Doppler Effect for Light — Question 5

Question 5

An oblique direction can cancel the net frequency shift

A monochromatic source moves at $v=0.60c$. For a source of rest frequency $f_0$ moving at +v, use $f=f_0/[\gamma(1-\beta\cos\theta)]$, where $\beta=v/c$ and $\theta$ is the photon propagation angle from +x in the laboratory.

1. Find the laboratory angle at which the received frequency equals $f_0$.

2. Find the corresponding source-frame angle using aberration.

Question 5 - Solution

An oblique direction can cancel the net frequency shift

(a) Method and calculation

$$1=\frac1{\gamma(1-\beta\cos\theta)}\quad\Rightarrow\quad\cos\theta=\frac{1-1/\gamma}{\beta}.$$$$\boxed{\cos\theta=1/3},\qquad\boxed{\theta=70.5288^\circ}.$$

(b) Interpretation and further calculation

$$\cos\theta\prime=\frac{1/3-3/5}{1-(3/5)(1/3)}=-1/3.$$$$\boxed{\theta\prime=109.4712^\circ}.$$

Physical check

Zero net shift does not imply zero relative speed; angular propagation and time dilation can cancel.