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Optics of Moving Sources

Problem 5.225

A source of light moves with velocity vv relative to a receiver. Demonstrate that for v≪cv \ll c the fractional variation of frequency of light is defined by Eq. (5.6a).

Reveal Answer
5.225. First of all note that when v≪c,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 T0.T_{0} . Then in the reference frame fixed to the receiver the distance between two successive pulses is equal to λ=cT0−vrT0,\lambda=c T_{0}-v_{r} T_{0}, when measured along the observation line. Here vrv_{r} is the projection of the source velocity on the observation line (vr=vcos⁡θ).\left(v_{r}=v \cos \theta\right) . The frequency of received pulses v=c/λ=v0/(1−vr/c),v=c / \lambda=v_{0} /\left(1-v_{r} / c\right), where v0=1/T0.v_{0}=1 / T_{0} . Hence (v−v0)/v0=(v/c)cos⁡θ\left(v-v_{0}\right) / v_{0}=(v / c) \cos \theta
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