7.9 The Doppler Effect

Consider a source at rest emitting spherical waves with frequency $f_{s}$ and speed $v$ . What does a detector towards the source with speed $v_{d}$ see? In this situation, the wavefronts approach the detector with speed $v+v_{d}$ , so the frequency as measured by the detector is

f_{d}=\frac{v+v_{d}}{\lambda}=f_{s}\frac{v+v_{d}}{v}.

(7.148)

If the detector is moving away from the source, then the observed frequency will be

f_{d}=f_{s}\frac{v-v_{d}}{v}.

(7.149)

This change in frequency due to a difference in velocity is known as the Doppler effect.

What about a moving source and a stationary detector? During one period $T_{s}$ of the source emitting, the waves will propagate a distance $vT_{s}$ . The source itself moves a distance $v_{s}T_{s}$ . If the source is moving towards the detector, then the observed wavelength is $\lambda_{d}=vT_{s}-v_{s}T_{s}=T_{s}(v-v_{s})$ , which gives an observed frequency of

f_{d}=\frac{v}{T_{s}(v-v_{s})}=f_{s}\frac{v}{v-v_{s}}.

(7.150)

If the source is moving away from the detector, then we get

f_{d}=f_{s}\frac{v}{v+v_{s}}.

(7.151)

In the general case of a moving source and detector, we have

f_{d}=f_{s}\frac{v\pm v_{d}}{v\mp v_{s}},

(7.152)

where we have the $\frac{+}{-}$ case when the source and detector are moving closer and the $\frac{-}{+}$ case when they are moving apart.