Resonance
When Timing Beats Force
Small Pushes at the Right Moment
An old account tells of a single bell ringer who pulled the rope at just the right moments. He set a bell swinging so hard that when four or even six men seized the rope to stop it, they were all lifted off the ground. One man’s well-timed pulls had beaten the strength of six.
The recipe is simple. Blow on a heavy pendulum hanging at rest, and it hardly moves. Blow again each time it has turned and begun to swing away from you, and by repeating the puffs in time with its swing you can give it considerable motion. Blow while it is coming toward you, and the puff hinders it instead.
The reason lies in the energy each push hands over. A push adds energy only while the body it pushes is moving the way it is pushed. A puff on a ball moving away from you speeds it up, and a puff on a ball coming toward you slows it down. Timed to the swing, every puff lands on the right side, and each adds a little to all the ones before. Between puffs the ball loses only what the air and the bending of its wires take away, so the small gifts pile up. Pushes at random land about as often on the wrong side as on the right one, so they mostly undo each other. A hundred random puffs build a swing only about ten times as wide as one puff gives. A hundred in step build one a hundred times as wide.
A Rhythm of Its Own
For the pushes to stay in step, the swing has to keep a steady beat while it grows, and a pendulum nearly does. The time of one swing is set by the length of the cord and the pull of gravity: make the cord four times as long and each swing takes twice as long. It does not depend on the weight, and for small swings it hardly depends on how wide the swing is. A wider swing has farther to go, but it also falls more steeply and moves faster, and the two very nearly cancel.
It is easy to push this too far and say that a swing takes the same time however far out it starts, even 90 degrees from straight down. It does not: a pendulum let go from level with its pivot takes about 18 percent longer over each swing than one swinging through a few degrees. The difference shrinks fast for smaller swings, and that is what made the pendulum the best clock in the world for nearly three centuries.
Anything that springs back toward a resting shape, and overshoots it because it has mass, swings with a rhythm of its own, called its natural frequency. A stiffer spring makes it quicker and more mass makes it slower. A guitar string has one, and so do a wine glass, a bridge, a tall building and the air in a bottle. Most things have several, one for each pattern in which they can vibrate. Tap a thing and you hear them: that is its note.
Hang a row of light pendulums of different lengths from a bar that is itself hung so it can sway, with a heavy one among them, and set the heavy one swinging. Every time it swings it tugs the bar, so each pendulum on the bar gets the same small tugs at the same rhythm. Only the one as long as the heavy one has the same rhythm of its own, keeps in step with the tugs, and builds up a big swing.
Watch the one that answers. It swings a quarter of a beat behind the heavy one, and that is no accident: it is the timing at which every tug meets it moving the way the tug pulls, as the puffs met the pendulum. The shorter pendulums swing along with the heavy one and the longer ones against it, and neither kind builds.
Even a sea has a rhythm of its own. The water of the Gulf of Maine and the Bay of Fundy, on the Atlantic coast of North America, sloshes from end to end in about 13 hours. That is close to the 12 hours and 25 minutes between one high tide and the next, so the ocean’s tide pushes it nearly in step. The Bay of Fundy has some of the highest tides on Earth, up to about 16 meters from low water to high, rivaled only by Ungava Bay in northern Quebec.
How High It Can Climb
If every push in step adds energy, why does a swing not climb forever? Because every swing also loses a little. The air drags on it and its ropes rub where they hang, and the wider the swing, the more it loses each time. It stops growing when each push only makes up what one swing loses.
So the height is decided by how little the swing loses, and there is a simple test for that. Set it going, leave it alone, and count how many times it swings before it fades. The longer a thing rings on its own, the larger a small, steady rhythm can make it.
The same count decides how fussy it is about the rhythm. A long ringer builds up only from pushes that stay in step for all those swings. A rhythm off by even a little drifts out of step over that many swings, until its pushes land on the wrong side and take back what the earlier ones gave. So a thing that rings long answers only its own note, and very exactly. A thing that rings briefly answers a wide range of rhythms, and never strongly.
A wine glass is a long ringer. In laboratory tests, good wine glasses rang at about 630 vibrations a second and took about two seconds to fade to a third, well over a thousand swings of the rim. Each swing bends the rim into an oval, first one way, then the other. Sound at the glass’s own note pushes the rim in step and builds the bending up until the glass breaks. The note has to stay within a fraction of a vibration a second of the glass’s own, for seconds at a time.
The scene slows the glass down and draws its bending far larger than life. It can be done with a voice. A singer has done it on camera, with no amplifier, at about 105 decibels. It took a dozen glasses or more: each breaks at its own weakest flaw, and two that look the same can differ a great deal. In the laboratory tests, the sound that broke a glass delivered only a fraction of a watt to it.
Bridges have rhythms too, and so do marching soldiers. Once, 74 soldiers marching four abreast over a suspension bridge in England felt it begin to shake in time with their steps. Some of them, enjoying it, stepped so as to make it shake more. Then a bolt in one of the stay-chains snapped where it was fixed into the masonry of its anchor in the ground. The bolt had been badly forged. That corner of the deck fell about five meters into the river, throwing some forty men into the water or against the chains. Twenty were hurt and none was killed. The army then ordered troops to break step when crossing bridges, and signs on London’s Albert Bridge still tell them to.
Pushes Timed by the Motion
A parent pushing a swing does not count beats. They push whenever the swing comes back to their hands, so the swing itself times every push, and the pushes stay in step even as its rhythm drifts. The same trick lets a steady supply of energy, with no rhythm in it at all, keep something swinging. A bow drawn steadily across a violin string keeps it sounding at its own note: the string’s own motion sets when the hair grips it and when it slips. Physicists call such a thing a self-oscillator.
A clock is the plainest one. A weight on a cord pulls steadily on a toothed wheel. The pendulum rocks a small anchor over the wheel that lets it turn one tooth at a time, and each tooth, as it slips past, gives the pendulum a small push in the direction it is already moving. The weight pushes all the time, and the pendulum’s swing sets when its push gets through.
Wind can do the same, and once it tore a bridge apart. The Tacoma Narrows Bridge in Washington State was so flexible that it was nicknamed Galloping Gertie soon after it opened. One morning, in a steady wind of about 68 kilometers an hour, its deck began to twist, one edge rising as the other fell, about twelve times a minute. Within minutes the twisting was huge, and it went on for nearly an hour, until a long stretch of the middle of the deck broke away and fell into the water.
The usual story calls it resonance, as if gusts had come in step with the bridge. But the wind was steady. The twisting deck changed how the wind pushed on it, so that each twist made the wind push the next one harder: the bridge timed the wind’s push itself, as a violin string times the bow’s. Engineers call this flutter.
London’s Millennium Bridge, a footbridge over the Thames, closed two days after it opened. As crowds crossed it, it swayed from side to side by up to seven centimeters. A walker on a swaying deck shifts their feet to keep their balance, and those steps push the deck sideways in step with its own sway. With enough people on it, they fed in more energy than the bridge could lose. The engineers saw many walkers fall into step with the sway, and the story is usually told that way. A recent study found that the falling into step comes after the sway has grown. Each walker keeping their balance pushes with the sway on average, even when no two walk together.
When the motion times its own push, there is no outside rhythm to avoid. There are two cures. Take energy out faster than the motion feeds it in, or change the thing so that its motion feeds it less. The bridge rebuilt at Tacoma is far stiffer, with an open framework of steel under its deck that lets the wind through.
A Swing Against a Sway
The plain way to take energy out is a damper: a cylinder of oil, like a car’s shock absorber, that turns motion into warmth. The Millennium Bridge reopened with 37 of them fitted under and beside its deck.
The cleverer way turns resonance against itself. Hang a heavy pendulum near the top of a tall building and tune it to the building’s own sway. When the building begins to sway, it moves the pendulum’s hanging point at the pendulum’s own rhythm, and the pendulum’s swing builds up. It swings a quarter of a beat behind the building, like the pendulum on the bar, so it is always pulling against the way the building is moving. Dampers on the pendulum turn its swinging into warmth in their oil. The sway goes into the pendulum and ends there.
Taipei 101, a tower of 101 floors in Taiwan, carries such a pendulum: a steel ball 5.5 meters across, weighing 660 metric tons, hung on eight cables between the 87th and the 92nd floors. During one typhoon it swung a meter from its center, the farthest it is known to have moved.
The Sharpest Notes
The fussiness that makes resonance dangerous also makes it useful. Something that answers only its own rhythm can pick that rhythm out of a crowd. Every station on the dial reaches a radio’s antenna at once. In the simplest radio, a coil and a capacitor that ring at one frequency answer one station strongly and the rest hardly at all, and turning the dial changes the frequency they ring at.
Something that keeps its own rhythm well keeps time. The pushes that keep a clock going always disturb it a little, and the longer its swinging part would ring on its own, the less they can pull it off its beat. A quartz watch keeps a tiny tuning fork of quartz ringing 32,768 times a second, sealed in a vacuum, where it would ring for many thousands of swings on its own. Halved fifteen times, its count gives one tick a second. The second itself is defined by a cesium atom: by the microwaves that switch it between two of its states, 9,192,631,770 waves to the second, a rhythm set by the energy between the two. The best clocks today use atoms switched by light, which waves hundreds of trillions of times a second or more, and are trusted to better than one part in 10¹⁸.
The rule that ties how long a thing rings to how sharply it answers holds even where nothing swings that anyone can see. In a large collider, electrons were made to meet their antimatter twins, positrons. They turned into other particles far more often at one collision energy than at any other: a peak about one thirty-sixth as wide as the energy it sits at. The peak is the Z boson, a particle that decays almost as soon as it forms. A peak that broad is what a bell would give if it rang only about a dozen times, and the Z lives about 2.6 × 10⁻²⁵ seconds. How wide a resonance is and how long it lasts are one fact, from a bell rope to a particle.
Where the Steady Beat Stops
Most of this page rests on one thing: that a swinging thing keeps the same rhythm however wide it swings. For small swings that is very nearly true. For big ones it is not. A pendulum’s beat slows as its swing widens, and a spring that stiffens as it stretches beats faster. Drive such a thing at a fixed rhythm, and as its swing grows its own rhythm drifts away from the drive, so the pushes slip out of step and the growth stops early. Over a range of rhythms the same steady drive can then keep either a small swing or a large one going, and which you get depends on what came before. Nudge the drive past the end of that range and the swing jumps suddenly from one to the other.
There is a way round it, in principle. A bob running along a curve called a cycloid takes the same time however wide it swings, and pendulums were once fitted with curved cheeks that bent the cord so the bob would follow one. The rubbing of the cord on the cheeks cost more than the curve gained, and clockmakers kept the swing small instead, to a few degrees. Drive a pendulum hard enough, at the right rhythm, and it can lose its beat altogether. It swings over the top and back in a pattern that never repeats, and a difference too small to measure in how it starts changes everything it does later. Resonance is what happens at the gentle end of that range.
