Illustrated Physics
physics, drawn
Concept

Damping — where it appears

Loss that removes energy from an oscillation, deciding how sharply a system responds and how deep a modulation must be before anything grows. The ratio of the damping to the critical value separates a system that returns to rest without overshooting from one that rings, and its reciprocal is the quality factor.

Named by 19 essays across 6 fields — each of them below, with the objects they name alongside it.

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.

waves · Resonance 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.

waves · Resonance 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.

The 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.

mechanics · Pendulum A metre of tube, with and without the metal. The magnet's position against time over a 1.00 m drop, integrated from Newton's second law with the eddy-current drag included, and the same fall with the drag switched off. The braked descent takes 2.09 s against 0.45 s in free fall. After a transient of about 50 milliseconds — the mass divided by the drag coefficient, and the only timescale in the problem — the trace is a straight line, which is to say the magnet is falling at a constant 49.0 cm/s. Every joule of the 118 mJ of gravitational energy released has gone into resistive heating of the wall, and none into the magnet's kinetic energy, since that is the same at the bottom as it was a moment after the start.

The magnet that falls slowly

Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.

electromagnetism · Induction 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.

waves · Resonance The same steady pull, twice. Spring force divided by the normal load, against time, with the far end drawn away at 100 µm/s in both traces and every property of the contact the same. The soft holder at 0.4 k_c produces events every 3.75 s, each of which reaches 0.94 m/s — 9.4e+3 times the speed it is being pulled at. The stiff one at 1.6 k_c settles to a straight line and stays on it.

The chatter a stiffer holder removes

A brake squeals, a bow sounds a string, a fault slips in jerks. The usual explanation is that static friction exceeds kinetic friction — and that explanation, taken seriously, predicts the jerking would happen no matter how the thing were held. It does not, and what decides is a length nobody mentions.

mechanics · Friction Water's permittivity across six decades of frequency. The real and imaginary parts of water's permittivity against frequency, on logarithmic axes, from 10 to 16 in powers of ten hertz. ε′ begins at 80.1 — the number a textbook prints — falls through the Debye relaxation near twenty gigahertz, is dragged down again by the librational, bending and stretching bands of the molecule, and settles at 1.777 in the visible, whose square root is 1.3330: water's refractive index. The identity n = √ε_r is exact and is an identity between two numbers taken at the same frequency; using the static permittivity in it predicts an index of 8.9 and is the most instructive wrong answer in the subject. The curve is a Debye term and four Lorentz oscillators, with two parameters solved so that the two plateaus are the measured ones rather than fitted by eye.

The constant that depends on how fast it is asked

Water's relative permittivity is 80.1. Its refractive index is 1.333, and the square of 1.333 is 1.777. The identity n = √ε_r is exact, the two numbers differ by a factor of forty-five, and nothing is wrong with either — because a permittivity is a function of frequency and the two measurements were made eight orders of magnitude apart.

electromagnetism · Dielectrics 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.

waves · Resonance The energy a spin is allowed, and the only direction it can go. A body with principal moments 1.000, 3.000, 4.000 spinning with a fixed angular momentum. Every possible motion has an energy somewhere in the band drawn here, and the three marks are the three principal-axis spins: energy L²/2I, so the largest moment of inertia gives the smallest energy. The band runs from 0.1250 to 0.5000 in units where the momentum is one — a ratio of 4.0. Anything inside the body that flexes and warms takes energy out and leaves the momentum untouched, so the state can only move leftwards along this band, and there is exactly one place for it to stop. A spin about the axis of least inertia is at the far right: it is a perfectly good solution of the equations of motion, stable against small disturbances in a perfectly rigid body, and it sits at the top of a hill the smallest leak will roll it off.

The axis a leak of energy chooses

A body spinning with nothing pushing on it keeps its angular momentum exactly, and keeps its kinetic energy only while nothing inside it flexes. At fixed momentum the energy is least for a spin about the axis of greatest inertia — so any leak, however small, has a destination. The first American satellite found this out in orbit.

mechanics · Rotation 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.

waves · Resonance A few cycles, and everything about them is two numbers. The strain radiated by a remnant of 62 solar masses spinning at 0.68 of its maximum, as it settles down, with the decaying envelope of its fundamental mode drawn over it. The fundamental rings at 274 hertz and decays in 3.7 milliseconds, which is 1.0 cycles — this is not a bell and it does not sustain. Every frequency and every decay time in the sum is fixed by the mass and the spin alone; nothing about what made the remnant survives into them. What does depend on the collision is how loudly each mode is excited, and the relative amplitudes here are the rough values a merger of two comparable masses produces rather than a prediction.

A few cycles that are only mass and spin

After the orbit is gone there is one object left, distorted, and it settles down by radiating at frequencies that belong to it rather than to the collision. For a black hole those frequencies are fixed by the mass and the spin and by nothing else — so the first mode measured is a measurement and every mode after it is a test, and the test is that four curves in one plane pass through one point.

astrophysics · Gravitational waves The same thermal motion, spread three ways. How the thermal jiggling of a spring-mounted mass is distributed over frequency, for quality factors 3, 30, 300, with frequency in units of the resonance and the spectrum in units that make the area under each curve its mean-square displacement. All three curves enclose exactly the same area, kT/k — 1.000, 1.000, 1.000 of it in the units drawn — because equipartition fixes the energy in the spring and not how fast it changes. What the damping chooses is where that motion goes: a high Q concentrates it in a peak 100 times taller than the low-Q one, and leaves the rest of the spectrum 100 times quieter at 10 times the resonance frequency.

The 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.

thermodynamics · Equipartition The shear a pushed contact carries, and where it gives. Across a Hertzian contact of radius a, the normal pressure as a fraction of its peak, which is also the friction limit μp in units of μp₀ (dashed), and the shear traction for a sideways push of 0.3, 0.6, 0.9 times the load that would slide it, μP. Where the shear reaches the dashed ceiling the surfaces slip; inside, they stick. At 0.3 μP the stuck disc has a radius of 0.89a, so 21 per cent of the contact area is slipping; at 0.6 μP the stuck disc has a radius of 0.74a, so 46 per cent of the contact area is slipping; at 0.9 μP the stuck disc has a radius of 0.46a, so 78 per cent of the contact area is slipping. The shear is highest just inside the slipping ring and, in an elastic contact that simply stuck everywhere, would be infinite at the rim: slip at the edge is what removes that singularity.

The contact that slips before it slides

Static friction is supposed to hold perfectly until the load reaches μ times the normal force and then give way. Two elastic bodies pressed together do something else. Push a ball sideways on a flat with any force at all and a ring at the edge of their contact is already slipping, while the centre stays stuck. As the push grows the ring spreads inward, and gross sliding is only the moment the stuck centre shrinks to nothing. Shake the push back and forth and the ring slips each way on every cycle. The contact then loses energy though it never slides, as the cube of how hard it is shaken.

mechanics · Friction Two ways for a swing to die. A mass on a spring released 10.6 dead-band widths from rest, against time in units of 1/ω. Red: sliding on a dry surface, whose friction is a constant force; each swing is exactly a cosine, and the turning points fall in a straight line by two dead-band widths per half-cycle until the mass stops, 2.5 cycles after release, at −0.60. Blue: the same mass with viscous damping (damping ratio 0.05), whose force is proportional to speed; the swings shrink by the same fraction each cycle, along an exponential, and never stop. The period is the same in both and the same as with no damping at all to within the viscous case's 0.13 per cent.

The swing that dies in a straight line

A pendulum in air loses the same fraction of its swing every cycle, so its amplitude falls along an exponential and, in principle, never quite reaches zero. A mass on a spring sliding over a dry surface does something else. Its friction is a fixed force, so it loses a fixed amount of amplitude every half-swing — the swings shrink in a straight line — and it stops completely, after a countable number of swings, somewhere inside a band around its natural position whose width is the friction divided by the stiffness. Where in that band it stops depends on where it started, which is why a barometer with friction in it is tapped before it is read.

mechanics · Friction A governor that hunts. The speed of an engine with a centrifugal governor after its load is suddenly increased, against time in units of √(ℓ/g), for three amounts of friction in the governor's linkage: damping 0, 0.25, 0.6, against Vyshnegradsky's minimum for this governor, 0.397. With no friction the speed oscillates about its new, slightly lower value with a growing swing — the governor hunts, opening and closing the throttle out of step with the engine — until the arms hit their stops and the curve ends. At damping 0.25, below the minimum, it still hunts, more slowly. At damping 0.6, above the minimum, the oscillation dies away and the engine settles at its new speed. The friction that seems to spoil the mechanism is what makes it work.

The governor that needed friction to work

James Watt's steam engines were kept at a steady speed by two heavy balls on hinged arms, spun by the engine itself: run faster and the balls fly out and close the steam valve; run slower and they drop and open it. The balls stand at a height set by the speed alone. As engines grew and governors were made more sensitive, many began to hunt — speeding and slowing in a growing rhythm, the governor always too late. In 1868 James Clerk Maxwell wrote the equations down and showed why: a governor with no friction cannot settle at all, and the more precisely it is meant to hold the speed, the more friction it needs.

mechanics · Circular motion The suction that holds two wet plates together. The force needed to pull apart two discs 10 cm across, starting 10 micrometres apart with water between them, at constant opening speeds of 1, 10 and 100 micrometres a second, against time; the dashed line is the largest suction the water can transmit, ½πR²pₐₜₘ = 398 N, reached when the pressure at the centre falls to zero and the water cavitates. At 1 µm/s the force starts at 29.5 N and falls as the cube of the widening gap, to 10.7 N after four seconds, when the gap is 14 micrometres. At 100 µm/s the viscous law would demand 2.9 kN; the water cannot pull that hard, and the force sits on the cavitation limit until the gap has opened enough for the viscous suction to fall below it.

The wet plates that hold on when pulled

Lift a wet glass plate straight up off another and it resists with a force far beyond its weight; slide it sideways and it comes away at a touch. The grip is not surface tension. It is the water that must rush into the widening gap, and the suction it takes to draw it in grows as the inverse cube of the gap — until the water tears open and the atmosphere sets the limit. The same cube makes a flat base float on a wet table where a ball would settle, and gives a micro-machined accelerometer exactly the damping it needs from a few micrometres of air.

fluids · Viscosity A light panel that blocks one band as a heavy one would. Sound transmission loss at normal incidence against frequency, on a logarithmic frequency axis. Dashed: a limp panel of 2 kg/m², following the mass law, 13.4 dB at 300 Hz. Dotted: the same panel made 3 kg/m² by adding 1 kg/m² of plain mass, 16.8 dB. Solid: the 2 kg/m² panel with the same added mass hung on springs tuned to 300 Hz (damping 0.02). It blocks 35 dB at 300 Hz — the loss the mass law would give a wall 26 kg/m² heavy — and just above, at 367 Hz, falls to 5.2 dB, far below even the bare panel's 15.1: there the panel lets sound through more easily than if the resonators were not there.

The wall made heavy only at one note

A wall stops sound by being hard to shake, and how hard a plain wall is to shake is its weight: that is the mass law, and it makes quiet walls heavy. Hang small masses on springs inside a light panel, tuned to one frequency, and near that frequency the panel becomes many times harder to shake than it weighs, because the hidden masses swing against it. A two-kilogram panel then blocks a tone as a wall thirteen times heavier would. The price comes just above the tuning, where the panel's dynamic mass passes through zero and it lets sound through more easily than if the masses were not there.

waves · Impedance The power a cavity holds depends on where its mirror is. The power circulating in a cavity of finesse 10,000, a metre long and lit at 1064 nm, as a fraction of its value on resonance, against the detuning of the laser from the cavity's resonance in units of the cavity's half-linewidth (7.5 kHz). Lengthening the cavity lowers its resonance and moves every point of the curve to the right: 38 half-linewidths for each nanometre of mirror travel. On the blue side (laser above resonance, right) that motion loses power, so the radiation force on the mirror falls as the mirror moves out — a restoring spring, steepest at δ/γ = 0.58 (dashed tangent). On the red side the same motion gains power and pushes harder: a spring of the opposite sign.

The mirror held on a spring of light

Light bouncing between two mirrors pushes them apart, and how hard it pushes depends on how long the gap is, because the gap decides how much light the cavity holds. A mirror free to move therefore sits on a spring made of light: with a kilowatt stored in a metre-long cavity it is four-fifths as stiff as a steel wire a millimetre thick. The spring has a flaw no steel spring has. Stored light takes time to rise and fall, so the push always arrives a little late, and a late spring either winds the mirror up or calms it, depending only on which side of the cavity's resonance the laser sits.

astrophysics · Radiation pressure The same damped motion in two phase planes. A cloud of damped oscillators that start on a small circle of positions and velocities, drawn at times 0, 2, 4, 6 (outlines) with the centre's path. Left: position against the physical momentum mẋ. The cloud spirals inward and shrinks: its area falls as e^(−γt), to 0.30 of its start by t = 6, as friction squeezes every motion towards rest. Right: position against the canonical momentum of the Caldirola–Kanai Lagrangian, p = e^(γt)mẋ. The same cloud keeps exactly its starting area while it is stretched out of shape, as Liouville's theorem requires of any Hamiltonian flow. The friction has been hidden in the definition of momentum: the canonical momentum grows exactly as fast as the real one decays.

The friction an action can hold only by changing its clock

The principle of least action generates every conservative law of motion from a single integral, and it has no place for friction: there is no ordinary Lagrangian whose equation of motion is a damped oscillator's. Multiply the Lagrangian by an exponential of time and friction appears — but only by redefining momentum so that it grows as fast as the real momentum decays. The phase-space area the principle conserves is then no longer the one that counts motions, and quantised, the damped oscillator becomes more sharply known than the uncertainty principle allows. Friction belongs to a system and the surroundings it loses energy to, and an action for the system alone can only fake it.

mechanics · Least action

Named alongside it

The objects these essays reach for when they reach for this one.

ResonanceQuality factorDissipationFrictionNormal modesSimple harmonic motionStabilityAmplitudeFeedbackFluctuation dissipationHysteresisInstability

All concepts