Doppler effect
When a source and a listener move relative to each other, the apparent frequency changes: higher when they come closer, lower when they move apart.
What is the Doppler effect?
Stand near a railway level crossing. As a train comes towards you, its horn sounds higher in pitch (shriller). As it passes and moves away, the pitch suddenly drops. The real frequency of the horn does not change. Only the frequency you hear, the apparent frequency, changes. Doppler effect: the apparent change in the frequency of sound due to the relative motion between the source and the listener. When the source comes closer, the waves are squeezed together, so more waves reach you each second. When it moves away, the waves are stretched out, so fewer reach you. In the formulas, n is the actual frequency and n′ the apparent frequency. v is the speed of sound, vS the speed of the source and vL the speed of the listener.
Situations and formulas
| Situation | Apparent frequency n′ | Heard as |
|---|---|---|
| Source moves towards a listener at rest | n′ = [v / (v − vS)] × n | Higher |
| Source moves away from a listener at rest | n′ = [v / (v + vS)] × n | Lower |
| Listener moves towards a source at rest | n′ = [(v + vL) / v] × n | Higher |
| Listener moves away from a source at rest | n′ = [(v − vL) / v] × n | Lower |
| Both move towards each other | n′ = [(v + vL) / (v − vS)] × n | Higher |
| Both move away from each other | n′ = [(v − vL) / (v + vS)] × n | Lower |
| Both move the same way, source behind the listener | n′ = [(v − vL) / (v − vS)] × n | Higher if the source is faster |
| Both move the same way, listener behind the source | n′ = [(v + vL) / (v + vS)] × n | Higher if the listener is faster |
When there is no Doppler effect
- The source and the listener are both at rest.
- Both move with the same velocity in the same direction, so the distance between them stays constant.
- They move at right angles (perpendicular) to the line joining them.
- The listener moves in a circle with the source at the centre, so the distance does not change.
- Only the medium (air) moves, while the source and listener stay at rest.
Applications of the Doppler effect
- Speed guns: traffic police find the speed of a vehicle from the change in frequency of waves reflected by it.
- RADAR (RAdio Detection And Ranging): finds the speed and direction of aircraft.
- SONAR: finds the speed of submarines and ships under water.
- Tracking satellites: the change in frequency of a satellite's signals gives its speed and position.
- Doppler weather radar: tracks the movement of rain clouds and cyclones.
- Medicine: a Doppler scan measures the speed of blood flow in the heart and blood vessels.
Worked example: a train horn at a station
A train approaches a station at 30 m/s, sounding a horn of frequency 520 Hz. The speed of sound is 340 m/s. Find the frequency heard by a person on the platform (a) as the train approaches (b) after it passes and moves away.
- (a) Source moves towards a listener at rest: n′ = [v / (v − vS)] × n.
- n′ = [340 / (340 − 30)] × 520 = (340/310) × 520.
- n′ = 176,800 ÷ 310 ≈ 570.3 Hz, higher than 520 Hz.
- (b) Source moves away: n′ = [v / (v + vS)] × n.
- n′ = [340 / (340 + 30)] × 520 = 176,800 ÷ 370 ≈ 477.8 Hz, lower than 520 Hz.
(a) about 570.3 Hz (b) about 477.8 Hz, both rounded to 1 decimal place
An ambulance siren on Anna Salai sounds shriller as it rushes towards you and deeper once it has passed. The Doppler weather radar at Chennai uses the same idea to track cyclones over the Bay of Bengal.
2 marks: “What is the Doppler effect?” The apparent change in the frequency of sound due to relative motion between the source and the listener.
4 marks: “A listener on a bike moves at 17 m/s towards a stationary siren of frequency 400 Hz. Find the apparent frequency (v = 340 m/s).” n′ = [(v + vL)/v] × n = (357/340) × 400 = 1.05 × 400 = 420 Hz.
7 marks: “Explain the Doppler effect. Write the formulas for the different situations, state when it is not observed and give its applications.” Definition with the train horn; the eight cases with formulas, higher or lower; no effect when both at rest, same velocity, perpendicular motion or only the medium moves; uses: speed guns, RADAR, SONAR, satellites, weather radar.
Closer = Chirpier (higher pitch); Apart = All low. The source's speed sits in the denominator; the listener's speed sits on top.