The glass a voice can break

The frequency that gets an answer introduced resonance with the stock examples, among them a wine glass and a voice, and it moved on. It is a claim worth examining, since it has been the subject of an amount of folklore and a certain amount of fraud. Caruso is said to have done it, without any evidence. A television advertisement in the 1970s had Ella Fitzgerald break one with a recorded note, and it was widely doubted. In 2005 a rock vocal coach, Jaime Vendera, did it on camera for a popular science programme, with no amplifier, after a number of attempts and a selection of glasses. Measured at the glass, his voice was about 105 decibels. So it can be done. What is less often discussed is what it takes, and the answer is much stranger than a loud note at the right pitch.

Strike a glass with a fingernail, and it rings. That ring is mainly the lowest mode of the bowl, in which the circular rim is squeezed into an oval and then into the opposite oval, with four nodes round the rim. It usually has a frequency of a few hundred hertz, and for a thin wine glass in a good state it dies away over a second or two. That slow decay is the whole of the story. As the width that is a lifetime established, a long ring and a narrow response are the same fact. If a thing rings for about $Q/\pi$ cycles, then it responds to a force at its own frequency with $Q$ times the motion that a steady force would give, and it responds over a bandwidth of one part in $Q$.

A lever and its window #

A good crystal glass has a $Q$ near a thousand. Thin soda-lime glass with a decent shape might reach 500 or so, and a thick, rough tumbler perhaps a hundred or less. The gain is a real lever: at exact pitch, a force is effectively a thousand times greater. But look at how narrow the window is. For a mode at 600 Hz with $Q$ of 1000, the half-power width is 0.6 Hz in total. A musician thinks in cents, a hundredth of a semitone, and that makes the window about ±0.9 cents.

A cent is well below what most listeners can distinguish, and a very good singer would be doing well to hold within five or ten cents of a target for a short phrase. So the pitch has to be found with a precision no ear can use. And then it has to stay there.

Why is the window so narrow? The physical reasoning is a matter of phase. If a push is applied at just the wrong frequency, then over $Q$ cycles it gradually falls out of step with the rim. Soon the two are opposed, and the push is taking away what it put in. So the rim can accumulate energy only as long as the push stays in step over the whole time it stores that energy, and this is $Q$ cycles. It takes a difference of one part in $Q$ to fall out of step in that time.

A lever that has to be pulled for a while #

The glass does not reach its full motion at once.

At 600 Hz, a $Q$ of 1000 means a time constant of 0.53 s and a settling time of about a second and a half. So a voice cannot simply hit the note and have an effect. It must find the note and hold it for a second or more at the highest level it can manage. That is hard physically. A loud sustained note at a chosen pitch is a skill in itself, and the singer has to hold it within a window much narrower than ordinary intonation, for longer than a typical high-note attack. It is not surprising that the demonstration usually takes a few attempts.

The same ring-up is what gives a clue about how to get the pitch. Vendera and others tap the glass first, to hear the note. They sing near it and watch for the visible motion of the rim, or of a straw placed in the glass, for the sound level at the rim is high enough for the motion to be seen. The straw shows them which way to move. No ear is that exact, but a glass can tell by its response, and the singer is using it as a pitch detector, in a closed loop. A sound level meter or an electronic tuner would help only a bit, since it is the glass’s own frequency, and not the nominal note, that has to be matched. Glasses vary. Adding a little wine lowers the mode by a measurable amount.

That closed loop has the same structure as the clock that is pulled into step, with a person in place of a phase-locked oscillator. In practice the singer has to adjust at a rate that is slower than the ring-up time of the glass, since otherwise the glass does not have time to report what has changed.

The wobble that hardly matters #

A trained singer’s voice has vibrato: its pitch swings regularly, about five or six times a second, over a range often up to a quarter-tone or more. It would seem hopeless. The window is ±0.9 cents, and the vibrato takes the note ±50 cents away five times a second. Surely a voice with vibrato spends almost no time on resonance?

It would seem so, and the conclusion is wrong, and that gives the most interesting thing in this calculation.

The way to see it is to stop thinking of pitch as a thing that moves, and to look at the spectrum instead. A tone whose frequency swings sinusoidally is a frequency-modulated signal, the same as an FM radio station’s. Its spectrum is a set of discrete lines: a central line at the average frequency, and sidebands spaced by the wobble rate, each with a height set by a Bessel function of the modulation index, the depth divided by the rate. So the tone can be regarded as a sum of steady tones. The glass, with its resonance 0.6 Hz wide, has no way to know that those lines at ±5.5 Hz belong to the same voice. It responds only to the central line.

The central line’s share of amplitude is $J_0(\beta)$, with $\beta = \Delta f/f_m$. For a wobble of ±18 cents, $\Delta f$ is about 6 Hz, and with a rate of 5.5 per second the index is near 1.1, and $J_0$ is about 0.70; the direct integration gives 706 of a possible 1000. Most of the amplitude is left in the middle. So a moderately wobbling voice retains most of its effect, provided that its centre is correct. The time domain agrees with this: the numbers plotted come from integrating a mass and spring driven by a wandering tone, in which nothing about Bessel functions has been assumed. The curve, which is computed separately, matches them to within a few per cent above about a fifth of the peak.

It also predicts something strange. At an index of 2.4, the first zero of $J_0$, the central line vanishes altogether. A voice with a wobble of about ±38 cents, at that rate and at that frequency, leaves no energy at the frequency of the glass. A wider vibrato, ±60 cents, brings a share of about four-tenths back, with its sign reversed. That is an unusual prediction. The effect of a wobble of this size at one rate would be a marked loss, and a slightly deeper wobble would improve things. FM engineers have long used this same zero as a calibration method to set a transmitter’s deviation accurately, by adjusting a modulating tone until the carrier disappears. A wine glass and a radio station are on the same curve, and the mathematics of the fork inside a watch gives the same demand for a narrow, stable resonance. Note that a slow drift of pitch does not obey this at all. A wander slower than the ring-up rate is simply followed by the resonance. So a slow error in the voice’s mean, of a few cents, is the true enemy. The fast regular wobble that seems so damaging is not.

How loud it has to be #

There is still the matter of how much sound. A rough estimate can be made.

An effective mass of 10 grams at 600 Hz has a stiffness of about 140 kN/m. A sound level of 120 dB is 20 Pa, and if a third of that acts over ten square centimetres, the force is about 6 mN. Static deflection would be about 40 nm. With a $Q$ of 1000, that becomes 40 µm. A rim of a 35 mm radius with a wall of 1 mm, bent out by $x$ in the oval mode, has a surface strain of order $3xh/R^2$. Then 40 µm gives about $10^{-4}$. Strain at which glass breaks depends on what has happened to it. Flawless drawn fibres reach several per cent. A glass that has been handled has microscopic surface cracks and it typically fails at $10^{-3}$ or below, and the concentration of stress at scratches, which the stress that polarised light can see can display, may bring that down further under cyclic loading.

Hence a figure of 120 to 140 dB is what this rough sum gives, with a large uncertainty, and it is around 15 to 35 dB more than Vendera’s report. Part of that gap is the coupling: a voice is a small source a few centimetres from one side of a glass, and a large pressure gradient drives an oval mode far better than a uniform pressure. Part of it may be that real glasses have flaws. A chosen glass is often one that has already been observed to ring well, and to have a suitable fault. So the sum should not be taken as an exact threshold. But it does show why a higher $Q$ helps. Going from 300 to 1000 is worth 10 dB. To a singer near the limit of the human voice, whose maximum sustained level in the chest is near 120 dB in a very trained case and well below that for most, ten decibels is the difference between a demonstration that works and one that does not. And cyclic loading, thousands of cycles in one note, gives cracks the chance to grow, as a steady load does not.

A strong rim motion also has a side-effect. A strain large enough to matter means an amplitude large enough to see, and at that motion the rim moves a good fraction of a millimetre. The air around the rim has been pumped hard and it radiates, as well as doing a surprising amount of damping, so that $Q$ falls at very high amplitudes, a nonlinearity that this estimate leaves out. Any wine left in the glass also damps it; the ripples that answer at half the shaking can appear on the surface as a pattern of standing waves, and they carry away energy. A dry, clean glass is what is wanted.

A better glass is harder to hit #

The essay’s main argument lies in the numbers side by side.

Every increase in $Q$ brings more gain, and it costs precision and patience in exactly equal measure. The product of the window and the ring-up time is fixed, whatever the value of $Q$. It is not a limitation of the glass; it is the same trade between frequency and time as in sharpness has to be paid for. A resonance cannot be sharp in frequency without being long in time.

The curve for a steady voice crosses the window at a $Q$ near 173. Below that point, a voice that is held as steady as a good singer’s can use the whole of the gain, and pitch is not the limiting factor. Beyond it, the voice is unable to use all the $Q$ on offer unless it does something to find the centre. The actual gain available to a voice with a fixed mean error $\varepsilon$ cents is about a constant, as $Q$ rises, once the window is narrower than the error. That is an interesting result, since it implies that a voice with no feedback would break no better a glass than a modest one. It is the closed loop, using the visible straw, and a singer’s practice with a given glass, that makes a very high $Q$ usable. In this way a person can in effect narrow their own pitch error by watching.

A similar trade holds for anyone using a high $Q$ resonance as a detector. A high $Q$ instrument gives a larger response to a weak signal, in less bandwidth, and it needs a longer time to give a reading. The noise a high Q moves out of the way showed how a thermal noise power is shifted around in frequency, with the same total. A radio receiver, an atomic clock and an optical cavity for the absorption measured with a stopwatch all pay this bill, and a person singing into a glass is an inefficient member of the same family.

What the rim does not tell #

The figures treat the mode as a single mass on a spring, at one frequency. A real glass has a pair of degenerate oval modes at right angles, slightly split by imperfections of the rim, and a tone excites the one more strongly aligned with the sound. If they are split by more than the width of the resonance, as they often are, there are two separate peaks. Spinning the glass exchanges energy between them, the effect in the vibration that turns part of the way. So a singer may be trying to find one of two narrow windows and aim the voice along the corresponding axis. Moving the sound source around the rim changes the strength of coupling a lot, and it is possibly the largest unaccounted factor in the sound level needed. A uniform pressure on a symmetric glass would excite no oval mode at all, as it is the wrong shape. The figure of a third is a stand-in for this geometry.

The numbers also depend on a chosen mode frequency. Glasses from 300 to 1000 Hz are common, and the voice’s achievable level varies with pitch. A high female voice may reach the 800 Hz of a small glass with less trouble. Bigger glass has lower $Q$ at low frequencies, due to radiating less, and thus the $Q$ used here is not a universal number. Very high values of 2000 or more have been reported for good lead crystal; and values of $Q$ that are measured are often reduced by whatever is holding the glass, usually at the stem, in a way a hand can change.

The strain estimate takes a static failure strain, whereas fatigue is a matter of time and environment. Moisture on the surface accelerates crack growth in glass by stress corrosion, and thus humidity in the room affects it in ways no one has systematically measured for this particular demonstration. Accurate tests of the failure of real wine glasses under controlled tones, at measured strains, are rare, partly because glasses vary so much. Failure of a single glass is a statistical event, and it is unwise to generalise from a few tests of glasses that were chosen for being fragile.

Still open: how much of a demonstration is the glass #

There is no agreed account of how often an unamplified voice can do this. A few successful trials have been filmed, and a far greater number of attempts have failed, without those being recorded. How much of the difference lies in the voice and how much in the particular flawed glass has not been separated with any care, because it would require a measured strain gauge on many nominally identical glasses, broken one after another under a controlled tone, and then a human singer tested the same way. A loudspeaker can break glasses reliably at modest amplifier power, and that has been a popular classroom exercise, so the physics is not in doubt. What is open is how close to the limit of the human voice this actually lies. It may also matter that the sound has a strong second harmonic, and whether a pair of harmonics can do better than one tone, or the use of a slow chirp to pass through a resonance whose position is unknown, are the kind of questions acoustics students have tried at small scale.

A glass answers a tone with a gain of Q in a window of one part in Q, after a time of about Q cycles: at Q = 1000 and 600 Hz that means a thousandfold lever, ±0.9 cents and over a second’s hold, in a trade whose product cannot change; a fast vibrato hardly matters, because the glass hears only the central line of its spectrum, $J_0$ of the depth over the rate, and only a wobble near ±38 cents removes it. The folklore has loudness as the mechanism. The arithmetic has aim. The habit worth keeping is to ask what is actually being delivered to a narrow resonance, because something that appears to be moving a note around may leave the spectrum at the centre nearly untouched.