Illustrated Physics
physics, drawn
Concept

Doppler effect — where it appears

The change in the frequency of a wave when the source or the observer moves, by a factor built from the ratio of the speeds. A reflector shifts it twice, which is what a speed radar measures; for light the shift is the square root of (1+β)/(1−β) and is the same for source and observer alike.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Wavefronts from a moving source. Circles showing where each crest has reached, centred on where the source was when it emitted them. Ahead of the source the crests are closer together and the frequency heard is higher; behind, they are spread out and it is lower.

The note that changes on approach, and the two ways of getting it

A moving source and a moving listener produce different formulas for the same shift, because the medium is watching. The difference is small, real, and the reason light had to be treated differently.

waves · Doppler A reflection is shifted twice, and not by the same factor. A wave of unit frequency sent at a reflector closing at 90 km/h, in air. The target meets the fronts sooner than they arrive at a fixed point, so what reaches it is 1.072886 — the observer shift, which multiplies by (c + v)/c. It then re-emits what it received, and now it is a source running after its own wavefronts, which divides by (c − v)/c: the echo comes back at 1.157233. The two operations are different functions and only their product is symmetric, so the round trip is (c + v)/(c − v) rather than the square of either. The shift is 15.7233 per cent where a single one would be 7.2886 — a factor of two to first order in v/c and not exactly two at any speed. For light the same round trip is (1 + β)/(1 − β), which is the square of 1.000000083, and that number is the whole content of the relativistic Doppler effect rather than an approximation to it.

The shift a mirror gives twice

A moving reflector is a receiver and a source in one, so it shifts a wave twice — and the two shifts are different functions of the speed, which is why the round trip is not the square of either. Everything a speed radar does follows from that, including the two things it cannot do.

waves · Doppler One factor, defined by an experiment rather than by a transformation. A observer A stays at x = 0 and flashes a light every 1 second by their own clock. B recedes at 0.6c. The flashes are the diagonal lines; where each meets B's worldline is where B receives it. B's clock reads a longer gap between arrivals than A's read between departures, by the factor k = 2.0000, and it is the same factor between every consecutive pair — measured here off the drawn meetings rather than assumed. That single number is the whole apparatus. Nobody has written down a coordinate transformation, chosen a convention for distant simultaneity, or drawn a tilted axis; the only thing used is that light travels on the diagonals and that neither observer is special, so B's flashes reach A stretched by the same k. From it: γ = (k + 1/k)/2 = 1.2500, and β = (k² − 1)/(k² + 1) = 0.6000.

Everything from an exchange of pulses

Send a flash every second and ask how often the far observer receives them. That one measured ratio generates time dilation, the composition of velocities and the twin result, with no coordinate transformation written down anywhere and no convention chosen about what "at the same time" means far away.

relativity · Doppler The fringes a flow of water moves. The interference fringe shift against the speed of the water, for two tubes 1.5 m long, an index of 1.333, and light of 526 nm — Fizeau's apparatus. The beam is split, each half goes with the flow in one tube and against it in the other, and the two are recombined; the shift is the difference in transit time counted in wavelengths. Three predictions are drawn and they are not close together. If the water did not affect the light at all the shift would be zero, flat along the bottom. If the water carried the light with it completely the shift would be the steepest line. Fresnel's partial drag is the middle one, and at 7 m/s it gives 0.207 of a fringe — which is what was measured, to the accuracy of an eye reading a fringe pattern in 1851. The experiment therefore did not merely detect an effect; it chose between three quantitative possibilities that differ by factors of two, which is why a fraction of a fringe settled something.

The drag that was only an addition

Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.

relativity · Velocity addition Two pushes that add up to a friction. The force on a sodium-23 atom from each of two counter-propagating laser beams tuned 0.5 linewidths below resonance, at 0.1 of saturation each, and their sum, against the atom's velocity in units of the linewidth over the wavenumber. Each beam pushes along its own direction and is heard loudest by an atom moving towards it, because the Doppler shift brings the red-detuned light up into resonance. At rest the two pushes cancel exactly; moving, the atom scatters more from the beam ahead of it than from the one behind, and the difference points against the motion. Near zero velocity the sum is a straight line through the origin — a friction, with slope −0.0907 ħk² — and it is largest at 3.13 m/s, beyond which the atom has been Doppler-shifted out of resonance with both beams and the grip weakens. The damping time it implies for sodium-23's mass is 17.5 microseconds.

The friction made of light

Two laser beams pointed at each other push an atom both ways at once, and at rest the pushes cancel. Moving, the atom hears the beam ahead of it louder than the one behind, and the difference is a friction. The photons that supply the friction arrive one at a time, so they also kick — and the temperature where the two balance contains the width of a spectral line and nothing else.

astrophysics · Radiation pressure A blackbody in every direction, at a different temperature in each. The spectrum of a blackbody at 100 kelvin in its own frame, seen by an observer it is moving past at 0.5 of the speed of light, in 5 directions. Each curve is a Planck spectrum exactly — the Planck form survives a Doppler shift, with the temperature multiplied by the shift — and the temperatures run from 57.74 kelvin looking one way to 173.21 looking the other. So the body is a perfect blackbody in each direction and has no single temperature. A thermometer placed in the radiation reads something between, and what it reads depends on where it is put and on how much of the sky it sees — which is the reason a transformation law for temperature was argued about for sixty years without being found.

The body that has no temperature when it moves

Energy, momentum, length, duration and field strength all change when the observer moves. Temperature was argued about for sixty years, with three transformation laws proposed and each defended by people making no mistake. The resolution is that a moving blackbody is a perfect blackbody in every direction at a different temperature in each — so a thermometer's reading depends on where it is put, and the quantity the law was for is not there.

relativity · Relativistic thermodynamics

Named alongside it

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

MeasurementRadarRelativity principleVelocity additionAnisotropyBandwidthBeat frequencyBlackbodyBrownian motionConventionCosmic microwave backgroundDispersion

All concepts