Saramsha

No. 016  ·  2 October 2026

Why the Siren Drops

The horn is holding one note, 440 Hz, the whole way past — it has no idea you are there. What changes is the spacing of the crests it leaves in the air: crowded in front of it, stretched out behind. Turn the sound on and let it go by.

one crest of sound the source you where the sound you hear now left it

About this piece

The air keeps the score

Stand beside a road and the thing you notice about a passing siren is that it seems to sag — bright and urgent as it comes, flat and tired as it goes. The natural reading is that something about the siren changed. Nothing about the siren changed. If you were sitting in the ambulance you would hear one steady note from the moment it switched on to the moment it switched off, and that is the note the horn is making: the same note, all the way past you, the whole time.

What changes is the spacing of the crests. Sound in air is a train of pressure crests travelling outwards at about 343 metres a second — and crucially, at 343 metres a second through the air, regardless of how the thing that made them was moving. The air does not inherit the horn's motion. So each crest, once released, expands as a perfect sphere centred on the spot where it was born, and the next crest is born a little further down the road. Expanding circles with marching centres: that is the whole picture, and it is what you are looking at above. Ahead of the source the circles crowd together. Behind it they fall apart. Pitch is crest spacing — it is nothing else — so the crowded side sounds high and the stretched side sounds low.

The arithmetic takes one line. In a tenth of a second the horn emits a tenth of a second's worth of crests, and they end up spread not over the 34.3 m the sound has travelled but over 34.3 m minus however far the source moved in the same tenth of a second. So the wavelength ahead is squeezed from c/f₀ to (c−v)/f₀, and because you still receive them at speed c:

f = f₀ · c / (c − v)

Behind, flip the sign. At the jet setting above — Mach 0.5, around 620 km/h — that is 880 Hz coming and 293 Hz going: an octave and a fifth of difference out of a horn that never budged off its note.

The part that is genuinely surprising

Those two numbers are the limits, for a source heading straight at you or straight away. In real life it passes to one side, and then only the part of its velocity pointing along the line to you counts: v·cos θ. That angle opens out as it approaches, so the pitch is already easing down long before the source is level with you — which is why an ambulance sounds like it is losing heart on the approach.

And here is the detail almost every explanation skips. The moment the source is exactly beside you is not the moment you hear its true pitch. The sound arriving at you then is old: it left the source a while back, when the source was still coming on, so it still carries a raised pitch. The unshifted note reaches you d/c seconds after the thing has gone by — a seventh of a second for a pass at 50 metres. You can watch that happen above: the hollow marker on the line of travel is where the sound currently entering your ear was emitted, and it is always behind the source, never under it. The pitch you hear crosses 440 Hz only when that marker crosses the dashed line, which is later.

So the page does not compute the shift from where the source is. It solves for the retarded time — the instant te at which the crest now arriving was actually emitted, found from |P(te) − L| = c·(t − te), which for a source on a straight line at constant speed is a quadratic you can solve exactly — and then takes the honest form of the law, with ṙ the rate at which the gap was opening at that moment:

f = f₀ · c / (c + ṙ(te))

That is why the drop is a step rather than a slide when the pass is close, and a long lazy slide when it is far. Try the distance slider at the jet setting and watch the curve underneath stiffen up. The timescale of the transition is roughly d/v: pass at 20 metres at 620 km/h and the whole octave-and-a-half happens inside a tenth of a second, which is heard not as a glide but as a thwip.

Moving source, moving listener — not the same thing

Sound has a medium, and the medium is the referee. A source closing on you at 34 m/s gives f₀·c/(c−v) = 1.11 f₀. You closing on a parked source at the same 34 m/s gives f₀·(c+v)/c = 1.10 f₀. Same relative speed, different answers, and the difference is real and measurable: the air can tell which of you is actually moving through it. That is the one way sound's Doppler effect differs in kind from light's. Light has no air. Relativity insists the two cases cannot be distinguished, and sure enough the relativistic formula depends only on the relative velocity, with a factor of √(1−v²/c²) that has no counterpart here.

Past Mach 1

Push the speed slider past 1,235 km/h and the picture breaks in a specific, visible way. The source now outruns its own crests, so instead of nested circles you get circles the source has left behind, all tangent to one pair of lines — the Mach cone, with half-angle sin μ = c/v. Outside that cone there is no sound at all, because none has arrived yet. You stand in silence watching the thing come and go, and then the cone sweeps over you and every crest emitted over several seconds lands at once. That is the sonic boom: not an explosion, not the moment of "breaking" the sound barrier, just a great many crests arriving simultaneously. It follows the aircraft around the whole time it is supersonic, a bow wave dragged along the ground.

Afterwards, something peculiar: you hear two signals at once, because the quadratic now has two valid roots. One is the normal, receding, low-pitched one. The other comes from the approach — and because the source was covering ground faster than its own sound, that part of the signal arrives in reverse order, latest crest first. The page plays both, so a supersonic pass gives you a boom and then a pair of tones, one of them running backwards. It is not a bug in the model; it is what the model says, and it is what the air does.

Doppler, and a train full of trumpeters

Christian Doppler presented the idea to the Royal Bohemian Society in Prague in May 1842, in a paper about the colours of double stars. The principle was right and the application was wrong: he thought the shift explained why one star of a pair looks reddish and the other bluish, whereas the real shifts are thousands of times too small to change a colour you could see. His countryman Buys Ballot said as much in print — and then, in 1845, settled the acoustic half of the question in the most nineteenth-century way imaginable. He hired a locomotive on the new Utrecht–Amsterdam line, put a group of trumpeters on an open carriage holding a single sustained note, stationed musicians with good ears along the track, and ran the train past them at speed, over and over, while everybody wrote down the pitch they heard. The notes came out sharp on the approach and flat on the retreat, by about the predicted amount. Doppler was right about sound, and — once spectroscopy arrived and astronomers started measuring the positions of spectral lines rather than guessing at colours — right about light too, in a way he did not live to see.

The effect now does a great deal of quiet work. Weather radar reads the Doppler shift of its own echo to tell which way the rain inside a storm is moving, and so finds the rotation that precedes a tornado. An ultrasound probe listens to blood: red cells moving towards it shift the returned pitch up, and the shift is the flow speed. A speed gun gets the shift twice — once on the way to the car and once on the way back — so the return is offset by 2vf/c, which conveniently doubles its sensitivity. And almost every exoplanet found before the transit era was found by watching a star's spectral lines twitch back and forth by a few metres per second as an unseen companion pulled it about. Hubble's expanding universe is the same measurement at the largest available scale.

Everything here is computed in your browser from c = 343 m/s, the speed of sound in air at 20 °C — it goes as the square root of absolute temperature and barely cares about pressure. The horn is a 440 Hz sawtooth-ish tone; amplitude falls as 1/r. The view is widened as you raise the speed so that the pass always takes about seven seconds, and the crest interval tracks it, so roughly the same number of circles is in flight at every setting; when the clock is being run fast the top-right corner says so. ?kmh=1500, ?d=0.4 and ?t=0.62 set the speed, the pass distance and the point in the pass. Space pauses, S toggles the sound, ← → step through the pass.