All Problems Optics of Moving Sources
Problem 5.225
A source of light moves with velocity \(v\) relative to a receiver. Demonstrate that for \(v \ll c\) the fractional variation of frequency of light is defined by Eq. (5.6a).
5.225. First of all note that when \(v \ll c,\) the time rate is practically identical in the reference frames fixed to the source and to the receiver. Suppose that the source emits short pulses with the intervals \(T_{0} .\) Then in the reference frame fixed to the receiver the distance between two successive pulses is equal to \(\lambda=c T_{0}-v_{r} T_{0},\) when measured along the observation line. Here \(v_{r}\) is the projection of the source velocity on the observation line \(\left(v_{r}=v \cos \theta\right) .\) The frequency of received pulses \(v=c / \lambda=v_{0} /\left(1-v_{r} / c\right),\) where \(v_{0}=1 / T_{0} .\) Hence \(\left(v-v_{0}\right) / v_{0}=(v / c) \cos \theta\)