The glass that sings lower as it fills

The glass a voice can break followed a wine glass driven at its ringing frequency until it shattered, and found that its high quality factor, a thousand or so, is what lets a voice hold a note close enough to that frequency for long enough. The note itself was taken as given. But anyone who has run a wet finger round a glass’s rim, or tapped a row of glasses at a dinner table, knows that the note is not fixed. Pour in water and it falls. The glass harp and the verrophone, instruments made of tuned glasses, are tuned exactly this way: by filling each glass to the height that gives its note.

What is less familiar is how the note falls. Fill a glass steadily and the pitch barely moves at first; a glass half full rings little more than a semitone below an empty one. Then, over the last third or so, the pitch drops fast, and a full glass sounds most of an octave lower. The water near the top does almost all the work, and the reason says something about how the glass is moving when it rings.

The mode that makes the note #

A struck glass rings mostly in one mode: the rim distorts from a circle into an oval, then into an oval at right angles, back and forth. Seen from above, the rim moves in and out at four points, two opposite pairs swapping roles each half cycle, with four still points, nodes, between them. This is the $n = 2$ mode of a cylindrical shell, the simplest way a round wall can flex without stretching, and it is the lowest note the glass makes. Higher modes, with three, four or more waves round the rim, ring at higher frequencies and die away faster. A rim is a closed loop, so a whole number of waves has to fit round it, the same counting that only some notes fit found for a string fixed at both ends; the glass can have no mode with two and a half waves round it, and its partials are the integers from two upward, spaced not harmonically but roughly as the square of that integer, since a shell bending into more lobes bends more sharply for the same amplitude.

The wall does not move equally at all heights. At the base the glass is held by the stem and the thick bottom, and hardly moves at all. At the rim it moves most. In between, the motion grows with height, and the way it grows depends on the glass’s shape and the thickness of its wall.

That growth is the key to the fill curve. A liquid in the glass does not change how stiff the wall is; it adds mass. But it only adds mass where it has to move, and it only moves where the wall pushes it. Liquid in the bottom of the glass, next to a wall that is almost still, is almost not part of the vibration at all.

Rayleigh’s quotient #

The frequency of a vibrating system can be estimated by comparing its two energies. The third of itself a spring carries did this for a mass on a spring with mass of its own: assume the spring stretches uniformly, compute its kinetic energy, and find that it moves as if a third of its mass were added to the weight. The method is Rayleigh’s, and it gives the frequency as the square root of a stiffness over an effective mass, both computed from an assumed shape of motion. Getting the shape roughly right gives the frequency very accurately, because the frequency is insensitive to small errors in the shape.

For the glass, the stiffness is the wall’s resistance to bending and does not change when liquid is added. The effective mass is the wall’s own mass, weighted by how much each part moves, plus the liquid’s. The liquid inside a cylinder whose wall moves in the $n = 2$ pattern flows in a matching pattern, and its kinetic energy is that of a layer of liquid of thickness $R/n$ moving with the wall, where $R$ is the radius. So the frequency of a glass filled to a height $h$ is

$$\left(\frac{f_0}{f}\right)^2 = 1 + \frac{\rho_l R}{n\,(1 + 1/n^2)\,\rho_g a}\cdot\frac{\int_0^h W^2\,dz}{\int_0^H W^2\,dz},$$

where $W(z)$ is the wall’s motion at height $z$, $a$ the wall thickness, $H$ the height of the bowl, and the factor $1 + 1/n^2$ accounts for the wall’s motion along its own surface, which comes with the in-and-out motion in a mode that bends without stretching. The second fraction is the share of the wall’s motion, squared, that lies below the water level.

The fourth power #

If the wall’s motion grows as $(z/H)^p$, the integral gives $(h/H)^{2p+1}$, and the fill law is a power of the height. Anthony French measured wine glasses in 1983 and found that the pitch’s fall followed a fourth power of the fill height, which corresponds to $p = 3/2$: the wall moving as the height to the power one and a half.

The three curves show how sensitive the fill curve is to the shape. A wall moving in proportion to height would give a cubic law, lowering the note sooner; one moving as the square would give a fifth power, holding the note longer before dropping. The measured fourth power is therefore a statement about the glass’s wall, read from nothing but its pitch: fill it in steps, tap it at each, and the curve of pitch against height fixes the exponent of the wall’s motion. A tuning procedure has become a measurement of a mode shape.

The full glass’s frequency is set by the ratio of liquid to glass: $\rho_l R / (\rho_g a)$, the mass of water in a layer as thick as the radius, compared with the mass of the wall. For a glass seven centimetres across with a two-millimetre wall of glass of density 2,500 kilograms per cubic metre, the coefficient is 2.8, and a full glass rings at about half its empty frequency.

Where the tuning happens #

The musical consequence is that the useful range of each glass is near the top.

The first half of the water moves the note by little more than a semitone; the last quarter by about six. Players of the glass harp, who tune each glass by ear before a performance, know this without the formula: a few drops added to a nearly full glass shift its note noticeably, while a nearly empty glass can take a good splash before it moves. It also means a set of glasses must be chosen so that each glass’s useful range, from a third full to nearly full, covers the notes it has to play, which is why glass harps use glasses of many sizes rather than one size filled to different levels.

Benjamin Franklin’s glass armonica of 1761 solved the tuning problem differently. Its glass bowls were made in graded sizes, each blown and ground to its pitch, nested on a rotating spindle and played with wet fingers. No water was used to tune them, because the arithmetic here makes water an awkward tuning medium: it works well over a limited range and evaporates during a performance, slowly lowering the pitch of every glass, and the top quarter, where the tuning is most sensitive, is also where evaporation acts first.

A glass that weighs its contents #

The liquid enters the formula only through its density.

So a glass is a crude densitometer. Two identical glasses, one filled with water and one with vodka, ring at audibly different pitches, the spirit higher. The principle is the one the weight a frequency can find used to weigh small masses by their effect on a vibrating structure’s frequency, and the same principle runs the vibrating-tube densitometer used in breweries and laboratories, in which a U-shaped glass tube full of the sample is made to vibrate and its frequency gives the density to five or six figures. The wine glass is that instrument at its roughest.

Viscosity does not appear because at a few hundred hertz the liquid in a glass moves as an almost perfect fluid: the viscous layer at the wall is a fraction of a millimetre thick and carries little of the motion. Viscosity does damp the ringing, so a glass of syrup rings shorter as well as lower, but the pitch is set by mass.

Thin crystal and thick tumblers #

The wall’s thickness enters the ratio directly.

A fine crystal glass with a one-millimetre wall is light compared with its contents and drops more than an octave when filled. A thick tumbler drops much less. The thin glass is also the one with the higher quality factor and the purer, longer ring, which is why it is the one a voice can break and the one a glass harp is made from. Its sensitivity to its contents is the same lightness viewed from the other side.

There is a quiet link here to the dent that raises the note, where a small change to a vibrating body shifted its frequency by an amount weighted by how much the body moved at the place of the change. A dent near a node does nothing; a dent near an antinode does a lot. Water in a glass is a distributed version of the same perturbation, added mass weighted by the square of the motion, which is why the bottom of the glass, near the still stem, matters so little.

Tapped and blown: two notes going opposite ways #

A bottle makes the point sharply, because it can be played two ways. Tap a glass bottle with a spoon and it rings like the wine glass: its wall vibrates, the liquid inside adds mass where the wall moves, and filling it lowers the note. Blow across its mouth instead and it sounds a quite different note, made by the air inside bouncing in and out of the neck, and filling it raises that note. The note a bottle sings whatever its shape traced the blown note to Helmholtz’s resonator, in which the air in the neck is a mass and the air in the body a spring; adding water shrinks the body’s volume, stiffens the spring and raises the pitch.

So the same bottle, filled step by step, has one note that falls and another that rises. They come from different vibrating things, the glass in one case and the air in the other, and the liquid does opposite things to them: it is extra mass for the glass and lost volume for the air. A child filling a bottle and alternately tapping and blowing across it hears the two notes converge and pass each other, and the crossing is a demonstration that one object can carry two unrelated resonances.

Tuning by mass where it moves #

The rule that mass matters only where the body moves is the basis of tuning for most struck instruments. A bell founder tunes a bell after casting by turning metal from the inside of the bell on a lathe, at heights chosen for the partial to be adjusted: removing metal where a particular mode moves most lowers that mode’s frequency, while removing it near a node leaves that mode almost unchanged and adjusts another. The marimba maker carves an arch under each bar, thinning it in the middle to lower the fundamental and leaving the ends, where the fundamental hardly bends, to set the overtones. Both are applications of the same weighted sum that sets the glass’s note: a frequency depends on mass and stiffness weighted by the square of the mode’s motion, and a change made where the mode is still is no change to that mode.

The drum that has no harmonics explained why a drum’s overtones are not a harmonic series and why most percussion sounds inharmonic as a result. Tuning by mass is how instrument makers fight that: they cannot change the shapes of the modes much, but they can move mass to where one mode moves and another does not, and shift the frequencies separately until they fall into musically useful ratios. A wine glass offers only one degree of freedom, the height of the water, and so it can tune only its fundamental, which is why the glass harp is a gentle instrument of pure, single tones.

A tuning fork, which the frequency that gets an answer used as its standard example of a high-quality resonator, is tuned by the same principle with the opposite intent. Grinding metal from the tips of the prongs, where they move most, raises the frequency by removing mass from the most moving part; grinding at the base of the prongs, where the bending is greatest, lowers the frequency by reducing stiffness. Every fork has been tuned by one or the other, and the grinding marks show which.

What the picture cannot show #

The model treats the glass as a straight-sided cylinder with a uniform wall, held still at the base, with the liquid’s motion that of an ideal fluid in a cylinder. Real wine glasses are curved bowls, wider in the middle or tapering towards the rim, with walls thicker at the base. The curvature changes both the mode shape and the liquid’s added mass, and a tulip-shaped glass will have a different fill curve from a straight tumbler. The measured fourth power is a property of the glasses that were measured, not a law of all glasses.

The power-law mode shapes are assumptions chosen to show how the exponent controls the result, not solutions of the shell equations. A proper calculation would find the mode shape for the actual geometry, and would include the stem’s flexibility and the liquid’s free surface, whose own sloshing modes are at much lower frequencies and do not couple to the ringing. The values of wall thickness, radius and density are typical, and the absolute pitch of a glass, set by its stiffness, has not been computed at all: only the ratio to its empty pitch.

The model also ignores what happens near the rim. A glass filled to the brim has liquid touching the edge where the wall moves most, and the free surface there is driven by the vibration; rubbing a full glass makes visible standing ripples on the water’s surface, four-fold patterns at the rim’s frequency, which take a little energy from the ringing. Those ripples are a way of seeing the mode, and they appear in the photograph of a ringing glass’s surface at exactly the four antinodes.

Still open: how the liquid’s surface takes part #

A glass filled near the brim and set ringing by a wet finger develops ripples on the liquid’s surface, sharp at the four places where the wall moves most. At large amplitudes the surface can break into droplets that are thrown upward, and in some experiments a wine glass rubbed hard ejects a fine spray from those four points. How the energy passes from the ringing wall into surface waves, how strongly that damps the glass, and at what amplitude the surface becomes unstable, have been studied in a few experiments with high-speed cameras, but a complete description that predicts when a glass will start to throw droplets has not been worked out. It is a small problem in fluid–structure interaction with a domestic apparatus, and it remains partly open.

A glass rings by bending its wall into an oval, with the motion growing from the stem to the rim, and liquid lowers the note by adding mass weighted by the square of that motion, so the fall in pitch goes as the fill height to the power 2p+1 — the fourth power measured by French, a wall bending as the height to the 1.5 — 1.4 semitones at half full and 11.6 full for a 7 cm glass with a 2 mm wall, scaled by the liquid’s density over the glass’s. The habit worth carrying is to ask where a change is made relative to where the system moves, because a perturbation at a node is no perturbation at all.