The Doppler Effect

The sound of a passing siren

An ambulance races towards you. Its siren sounds high. The moment it passes, the pitch drops, and it stays low as the ambulance moves away. The siren itself never changes. What changes is the pitch you hear. This is the Doppler effect, named after Christian Doppler, who explained it in 1842.

A moving source

A source of frequency $f$ sends out one crest every period, $T = \dfrac{1}{f}$. Sound travels at $v$ through the air, whatever the source does.

If the source moves towards you at $v_s$, it chases its own crests. In one period each crest moves $vT$, but the source moves $v_sT$ after it. So the crests in front are only $(v - v_s)T$ apart: the wavelength is shorter, and you hear a higher frequency.

An example. A car horn of $400\ \text{Hz}$ comes towards you at $25\ \text{m/s}$, with sound at $340\ \text{m/s}$:

As it moves away: $f' = 400\ \text{Hz} \times \dfrac{340\ \text{m/s}}{365\ \text{m/s}} = 372.6\ \text{Hz}$. The pitch drops by almost $60\ \text{Hz}$ as it passes.

You should find that the crests crowd together in front and spread out behind, more and more as the source speeds up.

A moving listener

Now the source stays still and you move towards it at $v_o$. The wavelength in the air does not change, but you run into the crests. Each second you meet the crests that would pass you anyway, plus the extra ones you run into:

Both moving

Put the two together:

Choose the signs so that motion towards each other raises the pitch: $+v_o$ on top when the listener moves towards the source, and $-v_s$ below when the source moves towards the listener.

Where we use it

• Speed guns of the traffic police send out radio waves and measure the shift of the echo from a car.
• Bats and dolphins hear the shift in their echoes, and know whether their prey moves towards or away from them.
• Doppler ultrasound measures how fast blood flows in an artery.
• Light shows the effect too. Light from galaxies that move away from us is shifted to longer wavelengths, the red shift. This told astronomers that the universe is expanding.

Try it in Python

The program finds the pitch of a $700\ \text{Hz}$ siren on an ambulance at four speeds, coming and going.

Practise with new numbers

Remember these

• Motion towards each other raises the pitch; motion apart lowers it.
• A moving source changes the wavelength: $f' = f\,\dfrac{v}{v \mp v_s}$.
• A moving listener meets the crests more or less often: $f' = f\,\dfrac{v \pm v_o}{v}$.
• The source's own frequency never changes.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why does the pitch of a train's whistle drop as the train passes you?

2. A source of $500\ \text{Hz}$ moves towards you at $34\ \text{m/s}$. Sound travels at $340\ \text{m/s}$. What frequency do you hear?

3. You cycle towards a siren that stands still. Do you hear a higher or a lower pitch? Why?

4. A source moves towards a listener. Does the frequency of the source itself change?