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Doppler effect calculator (sound)

Perceived frequency when the source and the observer move, with a single sign convention, the source/observer asymmetry and the sound-barrier limit.

Source and medium

The emitted wave
Hz
m/s≈ 343 m/s in air at 20 °C.
The two motions
m/sExample: 30 m/s, an ambulance at 108 km/h.
m/s

Sign convention: positive = moving toward the other, negative = moving away.

Wave

the fronts and the comb of arrivals
f 440 Hz · v 343 m/s · source 30 m/s · observer 0 m/st = 0.152 ssourceobserveralong the motion λ′ 0.711 magainst the motion λ′ 0.848 m2 m1 px = 0.682 m/semission at the source · 1/f = 2.27 msarrival at the observer · 1/f′ = 2.07 ms10 fronts: 1.99 ms earlier05101520time since the first front (ms)
t = 0.152 s

Animation slowed sevenfold; the slider and the dimensions give real times and lengths.

Perceived frequency
f′ = f·(v + vobs)/(v − vsrc) = 440 × (343 + 0) ÷ (343 − 30) = 482.17 Hz
Spacings
λ = v/f = 0.77955 m; along the motion (v − vsrc)/f = 0.71136 m, against it (v + vsrc)/f = 0.84773 m
Cadences
emission 1/f = 2.2727 ms; arrival 1/f′ = 2.0739 ms

the fronts and the geometry of the mediumthe result: λ′ and the received cadence

Result

continuous computation
What the observer receives
f′482.17Hzhigher than the emission
Shift
42.173 Hz
Ratio
1.0958
The two wavelengths
Emitted wavelength
0.77955 m
Perceived wavelength
0.71136 m

At the same speed: 482.17 Hz if the source moves, 478.48 Hz if the observer moves.

Calculation, assumptions and sources

OverviewPerceived frequency f′ = f·(v + v_obs)/(v − v_src) when the source, the observer or both move, with a single sign convention (positive = approaching).

Perceived frequency f′ = f·(v + v_obs)/(v − v_src) when the source, the observer or both move, with a single sign convention (positive = approaching). Cases outside the formula are flagged: source at the sound barrier, observer receding faster than sound.

Method & formulasf′ = f · (v + v_obs) / (v − v_src)
f′ = f · (v + vobs) / (v − vsrc)convention: positive speed = approaching, negative = recedingwavelengths: λ′ = v / f′

An approaching source compresses the waves in the medium (the denominator shrinks). An approaching observer crosses unchanged waves more often (the numerator grows). The two cases differ: at 30 m/s on an A440, a moving source gives 482.2 Hz, a moving observer 478.5 Hz. At low speeds both tend to f·(1 + u/v).

The sign convention
positive = approaching, negative = receding, for the source and the observer alike.
Speed of sound (v)
≈ 343 m/s in air at 20 °C, about +0.6 m/s per degree; ≈ 1,480 m/s in water. The speeds in the formula are measured relative to the medium.
Sound barrier
source speed equal to v. The forward-emitted waves pile up on it (Mach cone, sonic boom) and the Doppler formula diverges.
Validity domainOne-dimensional acoustic model: motions along the source-observer line.

One-dimensional acoustic model: motions along the source-observer line. For a pass at a distance, only the radial component counts, and it varies continuously. The medium is assumed still; a constant wind is handled by adding its speed to both actors. The Doppler effect for light follows a different formula, relativistic, symmetric and medium-free; it is not covered.

Perceived frequency while approaching and recedingAt constant speed, the perceived frequency stays fixed during the whole approach (higher than f), then during the whole recession (lower); it changes at the moment of passing.

At constant speed, the perceived frequency stays fixed during the whole approach (higher than f), then during the whole recession (lower); it changes at the moment of passing. The impression of a pitch rising during the approach comes from the growing loudness.