Resonance — the series · Illustrated Physics

Response against driving frequency. The steady-state amplitude of a driven oscillator against driving frequency, at three damping ratios. Lighter damping gives a taller and narrower peak, and the peak sits slightly below the natural frequency.

The frequency that gets an answer, and the quarter cycle nobody mentions

Push an oscillator at its own frequency and the response grows enormously. The reason is not that the push is in step with the motion — at resonance it is a quarter cycle out, and that is precisely why it works.

part 1 · waves

Where modulating a system sets it going. The regions of the modulation plane in which an oscillator with a damping ratio of 0.02 will not stay still. The horizontal axis is the modulation frequency in units of the oscillator's own; the vertical is how deeply the stiffness is modulated. Inside a shaded wedge the state of rest is unstable and any disturbance grows exponentially; outside it, nothing happens at all. The wedges sit at modulation frequencies of twice, once and two-thirds of the natural frequency, and the first is much the widest — it opens at a depth of 8.0%, against 40.0% for the second. Each boundary is found by integrating one period of the modulation from two independent starts and asking whether the resulting map has a multiplier outside the unit circle, then bisecting on the depth; none of the shape is drawn by hand. Without damping every wedge would come to a point on the axis and there would be no threshold at all — the flat bottom of each is damping, and it is why a swing has to be pumped hard enough before it does anything.

The swing that is pumped, not pushed

Nobody pushes a swing they are sitting on. They stand up at the bottom and sit down at the ends, which changes the pendulum rather than forcing it — and does so twice per period. The equation that describes it has no forcing term at all, so standing still is always a solution, and what the pumping changes is whether standing still is stable.

part 2 · waves

The response of a machine with a 10% absorber bolted to it. The amplitude of the driven mass, in units of the deflection the same force would cause if applied slowly, against drive frequency in units of the machine's own natural frequency. Dashed: the machine alone, with its single resonance. Solid: the same machine with a second mass and spring attached, weighing 10 per cent of it and tuned to the same frequency. At that frequency the driven mass does not move at all — the amplitude is zero rather than small, and the absorber is moving 10.0 static deflections to make it so. What the device costs is the two new resonances it creates, at 0.854 and 1.171 times the original frequency, which are infinite in this undamped calculation and straddle the frequency the machine was protected at.

The mass that makes another stand still

Bolt a small mass on a spring to a machine that is shaking itself apart, tune it to the frequency that is doing the damage, and the machine stops moving. Not moves less — stops, exactly, at that one frequency. The price is two new resonances either side of it, and the whole device is a bet that the drive stays where it was put.

part 3 · waves

The same number read off a decay and off a linewidth. A lightly damped oscillator released and left alone, above, and the power spectrum of exactly those samples, below. The decay falls to 1/e of its starting amplitude after 8.0 cycles, which makes the quality factor π times that, or 25.0. The spectrum peaks at 1.0000 radians per second and falls to half its power 0.04001 radians per second wide, which makes the quality factor the peak divided by the width, or 25.0. The two disagree by 0.02 per cent, which is the resolution of the frequency grid rather than a difference in the physics. They cannot disagree by more, because they are the same statement: a resonance is narrow because its ringing is long, and the transform that turns one into the other is not an approximation but an identity. A measurement of either is a measurement of both — which is why a bell can be characterised by hitting it and listening, or by driving it and sweeping, and why the two instruments never argue.

The width that is a lifetime

Hit a bell and time how long it rings; drive it and measure how narrow its response is. The two numbers are the same number, and they cannot disagree — not because the physics conspires but because a decay and a linewidth are one function seen in two coordinate systems.

part 4 · waves

The resonance with a zero in it

Where a resonance is the only way through a system, the response is a symmetric peak. Where there is also a smooth path that does not care about the frequency, the two add before anything is squared — and the result is lopsided, with a frequency at which nothing gets through at all.

part 5 · waves

One width that falls to nothing. The decay rates of the two modes of a pair of resonances that leak into one shared channel at rates 0.1 and 0.05 and are coupled to each other with strength 0.2, against the detuning of the first from the second. The two rates always add to 0.15, the trace of the leak matrix, to 10⁻¹². Where the resonances are far apart each mode keeps roughly its own resonance's leak. Near them the leaks interfere, and at a detuning of 0.1414 — κ(γ₁ − γ₂)/√(γ₁γ₂) — the slower mode's decay rate is zero to within 10⁻¹², and the faster carries all 0.15. That mode sits at a frequency where the channel is open and does not leak into it: the two routes by which it could escape cancel.

The resonance that refuses to leak

A resonance that sits at a frequency where waves can escape has a width, because it leaks. Put two such resonances into the same channel and let them talk to each other, and at one precise detuning one of their combinations stops leaking altogether — a mode with no width, surrounded by a continuum it could escape into and does not. It cannot be seen from outside, it traps whatever energy lands in it, and if a symmetry is what forbids the leak, it survives any change that keeps the symmetry.

part 6 · waves

The clock that is pulled into step

In 1665, ill in bed, Christiaan Huygens noticed that two of his pendulum clocks hanging from the same beam were swinging in exact opposition, and that however he disturbed them they returned to it within half an hour. A clock is not a passive resonator waiting to be pushed; it keeps its own time, by its own mechanism. What a weak push can do to it is not change how strongly it swings but pull its phase — and within a range of frequencies set by the push's strength, pull it all the way into step. The same arithmetic locks lasers, pacemaker cells, power stations and the body's daily clock.

part 7 · waves

A bottle as a mass on a spring. A 750 ml bottle with a neck 8 cm long and 19 mm across, and its equivalent: the plug of air in the neck, 32 milligrams, moving in and out as a piston on the air in the body, which acts as a spring of stiffness 15 N/m because squeezing it raises its pressure. The plug's mass and the spring's stiffness give a frequency of 109 Hz, the note heard when air is blown across the mouth. The effective neck length includes about one and a half radii of air beyond its two ends that moves with the plug. Nothing in the body's air moves much — it is compressed and released almost uniformly, because the wavelength, three metres, is far larger than the bottle. That is what makes the resonator lumped: one mass, one spring, one note.

The note a bottle sings whatever its shape

Blow across the mouth of an empty wine bottle and it sounds a low note, about 109 hertz for a 750-millilitre bottle. Its wavelength is three metres, ten times the bottle's height, so it is not a standing wave in the bottle the way a flute's note is a standing wave in the flute. It is a mass bouncing on a spring: the plug of air in the neck, thirty milligrams of it, riding on the springiness of the air in the body. The note depends on the body's volume and the neck's size and on nothing about the body's shape, which is why a sphere, a cube and a tall cylinder of the same volume all sing the same note — and why a car driven with one window open throbs at eighteen hertz.

part 8 · waves

Too little noise, the right amount, too much. The particle's position over 8 cycles of the push (thin grey line, scaled to the wells), with noise of intensity 0.03, 0.085, 0.3 (top to bottom), the barrier being 0.25. With little noise it stays almost always in its starting well and only trembles: 0 hops in eight cycles. With noise 0.085 it hops 30 times, almost always once per half-cycle and almost always while the push is helping — a clean square wave following a push too weak to cause a single hop. With noise 0.3 it hops 285 times, at random, and the push is lost again. There is a best amount of noise, and it is not zero.

The noise that helps a weak signal through

A system that can sit in either of two states, nudged back and forth by a periodic push too weak to switch it, does nothing at all — until noise is added. Then it begins to switch, and with the right amount of noise it switches in step with the push, once each half-cycle, as though the push had been strong enough all along. Too little noise and it never switches; too much and it switches at random. The response peaks at a nonzero noise, which is why the effect is called a resonance, and why crayfish, paddlefish and human fingertips appear to use it.

part 9 · waves

A whisper that grows into a note. The amplitude of the tube's lowest mode against time, in units of the time sound takes to cross the tube (a period is two), started from a small disturbance, with the gauze at a quarter of the length, at 0.12 of it, and at three-quarters; heat-transfer strength K = 1. At L/4 the disturbance grows some fortyfold over about thirty-five periods and then stops growing, at an amplitude of 0.83 in these units, where the air's to-and-fro speed at the gauze is comparable to the steady upward draught and the heat transfer, which goes as the square root of the air's speed, can no longer follow it in proportion. At 0.12 the gauze is near the edge of the singing range and the sound grows only slowly. At 3L/4 the same disturbance dies away within a few periods.

The tube that sings when heated low down

Hold a wide metal tube upright, push a red-hot wire gauze a quarter of the way up inside it, and take the flame away. The tube begins to howl — a loud, steady note at its lowest resonance, lasting until the gauze cools. Put the gauze a quarter of the way down from the top instead and the tube is silent; lay the tube on its side and it stops. Heat is driving a sound, and whether it drives or damps depends on where the heat is released and when, in a rule Lord Rayleigh stated in 1878 that now governs rocket engines and power-station turbines.

part 10 · waves

A quiet resonator growing louder before it sings. The pressure in a resonator's 200 Hz mode over two and a half seconds, kicked continually by the same random noise, at three settings approaching the onset of self-sustained oscillation: the mode's natural decay rate falling from 20 to 6 to 2 per second. Each panel is on the same scale. Far from onset the noise is a faint, ragged hiss, each burst dying almost as soon as it starts. Closer, the bursts last longer and grow larger, because the mode forgets each kick more slowly and the kicks pile up; at the last setting the resonator hums in long swells of a nearly steady tone. Nothing has started oscillating on its own — every setting is below onset — but the noise has already begun to sound like the oscillation it is approaching.

The noise that warns a tube is about to sing

A resonator that is about to break into self-sustained oscillation — a combustor about to hum, a heated tube about to howl — is silent on average right up to the moment it starts. Its noise is not. As the margin of stability shrinks, the random hiss in the mode grows louder, remembers each kick for longer and narrows towards the coming tone, in proportions fixed by the margin itself. The instability can be heard before it exists, except in the case engines most often present, where noise starts the song before the warning has properly begun.

part 11 · waves

The parabola every quartz watch follows. The frequency error of a watch's quartz fork, in parts per million, against its temperature: a downward parabola, −0.034 ppm per degree squared, about a turnover at 25 °C, where the fork runs exactly at its nominal rate; with the right-hand scale in seconds a day, since one part per million is 0.086 s a day. On a wrist, at about 31 °C, the watch runs 1.2 ppm slow, 0.11 s a day; in a drawer at 10 °C, 7.7 ppm, 0.66 s a day; at 0 °C 1.8 s a day; at 60 °C 3.6. Every departure from the turnover, hotter or colder, makes it lose. The turnover is set by the angle at which the fork is cut from the crystal, and is chosen near the temperature of the place watches are worn — the parabola is a fact about quartz, and its vertex is a fact about people.

The fork inside a watch

Every quartz watch keeps time with a tuning fork three millimetres long, cut from a crystal of quartz and ringing thirty-two thousand seven hundred and sixty-eight times a second. Each number in that sentence is a decision. The frequency is two to the fifteenth, so that fifteen halvings make one tick a second, and it sits just above the range of human hearing. The shape is a fork because two tines swinging against each other leave their base still and lose almost no energy into it. And the fork's rate drifts with temperature along a parabola whose vertex is set near twenty-five degrees, which is why a watch worn on a wrist keeps better time than one left in a cold drawer.

part 12 · waves

A better glass is harder to hit. The half-width of the resonance in cents, and its ring-up time, against Q at 600 Hz. One falls as the other rises, and their product is fixed. A glass of Q = 100 has a window of ±8.6 cents, inside what a steady voice can hold, and it gains only a hundredfold. A glass of Q = 1000 gains ten times as much, and asks for ±0.9 cents held for over a second. Where a voice's steadiness crosses the window is about the Q at which a voice stops being able to use a further increase, and that is near Q = 173.

The glass a voice can break

A singer can shatter a wine glass, and it is not done by loudness. A good glass answers a tone a thousand times more strongly than a steady push, and it charges for that three times over: the note must be within about a cent, it must be held for over a second, and the glass must be flawed enough to fail. A voice's vibrato turns out to matter far less than its aim.

part 13 · waves