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The Doppler effect

Waves, sound and light · level 5, lesson 6 of 6

A question to start

An ambulance siren sounds higher as it races towards you and drops lower as it passes. The siren itself never changes. What changes the pitch you hear?

The idea

A moving source chases its own waves in front, packing them closer together, and leaves them stretched out behind. Closer waves arrive more often, so you hear a higher frequency as it comes towards you, and a lower one as it goes away. This is the Doppler effect.

For a source moving at speed \(u\) through a medium where waves travel at \(v\):

\[f_\text{towards} = f\,\frac{v}{v - u}, \qquad f_\text{away} = f\,\frac{v}{v + u}.\]
in front: higherbehind: lower
A siren of 700 Hz, moving at 20 m/s Mars air Earth air
Speed of sound about 240 m/s about 340 m/s
Heard as it comes towards you 764 Hz 744 Hz
Heard as it goes away 646 Hz 661 Hz

The shift is bigger on Mars, because the source's speed is a bigger share of the slower speed of sound.

Worked examples

1. Earth, towards. \(700 \times 340/(340 - 20) = 700 \times 1.0625 = 744\) Hz.

2. Earth, away. \(700 \times 340/360 = 661\) Hz.

3. Ingenuity on Mars. Suppose its 84 Hz hum comes towards the rover at 5 m/s: \(84 \times 240/235 = 85.8\) Hz. Going away: \(84 \times 240/245 = 82.3\) Hz. Scientists used shifts like this to measure the speed of sound on Mars.

Try it

  1. Why does a source coming towards you sound higher?
  2. A 500 Hz horn comes towards you at 30 m/s in Earth air. What frequency do you hear?
  3. The same horn moves away at 30 m/s. What do you hear now?
  4. Why is the Doppler shift bigger on Mars for the same speed?
Answers
  1. Its waves are packed closer together, so they reach you more often.
  2. \(500 \times 340/310 = 548\) Hz.
  3. \(500 \times 340/370 = 459\) Hz.
  4. The speed of sound is lower, so the source's speed is a bigger fraction of it.

Common mistake

"The source changes its pitch." The source sends out the same frequency the whole time. Only the frequency you receive changes, because of the movement.

Where it leads