Wavefronts from a moving source
At its defaults it draws 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.
doppler-wavefronts is one function in lib/figures/waves.js —
travelling, standing, adding and shifting. 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.
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.
A source moving faster than its own waves
The options are the ones The cone the source leaves behind passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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 angle measures the speed, and contains nothing else
The options are the ones The cone the source leaves behind passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Half-angle of the Mach cone against Mach number, with the marked values measured off the constructed wavefronts rather than computed: for each speed, six circles are laid down at the positions and radii the construction gives, and the tangent from the apex to each is taken. All six agree to a part in 10¹², because the ratio of a circle's radius to its distance from the apex is c/v for every one of them. Mach 1.2 gives 56.4°; Mach 1.6 gives 38.7°; Mach 2 gives 30.0°; Mach 3 gives 19.5°. The relation sin θ = 1/M has no dynamics in it at all — no pressure, no density, no shape of the object — so a photograph of a shock wave is a speedometer, and it is the only one that needs nothing on board.
A source moving faster than its own waves
The options are the ones The cone the source leaves behind passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
A source moving faster than its own waves
The options are the ones The cone the source leaves behind passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
Wavefronts from a moving source
The options are the ones The cone the source leaves behind passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What checks it
physicscheck asserts something about doppler-wavefronts 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 cone the source leaves behind
Take the Doppler construction past the speed of the wave and the wavefronts acquire an envelope. Its half-angle obeys sin θ = 1/M, an expression with no pressure, no density and no shape of the object in it — so a photograph of the cone is a speedometer. And the bang is not an event at the moment of crossing — it is a signature dragged along the ground for the whole of the flight.
WavesThe Doppler shift with nothing coming closer
Every Doppler shift so far has needed something to approach or recede. Light can be shifted in frequency by something that does neither: a surface spinning in place, or a polarising plate turning in a beam that passes straight through it. A beam whose wavefronts wind round its axis, ℓ times per wavelength, comes back from a surface spinning at Ω shifted by exactly ℓΩ, at every distance from the axis. A circularly polarised beam through a spinning half-wave plate comes out shifted by exactly 2Ω, because the plate takes two units of angular momentum from each photon and does work on it. The shift is tiny and perfectly measurable, and satellite navigation has to correct for it.
WavesThe loudspeaker that goes round
The swirling sound of a Hammond organ comes from a loudspeaker cabinet in which the speaker's horn turns, six or seven times a second, and the swirl is the Doppler effect — the horn's mouth swings towards and away from the listener at about six metres a second, and every note it plays rises and falls by a third of a semitone. But a pitch that rises and falls periodically is not a note being shifted: it is a note split into a comb of new frequencies, spaced by the rotation rate and spread in proportion to the note itself, so the turning horn leaves the bass nearly pure and blurs the treble into a shimmer. In a room each wall returns its own copy, swinging out of step with the others, and the copies beat against each other several times a turn.
WavesThe 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.
WavesThe 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.
RelativityThe shift that survives at right angles
For sound it matters which of the two is moving, and the two answers differ. For light there is one answer — their geometric mean — and a term with no classical counterpart at all: a source going past at closest approach, with its distance not changing, is still shifted.
RelativityThe sky that crowds into a cone
A boost does not only shift frequencies. It remaps directions, so half of everything a fast traveller can see is squeezed into a forward cone of half-angle about 1/γ — and because brightness carries four powers of the Doppler factor, what lies ahead is overwhelming and what lies behind has effectively gone.
WavesThe speeds a pulsed radar folds into one
A weather radar measures the wind inside a storm by the Doppler shift of its echoes, but it never hears the shift itself. It listens in pulses, and between pulses it notes only how far the echo's phase has turned — which is known only to within a whole turn. So speeds that differ by a fixed amount give identical echoes, and every speed is folded into a window. Make the window wider by pulsing faster, and the echoes from distant rain arrive after the next pulse has gone and land at the wrong range. The two limits multiply to a constant the designer cannot touch.