Illustrated Physics
physics, drawn
Concept

Cavity — where it appears

A region bounded so that waves reflect back into it, supporting only those frequencies whose round trip is a whole number of wavelengths. How sharply it selects them is its quality factor, set by how little leaks out per round trip rather than by its size.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

What a thermal camera reads off a shiny surface. The temperature a camera calibrated for a black body reports, against the emissivity of the surface it is pointed at, for surfaces truly at 323 K, 373 K, 473 K in a room at 293 K. Every curve begins at the true temperature when the emissivity is one and ends at the room's temperature when it is zero, because a surface that emits nothing reflects everything and the camera is then looking at the room. polished aluminium at 0.05 reads 312 K; stainless steel, oxidised at 0.8 reads 451 K; matt black paint at 0.95 reads 468 K; human skin at 0.98 reads 471 K for a surface truly at 473 K. That is not a fault in the instrument. It follows from Kirchhoff's law — a poor emitter is a poor absorber and therefore a good reflector — so the shortfall in a shiny surface's own glow is made up almost exactly by whatever is reflected in it, and no measurement of the light leaving a surface can separate the two without knowing the emissivity in advance.

The glow that says nothing about the surface

A thermal camera pointed at a saucepan of boiling water reads a hundred degrees. Pointed at a polished aluminium block at the same temperature it reads about twenty-five, and the instrument is working perfectly. What it is measuring is emissivity as much as temperature — and a surface's emissivity is forced to equal its absorptivity, at every wavelength and every angle, by an argument with no physics of matter in it at all.

thermodynamics · Blackbody The one frequency a broken repeat lets through. Transmission through a quarter-wave stack whose repeat is broken once, on a logarithmic scale, against frequency in units of the quarter-wave design frequency. The perfect stack forbids this whole band; adding a single half-wave layer at the centre opens one line inside it, at exactly the middle of the gap, through which the stack transmits everything. With 4 pairs either side the line is 2.50e-3 wide, a quality factor of 400, on a background of 4.5e-4; With 6 pairs either side the line is 1.56e-4 wide, a quality factor of 6410, on a background of 6.3e-6; With 8 pairs either side the line is 1.00e-5 wide, a quality factor of 100000, on a background of 9.2e-8. The line narrows geometrically as the mirrors are made thicker, because the field inside the defect leaks out through a barrier whose transmission falls exponentially with its thickness — so the useful quantity of a band gap turns out to be not what it excludes but how well it can trap what a single flaw is allowed to hold.

The mode that lives in the mistake

A perfect stack of alternating layers refuses a whole band of frequencies — not weakly, not with loss, but not at all. Break the repeat once, by inserting a single layer of the wrong thickness, and exactly one frequency inside that band passes through the whole stack with a transmittance of one. The useful thing about a forbidden band turns out to be not what it excludes but what a single flaw is thereby allowed to hold.

waves · Periodic media Where a small weight is felt, and where it is not. The fractional change in the frequency of harmonic 3 of a stretched string when a point mass of 0.06 of the string's own mass is placed at each position along it. The solid curve is the exact answer, found by solving the string's frequency equation for the loaded string at each position; the dashed curve is the first-order prediction, minus the mass fraction times the square of the mode shape. The two agree to 10.4 per cent at the antinode, where the shift is largest. Every node is a place where the exact shift is zero to better than a part in a thousand million, and that is not an approximation: a mass at a node is never moved by the mode, so it takes no part in the motion and cannot change its rate. Between the nodes the shift follows the square of the displacement, which is the square of the amplitude — the mass is felt in proportion to the kinetic energy the mode was already keeping there.

The dent that raises the note

Push a wall of a resonator inwards and the pitch goes up or down depending entirely on where the wall is pushed. A mass added at a node changes nothing at all; the same mass at an antinode changes as much as it can. One rule covers a loaded string, a tuned microwave cavity and a bead drawn through a resonator to read out its field.

waves · Standing waves A smooth swing that sharpens into pulses. The shape of a string whose far end moves back and forth by 1% of its length at 2 times its fundamental frequency, started swinging in its fundamental, after 0, 10, 20, 30, 40 round trips of a wave. The smooth arch gathers into two ever-narrower pulses running in opposite directions, drawn at the moments they cross in the middle, with their height unchanged and their sides ever steeper; each reaches the moving end just as it moves inwards fastest. The energy has become 1.00, 1.09, 3.15, 10.9, 38.3 times the starting value. Every shape is computed exactly from the moving end, with no approximation to the wall's motion.

The string whose end will not keep still

Move one end of a vibrating string slowly and each mode simply retunes. Move it quickly, at twice the note the string is sounding, and something else happens: the smooth swing gathers into pulses, the energy grows by the Doppler factor of the moving end on every round trip, and it climbs through the modes without stopping — because the modes of a string are evenly spaced, and a wall that couples one pair couples them all.

waves · Standing waves

Named alongside it

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

Boundary conditionNormal modeStanding waveAbsorptionAntinodeBand gapBlackbodyBloch waveDefect stateDegeneracyDetailed balanceDoppler effect

All concepts