Doppler Effect

How the model works

Each wavefront spreads at the speed of sound c from the point where the source was when it emitted it. If the source moves, successive wavefronts are centred at different points, so they crowd together in front and spread out behind.

For motion along the line joining them, the received frequency is

f' = f · (c + v_L) / (c − v_S)

where v_L is positive when the listener moves towards the source and v_S is positive when the source moves towards the listener. The simulator uses the components of velocity along that line, so it handles passing motion too: the pitch drops as the source goes by.

When v_S ≥ c, the wavefronts pile up into a cone, the Mach cone or sonic boom. The formula breaks down there.

Model: still air, uniform sound speed, point source, motion in a plane. Audio is not played; the frequencies are calculated.

Guided experiments

1. Approaching source. Source moves right at 100 m/s towards a still listener (c = 343 m/s, f = 440 Hz). Prediction: f' = 440 × 343/243 ≈ 621 Hz ahead of the source. Approaching siren
2. Moving listener instead. Source still, listener moves left at 100 m/s towards it. Prediction: f' = 440 × 443/343 ≈ 568 Hz. Not the same as the moving-source case, because the air is the reference frame. Moving listener
3. Break the sound barrier. Source at 1.4 × c. Prediction: wavefronts form a cone with half-angle arcsin(1/1.4) ≈ 45.6°. The listener hears nothing until the cone sweeps past. Mach 1.4

Challenge

An ambulance's siren is 700 Hz. You hear 750 Hz as it approaches. How fast is it going (c = 343 m/s)? What will you hear after it passes?

References

• OpenStax, University Physics Volume 1, §17.8 “The Doppler Effect” and §17.9 “Shock Waves”.