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 11 essays across 4 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

Named alongside it

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

ResonanceDissipationNormal modesQuality factorSimple harmonic motionAmplitudeInstabilityPhase lagRelaxationRestoring forceStabilityAbsorption

All concepts