One equation, and the number that decides what it does
At its defaults it draws one equation, and the number that decides what it does. The same oscillator released from rest at one unit of displacement, with 4 different amounts of velocity-proportional loss. Every curve is a numerical integration of a single equation with no case analysis in it; what changes between them is one dimensionless number, the damping ratio. Below one, the two exponents are a complex conjugate pair and the motion crosses zero over and over inside a decaying envelope. At one they collide into a real double root and the trace reaches the axis and stops. Above one they are two distinct real numbers, the slower of them dominates, and the return is sluggish — an overdamped system takes longer to come back than a critically damped one, which is the thing the word 'over' is doing in its name. The ratios drawn are 0.15, 0.5, 1, 2, and the first zero crossing happens at 1.75 s for ζ = 0.15, 2.42 s for ζ = 0.5, never for ζ = 1, never for ζ = 2.
damped-response is one function in lib/figures/mechanics.js —
motion, force, energy and rotation. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
The same oscillator released from rest at one unit of displacement, with 4 different amounts of velocity-proportional loss. Every curve is a numerical integration of a single equation with no case analysis in it; what changes between them is one dimensionless number, the damping ratio. Below one, the two exponents are a complex conjugate pair and the motion crosses zero over and over inside a decaying envelope. At one they collide into a real double root and the trace reaches the axis and stops. Above one they are two distinct real numbers, the slower of them dominates, and the return is sluggish — an overdamped system takes longer to come back than a critically damped one, which is the thing the word 'over' is doing in its name. The ratios drawn are 0.15, 0.5, 1, 2, and the first zero crossing happens at 1.75 s for ζ = 0.15, 2.42 s for ζ = 0.5, never for ζ = 1, never for ζ = 2.
A clock that takes the drive's frequency over a whole range
The options are the ones The clock that is pulled into step passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The frequency a self-sustained oscillator actually runs at, relative to the frequency of a weak drive, against how far its own natural frequency is from the drive's, both in units of the locking strength K. Without the drive it would lie on the dashed diagonal, keeping its own frequency. With it, every natural frequency within ±K of the drive's is pulled all the way in: the oscillator runs at exactly the drive's frequency, a flat plateau where it has abandoned its own. Outside the plateau it runs free again but not at its own frequency; it is pulled towards the drive, its offset shrinking from the detuning to the square root of (detuning squared minus K squared) — 0.66 K instead of 1.2 K just outside, 2.83 K instead of 3 K further out.
Just outside the locking range, the phase slips in jumps
The options are the ones The clock that is pulled into step passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The phase of a driven oscillator relative to its drive over time, for natural frequencies 0.8, 1.05, 1.5 K from the drive's. Inside the locking range (0.8 K) the phase settles at a fixed lag, 0.93 rad, and stays there. Just outside it (1.05 K) the phase lingers for a long time near a quarter turn, where the drive almost holds it, then slips a whole turn quickly and lingers again: a staircase, one step every 19.6 time units. Further out (1.5 K) the steps blur into a steady climb at 1.12 K, slower than the 1.5 K it would have without the drive. A listener to the beat between the two hears pulses, not a smooth tone, near the edge of locking.
The locking range widens with the push and narrows with the Q
The options are the ones The clock that is pulled into step passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The range of natural frequencies, as a fraction of the frequency itself, over which an oscillator locks to a drive, against the drive's strength relative to the oscillator's own amplitude, for quality factors 10, 100, 1000 — Adler's half-width, the natural frequency over 2Q times the drive's amplitude over the oscillator's. Each region is a wedge, widening in proportion to the drive: with a drive a twentieth of the oscillator's own amplitude, the lock extends 0.25 per cent either side at Q = 10, 0.025 per cent at Q = 100 and 25 parts per million at Q = 1000. A sharply tuned oscillator is hard to pull, because its own resonance holds its frequency; a loosely tuned one follows almost any nearby push. The wedges are the simplest of the regions called Arnold tongues.
A body clock pulled back into step by the daylight
The options are the ones The clock that is pulled into step passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The offset between an internal daily clock and local daylight after a sudden shift of 3, 6, 9, 11.5 hours, if the daylight can pull the clock by up to 2 hours a day: Adler's equation with the day as the drive. After 3 hours it takes 4.8 days to come within a quarter of an hour; after 6 hours it takes 6.5 days to come within a quarter of an hour; after 9 hours it takes 8.2 days to come within a quarter of an hour; after 11.5 hours it takes 11.7 days to come within a quarter of an hour. Near half a day of shift the recovery slows dramatically: the clock sits close to the unstable point where the daylight's pull on it is nearly zero, and it lingers there before sliding one way or the other. A shift of eleven and a half hours takes nearly twice as long to recover from as one of six, although it is less than twice as large — a feature of the equation, whose pull vanishes at half a day, and one that the real body clock, with its own asymmetries, shares only roughly.
Two clocks that pull each other
The options are the ones The clock that is pulled into step passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The phase difference between two self-sustained oscillators that pull on each other with equal strength k = 0.5, whose own frequencies differ by 0.8 and 1.2 in the same units. Neither is the master; each is drawn towards the other, and their difference obeys Adler's equation with twice the coupling. With a spread of 0.8, less than 2k, they lock at a fixed phase difference of 0.927 rad and run at the average of their frequencies. With 1.2, more than 2k, they slip, a turn every 9.5 time units. Christiaan Huygens saw two of his pendulum clocks, hung from one beam, fall into step in 1665 and called it an odd kind of sympathy; the beam's slight give was the coupling.
What checks it
physicscheck asserts something about damped-response that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
ThermodynamicsThe noise a high Q moves out of the way
Anything held by a spring at a temperature jiggles, and equipartition says exactly how much: the mean square of its displacement is kT divided by the spring's stiffness, whatever the spring is made of and however it is damped. That looks like a floor no measurement can go below. It is a floor on the total, and the total can be moved. An oscillator with little damping puts almost all of its thermal motion into a narrow peak at its own frequency and leaves every other frequency quiet — which is why the mirrors of gravitational-wave detectors hang from fibres of glass that ring for minutes.
WavesThe 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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
MechanicsThe three ways of coming to rest
An oscillator with friction in it can swing and fade, arrive and stop, or crawl back so slowly it looks stuck. One equation and one number decide which. The number is not what most people would guess, and neither is the value that returns fastest.
WavesThe 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.
WavesThe 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.