, it deflects the jet outward; when it moves in, the jet is deflected in and blows into the bottle, pushing the plug further in. With the right speed of jet and distance to the edge, the jet’s push arrives in step with the plug’s motion and feeds it energy every cycle.

The bottle and the jet together are a self-sustained oscillator of the kind the clock that is pulled into step described: an amplitude set by the balance between what the jet supplies and what friction and radiation take away, and a frequency set almost entirely by the resonator, which the jet follows. Blow too hard and the jet locks onto a higher mode of the body instead, and the bottle jumps to a squeal well above its Helmholtz note; that higher note, unlike the low one, does depend on the bottle’s shape. Every flute, recorder and organ flue pipe works the same way, with an air jet driving a resonator, but there the resonator is a pipe of standing waves rather than a mass on a spring.

The sea in a shell #

The best-known Helmholtz resonance is the one nobody blows. A shell, a cupped hand or an empty cup held over the ear seems to hold the sound of the sea. What it holds is the ordinary background noise of the room, which contains a little of every frequency, filtered by the cavity’s resonance: the frequencies near the cavity’s own are amplified and the rest are not, and broadband noise passed through a broad resonance has the rushing, rising-and-falling quality of surf. A larger shell, or a hand cupped more loosely, has a larger volume and a lower resonance and sounds like a deeper sea. In a soundproofed room the sea disappears, because there is no noise to filter — which is how the explanation was settled, against the older idea that the sound was the listener’s own blood.

How sharply it answers #

A bottle’s resonance has a width, and the width is set by how fast the oscillation loses energy. Some is radiated as sound from the mouth, which is the part that is heard; some is lost to friction, as the plug of air rubs along the inside of the neck in a thin boundary layer. For a bottle with a narrow neck the friction dominates, and the quality factor — the number of cycles the bottle rings before its energy falls substantially, as the width that is a lifetime found — is a few tens, giving a peak a few hertz wide. A wider, shorter neck rubs less and radiates more.

Helmholtz used exactly this sharpness. In the 1860s he had a set of brass resonators made, spheres of graded sizes each with a short neck and a small nipple on the far side that fitted into the ear. Held to the ear, each one picked out, from a complex sound, the single component that lay inside its narrow peak and made it loud, leaving the rest faint. With a set of them he analysed the tones of musical instruments and of vowels into their harmonics, and showed that the quality of a sound — what makes a violin’s A different from a clarinet’s — lies in the proportions of its partials. The resonators were the first acoustic spectrum analyser.

The same note, from a whistle to a room #

The formula applies at every scale where a cavity is small compared with its wavelength. An ocarina, the egg-shaped vessel flute, is a Helmholtz resonator whose note is raised by opening finger holes, each of which adds a second neck in parallel with the mouth. Since the holes act as necks rather than as positions along a pipe, the note depends on the total area of the holes opened and not on which ones: an ocarina player can produce the same note with different fingerings, which no flute player can. A bass-reflex loudspeaker encloses its driver in a box with a tuned port, a deliberate Helmholtz resonator at thirty or forty hertz that reinforces the lowest notes the small driver cannot push alone.

At larger scales the resonance is felt rather than heard. A car’s cabin, three cubic metres of air, with one window open is a Helmholtz resonator tuned to around eighteen hertz, and the turbulent air rushing past the open window, shedding eddies at a rate that rises with speed, drives it hard when the two match: the familiar throbbing that fills the cabin at certain speeds and stops when a second window is opened, which changes the neck and detunes it. A room with a door ajar resonates at a few hertz, below hearing, and buildings with large open atriums have needed treatment for the same throb driven by wind. Harbours with narrow entrances oscillate the same way, with the harbour’s water as the spring and the water in the entrance as the mass, at periods of minutes.

A resonator that removes a note #

The oscillation can also be used to take sound away. A Helmholtz resonator connected to the side of a duct, with its neck opening into the duct’s wall, presents the duct with a branch that is very easy to push air into at the resonator’s own frequency and hard at others. At resonance, sound travelling along the duct finds a path of almost no impedance to one side and is reflected back where it came from instead of passing on: the transmission falls to nearly zero at one frequency, the shape the resonance with a zero in it found when a resonance sits beside a smooth path. Exhaust silencers, the intake systems of engines and the ventilation ducts of buildings use side-branch resonators to remove particular troublesome tones, in the same way that the mass that makes another stand still found a tuned mass removing a machine’s vibration at one frequency.

What the pictures cannot show #

The figures use the simplest lumped model, with an end correction of one and a half neck radii in total; the true correction depends on whether each end of the neck is flanged, on the shape of the transition into the body and slightly on frequency, and changes the computed note by several per cent. The quality factors are illustrative: a real bottle’s depends on its neck’s roughness and on how it is driven. The shape comparison treats each body’s own modes in the simplest way, which is enough to show they lie far above the Helmholtz note but not to give them precisely. And the dimensions of the car, the room, the loudspeaker and the ocarina are rough; each real example has its own openings and wall compliance, which lower the frequency when the walls themselves give, as a car’s windows and panels do.

Still open: where the friction goes in the smallest resonators #

Helmholtz resonators shrink to sizes where the neck’s boundary layer is as thick as the neck itself — sub-millimetre necks at kilohertz frequencies — and there the friction no longer merely broadens the peak but sets its frequency and decides whether the resonator works at all. Arrays of such tiny resonators, built into panels, make acoustic metamaterials that absorb sound over chosen bands from a structure much thinner than the wavelength it absorbs, which ordinary absorbers cannot do. How thin a panel can be made while still absorbing a low frequency completely, and how to spread the absorption over a wide band without losing its strength, are questions about the balance of radiation, friction and stiffness in exactly this oscillator, now pushed to the limits of its lumped description.

The habit worth carrying away is to ask whether a resonator is a wave or a lumped element before asking what sets its pitch. A cavity much smaller than its wavelength resonates as a mass of air in its opening on the spring of the air inside: $f = (c/2\pi)\sqrt{A/VL'}$, 109 Hz for a wine bottle, the same for any shape of the same volume, rising as $1/\sqrt{V}$ as it fills. The shape matters only when the body is large enough to slosh, and the same formula throbs in a car at eighteen hertz.