The Doppler Effect, Measured
By Lucian — builder & engineer, LK Forge
Everyone has heard it: an ambulance races past and its siren sags from a high whine to a low drone in a single instant. That drop is a number, and a musical one. We took the moving-source physics from the Doppler simulator and worked out exactly how far the pitch falls — and what happens when the source is fast enough to catch its own sound.
· 6 min read · exact from f = f0·v/(v∓vₛ)
The split widens with speed
Approaching, the waves bunch and the pitch rises; receding, they stretch and it falls. The two branches start together at the rest pitch and pull apart as the source speeds up — but not symmetrically. The receding branch flattens toward half the frequency, while the approaching branch curves upward and heads for the ceiling as the source nears the speed of sound.
Observed frequency vs source speed. The approaching pitch curves toward infinity at the 343 m/s sound barrier; the receding pitch sags toward 220 Hz. The gap between them is what you hear as the drop.
The drop, in Hz and in music
Frequencies are easier to feel as musical intervals. At walking-past-fast speeds the drop is a couple of semitones; at highway speed it is a clear minor third; a racing car crossing at 60 m/s drops a tritone; and by 200 m/s the pitch falls two full octaves from approach to retreat.
| Source speed | Approaching | Receding | Drop | In semitones |
|---|---|---|---|---|
| 10 m/s (36 km/h) | 453.2 Hz | 427.5 Hz | 26 Hz | 1.01 st |
| 30 m/s (108 km/h) | 482.2 Hz | 404.6 Hz | 78 Hz | 3.04 st |
| 60 m/s (216 km/h) | 533.3 Hz | 374.5 Hz | 159 Hz | 6.12 st |
| 100 m/s (360 km/h) | 621.1 Hz | 340.7 Hz | 280 Hz | 10.4 st |
| 200 m/s (720 km/h) | 1055.4 Hz | 277.9 Hz | 778 Hz | 23.1 st |
440 Hz source, stationary listener, speed of sound 343 m/s. Semitones = 12·log₂(approach ÷ recede). 12 semitones is one octave; a whole octave of drop needs a source doing about 114 m/s.
Past the sound barrier: the shock cone
Right at Mach 1 the approaching pitch is infinite because the source keeps pace with its own wavefronts — they stack into a single wall of pressure, the sonic boom. Go faster and the source leaves its sound behind, and all those fronts fall on a cone dragging behind it. The cone’s half-angle is arcsin(1/Mach): a right angle at Mach 1, closing to a sharp point as the speed climbs.
Shock-cone half-angle μ = arcsin(1/Mach). The faster the source, the tighter the cone dragging its boom behind it.
Reproduce it yourself
Two fractions give the whole thing. The pitch drop as a source passes is just the ratio of the two, turned into semitones.
const v = 343, f0 = 440, vs = 30; // m/s, Hz, source speed
const approach = f0 * v / (v - vs); // 482.2 Hz (-> infinity as vs -> v)
const recede = f0 * v / (v + vs); // 404.6 Hz
const drop_st = 12 * Math.log2(approach / recede); // 3.04 semitones
// beyond Mach 1: shock-cone half-angle
const coneDeg = M => Math.asin(1 / M) * 180 / Math.PI; // coneDeg(2) = 30The frequency and cone formulas are lifted straight from the Doppler solver; the speed sweep is the small script behind this post.
Watch the waves bunch
Open the simulator, drag the source speed up, and watch the wavefronts crowd ahead and stretch behind — then push past Mach 1 and see the shock cone snap into place.