# A-level Physics (Advancing Physics)/Doppler Effect

The Doppler effect is a change in the frequency of a wave which occurs if one is in a different frame of reference from the emitter of the wave. Relative to us, we observe such a change if an emitter of a wave is moving relative to us.

All waves travels in a medium. So, they have a velocity relative to this medium v. They also have a velocity relative to their source v_{s} and a velocity relative to the place where they are received v_{r}. The frequency at which they are received f is related to the frequency of transmission f_{0} by the formula:

The Doppler effect can be used to measure the velocity at which a star is moving away from or towards us by comparing the wavelength receive λ with the wavelength we would expect a star of that type to emit λ_{0}. Since the speed of light c is constant regardless of reference medium:

Therefore:

and

By substitution:

In this case, v is the speed of light, so v = c. Relative to us, we are stationary, so v_{r} = 0. So:

If we call the change in wavelength due to Doppler shift Δλ, we know that λ = λ_{0} + Δλ. Therefore:

So, the important result you need to know is that:

This value is known as the red-shift of a star, denoted z. If z is positive, the star is moving away from us - the wavelength is shifted up towards the 'red' end of the electromagnetic spectrum. If z is negative, the star is moving towards us. This is known as blue shift. Note that we have assumed that v is much smaller than c. Otherwise, special relativity makes a significant difference to the formula.

## Questions[edit]

1. M31 (the Andromeda galaxy) is approaching us at about 120kms^{−1}. What is its red-shift?

2. Some light from M31 reaches us with a wavelength of 590 nm. What is its wavelength, relative to M31?

3. Some light has a wavelength, relative to M31, of 480 nm. What is its wavelength, relative to us?

4. A quasar emits electromagnetic radiation at a wavelength of 121.6 nm. If, relative to us, this wavelength is red-shifted 0.2 nm, what is the velocity of recession of the quasar?