A Close Pass, a Smooth Drop
A siren passing close by drops rapidly but smoothly in pitch. Far before and after the pass, the high and low notes are nearly steady. Near plateaus joined by a steep, smooth bend. Send it past your ear and listen.
Ready. Press “Send it past”.
Watch the rings. Ahead of the source they pile up — each new crest is launched from a little closer to you, so they arrive bunched: a higher pitch. Behind, they spread out: a lower pitch. The source's own note never changes. Only the spacing you receive does.
Now watch the graph, not the rings. For a close pass, the pitch is nearly steady and high far before the pass, falls smoothly through the source's true note, then is nearly steady and low far after it. The closer it passes, the sharper that fall: at a few metres it's almost a cliff; from far away it's a gentle ramp.
The time sound takes
There's a delay built into hearing. The note that marks closest approach — the exact instant the pitch equals the siren's real pitch — leaves the source when it's directly level with you, but the sound still has to cross the gap to reach your ear. So you hear “the middle” a fraction of a second after the source is actually nearest you: a lag of exactly the gap divided by the speed of sound. At 10 m that's 29 ms. The dot is already pulling away by the time your ears report the turn.
The check — every number here, recomputed live
The pitch you receive from a wavefront is set by how fast the gap to the source is changing when that wavefront leaves. In one line (v = speed of sound, vs = source speed, vr = how fast the gap is opening):
- At your current settings (30.0 m/s, f₀ = 1000 Hz, v = 343 m/s): approaching 1095.85 Hz, receding 919.57 Hz, and exactly 1000 Hz at closest approach at emission (the gap is momentarily not changing); that sound arrives b/v later.
- The plateau ratio is 1.1917 = (v+vs)/(v−vs), a drop of 3.04 semitones — and that ratio doesn't depend on how close it passes, only on the speed.
- The drop is not symmetric: +1.58 semitones up to the high plateau but only −1.45 down to the low one. The Hz spread above f₀ is wider than below.
- You hear closest approach 29.2 ms late — the gap (10 m) divided by v.
The closed form above is checked against a second, independent method that knows nothing of it: a wavefront-counting simulation that just emits a crest every 1/f₀ seconds from the moving source, propagates each at v, and times the arrivals. The two agree to a maximum relative error of 3×10⁻⁸ across 6,000 samples. Both live in research/doppler-siren/verify.mjs — run it yourself.
A broader pass, a broader bend
For a source that passes close and fast, most of the change is compressed into a brief, smooth transition. Push the “how close” slider out to 50–60 m: at 108 km/h the 10%–90% transition takes about 4.43–5.31 s of this six-second view. The bend broadens and the near-flat ends shorten. The distant high and low limits stay the same.
Honest apparatus
Idealizations, named: a point source in still air, no wind, no reflections or echoes, and the observer at rest (only the source moves — moving-observer Doppler has a slightly different form, f′ = f₀(v+vo)/v, not used here). Speed of sound taken as 343 m/s (dry air, 20 °C); it rises ~0.6 m/s per °C, which shifts the plateau Hz a little but not the shape. The rings you see are slowed — a real 1000 Hz source emits a thousand crests a second, far too many to draw, so the visual crest rate is reduced for the eye; the audio uses the real f₀ curve. The wavefront illustration is schematic: horizontal and vertical distances use different scales, so the drawn ring spacing and arrival times are not quantitative measurements. The drop is given in equal-tempered semitones, 12·log₂(ratio), a convention for naming musical interval size, not a claim about perception. Real sirens often wail (sweep f₀ on purpose); here f₀ is held fixed so the Doppler shift is the only thing moving the pitch.
Corrections
Corrected 2026-10-14: The page said the pitch does not slide and called the transition a step, not a slope. The model gives a smooth drop that is rapid for a close pass. The film source is corrected, but the embedded film still carries the earlier wording; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The page offered build.sh as a fresh-checkout reproduction command without naming its prerequisites. It requires Node.js, Playwright, matching Chromium and FFmpeg with H.264 and AAC encoders; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The heading called propagation delay the bit nobody mentions. Michael Fowler explicitly describes this delay in his Doppler Effect notes; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The page said the 50–60 m setting shows long flat ends. At 108 km/h, the 10%–90% transition takes about 4.43–5.31 s of the displayed 6 s, leaving short ends; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The page called the wavefront geometry exact at any rate. The illustration uses different horizontal and vertical scales and is schematic; found by the assay line, 2026-10-14.
Corrected 2026-10-14: The metadata and a film caption placed the heard 1000 Hz crossing at closest approach. That sound is emitted at closest approach and heard b/v later, about 29.2 ms at 10 m. The film source is corrected, but the embedded film and its poster retain the earlier caption; found by the assay line, 2026-10-14.
Sources: Michael Fowler, The Doppler Effect; Matthew Schwartz, Lecture 21, section 2; research/doppler-siren/film/engine.mjs; research/doppler-siren/film/build.sh; this page's sceneDraw.