# The expanding Universe

## The Doppler effect

The Doppler effect is usually noticed when a vehicle with a siren approaches and moves away from an observer.

If a fire engine passes us we notice the pitch of the siren to be higher coming towards us and lower going away from us.

The apparent shift in frequency is due to the changing, as shown in the diagram, and the speed of the sound staying constant.

The observed frequency fo (o for observer) can be calculated if the frequency of the source fs (s for source), the speed of the source vs and the speed of sound v are known.

$f_{o}=f_{s} \left(\frac{v}{v\pm v_{s}} \right )$

The denominator is 'plus or minus' depending if the source is travelling towards or away from the observer:

• source travelling towards observer - subtract vs making the bottom of the fraction smaller to give an increased frequency
• travelling away from observer - add v and vs making the bottom of the fraction larger to give a decreased frequency

### Example

If a fire engine is travelling away at 18 ms-1 (40 miles per hour) sounding a siren of frequency 512 Hz what is the frequency heard by a stationary observer?

The speed of sound vs is given as 340 ms-1.

Put the known values into the Doppler equation. In this case add the speed of sound and the speed of the fire engine to make the denominator:

$f_{o}=512 \left(\frac{340}{340+18} \right )$

The terms in the brackets become a factor to multiply the source frequency.

$f_{o}=512 \times 0.9497$

$f_{o}=486Hz$

So there is a drop in pitch of 26 Hz as the fire engine passes and begins to move away.

Question

Do the occupants of a fire engine notice a doppler shift?

Not from their own siren as they are moving at the same speed.

If they were to hear a siren in a building as they passed, it would appear higher in pitch on approach and lower as they move away for similar reasons as above.