Resonance
Push a swing at the wrong moment and nothing happens. Push it at the right moment, over and over, and a child's arm lifts a body twice its weight into a three metre arc. Nothing about the pushes changed. Only their timing did. That trade, small effort arriving in step against large motion coming out, holds its shape all the way up the scale: a wine glass rim near 556 hertz, hydrogen inside a hospital magnet at 64 megahertz, and a caesium atom whose 9,192,631,770 beats a second are what a second now means.
The whole idea is in the swing
A playground swing has a rhythm of its own and will not be talked out of it. That rhythm comes from the length of the ropes and from gravity, and from almost nothing else. Two and a half metres of rope gives a round trip of about 3.2 seconds, which is 0.32 hertz, roughly a third of a trip every second. Put a heavier child on the same ropes and the rhythm does not change, because the extra weight pulling harder and the extra mass resisting being moved cancel each other out exactly. Shorten the ropes and it quickens. The size of the arc is the one that does bite: swing through a three metre arc on those ropes and the round trip stretches by about two per cent, which is why a swing pushed hard drifts slowly out of step with a pusher keeping strict time.
So the swing sets the timetable and you either meet it or you do not. Push as the seat is running away from you and your push adds to what is already there. Push as it comes back at you and you take the same amount out again. Halfway between, you do nothing at all. The strongest place to push is the bottom of the arc, where the seat is moving fastest, and that moment sits a quarter of a cycle away from the turning points at either end. Timing, not strength, is what makes the arc big.
Two more things follow from the same rope. A swing does not reach its full arc on the first push: it builds, adding a little on each trip until what you put in each cycle equals what the air and the bearings take out. And when you stop, it does not halt. It keeps going, smaller each trip, for a while.
Drive it yourself
Below is a mass on a spring whose own rhythm is one round trip a second. The top of the spring is being shaken up and down by a driver, and the two sliders set how fast the driver shakes and how much friction the rig carries. On the right, the curve is the answer the rig settles on for every drive speed at the current friction setting, drawn on a scale where each gridline is ten times the last.
The filled dot is what the rig is doing right now. The hollow ring on the curve is where it is heading. Watch the dot climb: on a low-friction setting it takes many seconds of pushing to get there, which is the build-up you feel in a swing made visible.
Three things worth finding with the sliders
Slower than its rhythm
The mass simply goes wherever the driver goes, moving together with it, arriving at the same time. There is no gain and no lag. Anything pushed far slower than its own rhythm behaves like a rigid stick.
Right on its rhythm
The arc climbs to many times the driver's movement, and with the friction light the mass settles close to a quarter of a cycle behind: the driver is moving fastest just as the mass reaches the end of its travel. Take friction out and the peak grows taller and narrower at the same time.
Faster than its rhythm
The mass cannot keep up and barely moves while the driver rattles away underneath it. That is not a failure. It is how a rubber engine mount, a camera stabiliser and a seismometer all work: hang the thing you care about below its own rhythm and the shaking stops arriving.
Every resonator carries a sharpness number
Engineers write that number as Q, or quality factor, and one number settles two separate questions. Divide it into the resonator's note and you get the width of the band of frequencies it answers to. Divide it by about three and you get the number of cycles it keeps going after the pushing stops, before its swing is down to a third of what it was.
So a blunt resonator answers to a wide band and dies quickly, and a sharp one answers to a hair's breadth and rings for a very long time. The two go together, and that is why the same number turns up in the design of a car's suspension, a radio's tuning and an atomic clock.
| Resonator | Sharpness | Cycles it keeps | Band it answers to | Why it sits there |
|---|---|---|---|---|
| A car's suspension | about 2 | under one | about half its note | set deliberately blunt: a car that kept bouncing over every bump would be unusable |
| A playground swing | tens | tens | a few per cent | rope friction and air drag, and a rider who is not trying to stop |
| A guitar string | about 1,000 | a few hundred | a tenth of a per cent | most of what it loses goes into the soundboard on purpose, which is the point of a guitar |
| A crystal wine glass rim | low thousands | several hundred | under a tenth of a per cent | glass is stiff and internally quiet, so a flicked rim rings for a second or two |
| A quartz watch crystal | around 100,000 | tens of thousands | a thousandth of a per cent | a tuning fork cut from a single crystal, sealed in vacuum, cut along a direction where temperature barely matters |
| A niobium accelerator cavity at 2 kelvin, two degrees above absolute zero, the bottom of the temperature scale | 3 × 1010 | 10 thousand million | 4 hundredths of a hertz out of 1.3 gigahertz | superconducting walls lose almost nothing, so microwaves stored in it fade over about 3.7 seconds |
| Strontium-87's clock transition | 4 × 1017 | 1017 | a millihertz out of 429 terahertz | a transition the atom is barely allowed to make at all, so it takes its time about making it |
The last row is worth sitting with. Counting to four hundred million billion at one a second would take about 13.6 billion years, close to the age of the universe. That is the ratio between the note strontium-87 sings and the width of the note. Jonathan Muniz, Dylan Young, Jarrod Cline and James Thompson at JILA in Colorado measured that width in 2021 by putting the atoms between two mirrors and listening to how they and the trapped light answer each other, resolving it to 30 millionths of a hertz.
The niobium figure comes off a production line rather than a specialist bench. Anna Grassellino and colleagues at Fermilab reported quality factors near 3 × 1010 in 1.3 gigahertz nine-cell cavities in 2015, cavities built for the LCLS-II light source and now running there. Fill one with microwaves, switch off the source, and the energy is still measurably sloshing about three and a half seconds later, having gone through close to five thousand million cycles on the way.
Sixteen steps of ten, one idea
The ladder below runs from a tenth of a hertz, slow enough to watch with the naked eye, up to a thousand million million hertz, which is red light. Sixteen steps of ten is a long way: the same jump takes you from a grain of sand to a distance out past Neptune's orbit. Every rung on it is the same arrangement, something with a rhythm of its own being pushed at that rhythm.
- What is resonating
- What it is used for
- For scale
A glass sings its own note
Run a wet finger round the rim of a wine glass and it hums. What is happening is that the rim is being pulled out of round, into an oval, and springing back, four times per cycle at four points around the circle. The pitch depends on how wide the rim is, how thin the glass is and how much wine is in it: fill the glass and the note drops, because the wine has to be shoved along with the glass.
Sing that note back at the glass loudly enough and the same flexing builds. The MythBusters team measured a crystal glass at about 556 hertz and, in a 2005 episode, the singer and vocal coach Jaime Vendera broke one with an unamplified voice at around 105 decibels, held for two to three seconds. At 556 hertz that is roughly fourteen hundred pushes arriving in step, and the arc built until the glass could not take the strain.
It is a fussy trick, which is exactly what the sharpness number predicts. A good glass has a Q in the low thousands, so it answers to a band well under a tenth of a per cent wide: sing a semitone off, about six per cent, and the rim still answers, at roughly a hundredth of its on-note swing for a Q of a thousand and a few thousandths of it for a Q of three thousand. That is a real movement, and it is nowhere near the strain that parts the glass. The demonstration is a test of pitch accuracy first and lung power second.
What the wind actually did at Tacoma Narrows
The Tacoma Narrows Bridge opened across Puget Sound on 1 July 1940 with a main span of 853 metres, at the time the third longest in the world. It was also strikingly slender: a deck about 11.9 metres wide carried on solid plate girders 2.4 metres deep, rather than the open trusses used on earlier long spans. It moved from the first day. Drivers reported the roadway rising and falling ahead of them, and it picked up a nickname.
On the morning of 7 November 1940, in a wind of about 68 kilometres an hour, the motion changed character. Instead of rippling up and down along its length, the deck began to twist, one side rising as the other fell, about one twist every five seconds. The edges of the roadway were travelling something like eight and a half metres up and down. It held that for around an hour and then came apart. Nobody was killed. A dog left in a car on the deck was.
A wind that kept time
The version in a great many textbooks is that eddies peeled off the deck at a steady rate, that rate happened to match the bridge's own rhythm, and the bridge was driven in step until it tore. It is a tidy story and it uses the right idea in the wrong place.
What the numbers say instead
Yusuf Billah and Robert Scanlan set out the arithmetic. The eddies shed by that deck at that wind speed came off near one hertz, about five times faster than the 0.2 hertz twist. There was no drumbeat at the bridge's own rate to be in step with.
What happened instead has a different shape, and it is the more interesting one. Once the deck starts twisting, its blunt edges tilt into the airflow, the flow separates off them, and the pressure that comes back depends on how fast the deck is already twisting. Beyond a certain wind speed the air is feeding energy in faster than the steel takes it out. The system's damping, the thing that normally makes a motion die away, goes past zero and turns negative. From there the motion grows on its own, with no timekeeper involved anywhere.
Engineers call it aeroelastic flutter. It is self-feeding, not forced. The wind was not a metronome; it was a negative brake. That distinction matters because the two failures have different cures: a forced resonance is fixed by moving the structure's note away from the drive, and flutter is fixed by changing the shape of the deck so the air stops handing energy back.
Why soldiers break step
On 12 April 1831 a detachment of 74 men of the 60th Rifle Corps marched back to barracks over the Broughton Suspension Bridge at Salford, four abreast. They felt the deck bouncing in time with their feet, and some of them found it entertaining enough to whistle along. A bolt at one of the chain anchorages gave way and about forty men went into the River Irwell. Twenty were hurt, some badly. Nobody drowned. The British Army issued the order to break step on bridges afterwards, and armies have kept it since.
Nineteen years later and rather worse: on 16 April 1850 a battalion crossing the Basse-Chaine suspension bridge at Angers in France went into the Maine when the bridge failed during a thunderstorm. 226 died. The inquiry put the failure on the storm, the loading, and corroded anchor cables, with the marching one factor among several rather than the cause on its own.
The modern instance of this has no soldiers in it at all, and it taught the profession something the 1831 story does not.
London, 10 June 2000
The Millennium Bridge opened as a footbridge across the Thames and swayed sideways with people on it. With something like two thousand walkers on the deck, the southern span moved at about 0.8 hertz and the central span at about 0.5 hertz, with sideways movements reaching around 70 millimetres. It closed two days later.
A walking person puts a foot down about twice a second, and each footfall pushes the body sideways as well as down, so the sideways push comes at about one per second. That is near the deck's rate but not equal to it, and the arithmetic of a simple forced resonance does not close. What Pat Dallard and the Arup team found instead was a loop with people inside it. A person on a deck that is rolling underfoot adjusts their gait to keep balance, and the easiest adjustment is to fall into time with the roll. So a small movement recruits walkers, the recruited walkers make the movement bigger, and the bigger movement recruits more of them. Above a critical number of people on the deck the loop closes on itself.
A feedback loop with people in it, not a crowd that happened to be in step. The crowd did not arrive marching. The bridge taught them.
The fix went in over the following year and a half: 37 viscous dampers, the kind that turn motion into warm oil, and 52 tuned mass dampers, which are weights on springs set to swing against the deck. Together they lifted the damping from about half a per cent of critical to around twenty per cent, and the bridge reopened in February 2002. It has carried crowds since without the wobble.
Footbridge design codes now carry a lateral pedestrian check because of it, which is the useful legacy: a class of loading that was not in anybody's calculation before 2000 is in everybody's calculation now.
A microwave oven does not hit water's note
The story most of us were told is that a microwave oven runs at the resonant frequency of water, and that is why it heats food and not the plate. Everything in that sentence is right except the resonance, and the actual mechanism is a better story anyway.
A water molecule carries a lopsided charge: the oxygen end is slightly negative, the two hydrogen ends slightly positive. An electric field pointing one way pulls it round to line up. The oven's magnetron makes a field at 2.45 gigahertz, which means it reverses direction 2.45 thousand million times a second, about as many reversals in one second as there are seconds in a 77 year life. The molecules start to turn to follow it, collide with their neighbours before they finish turning, and pass the energy on as jostling. The jostling is the heat.
An isolated water molecule, in vapour, genuinely does have sharp rotational notes, and they are nowhere near 2.45 gigahertz. The well-known ones sit at 22.235 gigahertz, 183 gigahertz and 557 gigahertz. Radio astronomers use the 22 gigahertz line to find water in star-forming clouds, and weather satellites use the 183 gigahertz line to weigh water vapour in the atmosphere. In liquid water those sharp lines are gone, because every molecule is hydrogen-bonded to its neighbours and gets knocked before it completes a turn. What is left is a broad hump in how strongly liquid water absorbs, peaking near 20 gigahertz at room temperature. The oven runs a factor of eight below the top of that hump.
Running below it is the design choice, not an accident. At the top of the hump the energy would be swallowed in the first few millimetres and a chop would be scorched outside and cold inside. At 2.45 gigahertz the depth at which the power has dropped to about a third is roughly five to fifteen millimetres in food, depending on what the food is, how salty it is and how warm it already is. Heating still runs outside in, but far enough in to be useful. Industrial microwave lines in the United States commonly run at 915 megahertz, where the penetration is roughly three times deeper again, which is exactly the same trade made for bigger pieces. A frequency chosen for how far it reaches into a roast dinner and for what the radio rules allow, not a note the water sings.
Nuclei that sing in a magnet
A hydrogen nucleus is a single proton, and it behaves like a very small magnet that is also spinning. Put it in a strong magnetic field and it does not simply swing round and point along the field, the way a compass needle does. It wheels around the field direction like a leaning spinning top, and the rate at which it wheels depends on the field strength and on what sort of nucleus it is, and on almost nothing else.
For hydrogen that rate is 42.58 megahertz for every tesla of field. A tesla is the unit magnetic field strength comes in, and the Earth's own field is about one twenty-thousandth of one. Australian hospital scanners mostly run at 1.5 or 3 tesla, which puts hydrogen at 63.9 megahertz and 127.7 megahertz. Those two numbers sit either side of the FM radio band, one just below 87.5 and one just above 108. Counting the 1.5 tesla wheeling at one a second would take about two years.
Send in a radio pulse at exactly that frequency and the nuclei tip over together, in step. Little else in the body answers, because sodium, phosphorus and carbon all wheel at different rates in the same field. Switch the pulse off and the hydrogen wheels back towards the field, giving off a faint radio note at the same frequency as it goes, and how quickly that note fades depends on what the water molecule it belongs to is sitting in. Fat, muscle, grey matter and a tumour all fade differently, which is the contrast in the picture.
Turning that into an image takes one more trick. Add a small extra field that grows steadily along the length of the body, and now the note is different at every position, so the pitch of the returning signal says where it came from. That is Paul Lauterbur's step, published in Nature in 1973, with Peter Mansfield adding the fast readout that made scans a matter of minutes. The pair took the 2003 Nobel Prize in Physiology or Medicine for it. Raymond Damadian had reported in Science in 1971 that tumour tissue and normal tissue relax at measurably different rates. The physics underneath goes back to Isidor Rabi's molecular beam work in 1938 and to Felix Bloch at Stanford and Edward Purcell at Harvard, who both detected the effect in bulk matter in 1946 and shared the 1952 Nobel Prize in Physics.
Sound in a crystal, counted
A crystal is atoms sitting on a regular grid, each held to its neighbours by bonds that behave a good deal like springs. Push one atom and the push travels to the next, and to the next: that is sound, and inside a solid it moves fast. Because the grid is regular and the crystal has edges, only certain patterns of vibration fit across it, which is the guitar string rule applied in three dimensions.
Quantum mechanics adds one thing. Each of those vibration patterns gains and loses energy in whole steps, and the size of one step is that pattern's frequency multiplied by Planck's constant, the small fixed number that sets the size of every quantum step. That is the premise the counting rests on, and it is what makes counting work at all. It turns out to be far easier to keep track of the steps as though they were particles moving about, so that is what physicists do. A phonon is one step of one vibration pattern: a unit of sound in a solid, counted the way light gets counted in photons. Nothing is being claimed about tiny balls flying around. It is a bookkeeping choice that happens to make the sums work.
The reason to bother is heat. In a metal, most heat rides on the loose electrons, which is why a steel spoon in hot soup burns your fingers. In an insulator or a semiconductor there are few loose electrons available, so heat has to travel as vibration, which means as phonons. Diamond does this better than any other bulk material in ordinary use at room temperature: about 2,000 watts through each metre for every degree Celsius of temperature difference, which a datasheet writes as 2,000 watts per metre per kelvin, since a step of one kelvin and a step of one degree Celsius are the same size. That is roughly five times copper, while being an electrical insulator. Carbon atoms are light and their bonds are stiff, so the vibrations run at around 14 kilometres a second averaged over the modes, some forty times the speed of sound in air, and they travel a long way before anything scatters them.
The highest note a diamond lattice will hold is about 39.9 terahertz, which chemists see as a single sharp line in scattered laser light, the fingerprint they use to identify diamond. Counting those vibrations at one a second would take 1.3 million years.
The second is a resonance
A caesium-133 atom has one electron in its outermost shell, and both that electron and the nucleus behave like small magnets. The two can sit either lined up or opposed, and the energy difference between those two arrangements is very small. Shine radio waves of exactly the right frequency on the atom and it flips from one arrangement to the other. That frequency is 9,192,631,770 hertz. Counting the beats at one a second would take 291 years.
Since the 13th General Conference on Weights and Measures in 1967, that is the definition of the second: 9,192,631,770 periods of that radiation. Not a fraction of a day, because the Earth's spin wanders by a millisecond here and there. A definition, not a measurement. Caesium does not have that frequency because somebody carefully measured it. The hertz is now defined so that it does, and other frequencies are compared back to it.
Louis Essen and Jack Parry built the first practical caesium standard at the National Physical Laboratory in 1955, good to roughly a second in three hundred years, which was about thirty times better than the quartz clocks of the day. A modern caesium fountain works differently: it gathers a ball of atoms, cools them with laser light until they are barely moving, tosses the ball up through a microwave cavity and catches it coming back down, so the atoms spend about a second in the field on the way through. A longer look gives a narrower answer, and the resonance comes back about a hertz wide out of 9.19 gigahertz. The evaluated uncertainty of one such fountain, NIST-F2 in Colorado, is around one part in 1016.
Light that agrees with itself
A laser is a resonance in a box made of mirrors. Two mirrors face each other, a material that can amplify light sits between them, and light bounces back and forth. Only the wavelengths that fit a whole number of half waves between the mirrors come back in step with themselves after a round trip and grow. Everything else cancels itself out and dies.
A helium neon tube 30 centimetres long holds about 948,000 half waves of red light between its mirrors, and the frequencies that fit are spaced 500 megahertz apart. The red line itself is at 632.8 nanometres, which is 473.8 million million hertz. Counting those crests at one a second would take 15 million years.
Two separate resonances have to agree for anything to come out. The neon atoms will only amplify a narrow set of wavelengths, set by the energy steps inside the atom. The cavity will only sustain wavelengths that fit between its mirrors. The laser runs where the two lists overlap, which is why a laser's colour is so precise and why nudging one mirror by a fraction of a wavelength changes what it does.
What would move the question
Six measurements, each with instruments that exist and a number waiting at the end.
Put a number on the crowd loop
The lateral pedestrian factor in current footbridge codes rests on a small number of instrumented structures. Force plates on a deck that can be driven sideways under a paying crowd, with overhead cameras tracking gait, would return how strongly people lock on as a function of how much the deck is already moving. That curve is what the codes are approximating.
Check flutter at full scale
Critical wind speeds for long spans come from deck sections tested at model scale in a tunnel. Anemometers and accelerometers left on a finished bridge for a decade would let the predicted onset speed be compared against the same deck at full size, in real turbulence, with real traffic on it.
Cross the phonon's free path
Measure thermal conductivity in a thin membrane as the spacing of a drilled pattern is stepped down through the distance a phonon travels between scatters. Where the curve bends tells you how much of heat flow is carried by which vibration wavelengths, which is the missing input for designing thermoelectrics on purpose rather than by trial.
Race the optical clocks
Compare strontium, ytterbium and aluminium ion clocks against each other at the 10-18 level over years. Each pair tests a different combination of the constants of nature, so a slow drift in any of them would show up as a ratio that will not sit still.
Watch a glass break properly
Drive a wine glass rim to failure with a loudspeaker while a laser vibrometer maps how the whole rim is flexing and a high-speed camera runs at a hundred thousand frames a second. It would settle whether the crack starts where the flexing is greatest or at whatever flaw the glass already carried, which is the same question as fatigue in a bridge.
Drive a resonator in its ground state
Membranes and drums of a few micrometres have been cooled until they hold almost no vibration at all, and then driven. Doing the same with a gram-scale mirror would put an everyday object's resonance curve into the range where the measurement itself disturbs it, which is where the interesting arguments about quantum mechanics and gravity are waiting.
The pattern holds from a rope swing to a strontium atom, over sixteen steps of ten, without anyone having to change the idea. The open work is not whether resonance is real. It is that the sharpest resonators anybody has built are now sharp enough to notice things nobody has been able to look for before: whether the constants drift, whether a heavy object can be held still enough to ask quantum questions of it, and how far into a crowd or a wind or a lattice the same simple loop reaches before something else takes over.
Keep exploring
Cymatics
Drive a plate at one of its own notes and sand collects along the lines that are standing still. The same standing wave that makes hot and cold patches in an oven, drawn where you can see it.
Crystal Lab
Build a lattice and shake it. Which patterns of vibration fit across a repeating grid is the question that turns into phonons, heat flow and the colour of a gemstone.
Zero Point
Only certain waves fit between two mirrors, and fewer fit inside the gap than outside it. That difference is measured in piconewtons on a bench.
Sources
- K. Y. Billah and R. H. Scanlan, "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks", American Journal of Physics 59, 118, 1991.
- P. Dallard, A. J. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. M. Ridsdill Smith and M. Willford, "The London Millennium Footbridge", The Structural Engineer 79 (22), 2001.
- MythBusters, "Breaking Glass", Discovery Channel, 2005, with Jaime Vendera.
- Resolution 1 of the 13th General Conference on Weights and Measures, 1967, defining the second · bipm.org/en/committees/cg/cgpm/13-1967/resolution-1
- L. Essen and J. V. L. Parry, "The caesium resonator as a standard of frequency and time", Philosophical Transactions of the Royal Society A 250, 45, 1957 · royalsocietypublishing.org/doi/10.1098/rsta.1957.0010
- T. P. Heavner et al., "First accuracy evaluation of NIST-F2", Metrologia 51, 174, 2014.
- J. A. Muniz, D. J. Young, J. R. K. Cline and J. K. Thompson, cavity quantum electrodynamics determination of the natural linewidth of the strontium-87 millihertz clock transition, resolved to 30 microhertz, 2020 to 2021 · arxiv.org/abs/2007.07220
- A. Grassellino et al., "Nitrogen-doped 9-cell cavity performance in a test cryomodule for LCLS-II", Journal of Applied Physics 117, 023908, 2015 · arxiv.org/abs/1411.1659
- I. I. Rabi, J. R. Zacharias, S. Millman and P. Kusch, "A new method of measuring nuclear magnetic moment", Physical Review 53, 318, 1938; F. Bloch, W. W. Hansen and M. Packard, and independently E. M. Purcell, H. C. Torrey and R. V. Pound, Physical Review 69 and 70, 1946.
- R. Damadian, "Tumor detection by nuclear magnetic resonance", Science 171, 1151, 1971; P. C. Lauterbur, "Image formation by induced local interactions: examples employing nuclear magnetic resonance", Nature 242, 190, 1973.
- W. J. Ellison, "Permittivity of pure water at standard atmospheric pressure over the frequency range 0 to 25 THz and the temperature range 0 to 100 °C", Journal of Physical and Chemical Reference Data 36, 1, 2007.
- J. Tang, "Unlocking potentials of microwaves for food safety and quality", Journal of Food Science 80, E1776, 2015 · penetration depths at 915 and 2,450 megahertz.
- Contemporary press reports of the Broughton Suspension Bridge collapse, Salford, 12 April 1831; the French official inquiry into the Basse-Chaine bridge at Angers, 16 April 1850.
- Taipei Financial Center Corporation, published specifications for the Taipei 101 tuned mass damper; Honshu-Shikoku Bridge Authority, deck design of the Akashi Kaikyo Bridge.