Doppler effect
Doppler effect
If the source and the observer of a wave are in relative motion, the perceived frequency will differ from that without relative motion. This is the Doppler effect.
For example, the pitch of the sound of a horn changes as the vehicle moves past the observer — whilst it is approaching, its pitch is higher, and whilst it is moving away, it is lower.
The observed frequency will be
\[f = f_0 \cdot \frac{c + v_{\mathrm{o}}}{c - v_{\mathrm{s}}},\]
where \(f_0\) is the frequency the observer would observe if both they and the source were standing, \(c\) is the speed of propagation of the wave in the medium, \(v_{\mathrm{o}}\) is the speed of the observer and \(v_{\mathrm{s}}\) is the speed of the source.
The formula above uses the following sign conventions for the speeds:
- \(v_{\mathrm{o}}\) is positive if the observer moves towards the source and negative if it moves away from it;
- \(v_{\mathrm{s}}\) is positive if the source moves towards the observer and negative if it moves away from it.
The Doppler effect applies to light as well. For instance, there is a so-called red shift in the spectrum of the light of stars which recede from us: all lines are shifted towards the red, that is, towards longer wavelengths.
Proof for a stationary observer
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Let us derive the formula for the special case when the observer is stationary and the source moves towards it. As the source moves towards the observer, the wavelength perceived by the observer will be shorter than in the stationary case. Assume that the observer perceives a crest; the next crest they observe will not be \(\lambda\) distance away from the first one but closer, because the source has moved closer in the meantime by an amount \[v_{\mathrm{s}} T = \frac{v_{\mathrm{s}}}{f_0}.\] The perceived wavelength is thus \[\lambda = \lambda_0 - \frac{v_{\mathrm{s}}}{f_0}.\] Using the relationship between the speed of propagation and frequency, the perceived frequency will be \[f = \frac{c}{\lambda} = \frac{c}{\lambda - v_{\mathrm{s}} / f_0} = \frac{c}{c / f_0 - v_{\mathrm{s}} / f_0} = f_0 \frac{c}{c - v_{\mathrm{s}}}.\] This is the Doppler formula in the special case we assumed (\(v_{\mathrm{o}} = 0\)). |