5.6 The Doppler Effect

Light has no medium, so there is no preferred frame of reference. We can always transform the relative motion of source and observer into the rest frame of the observer, so we only need consider the case of a moving source, depicted in Fig. 4.3.
With an identical argument to that of Section 4.4.1, we have

$\begin{displaymath} \tau' = \tau(1 - \beta \cos \theta) \end{displaymath}$ (5.41)

where

$\begin{displaymath} \beta = \frac{v_s}{c} \end{displaymath}$ (5.42)
However, at relativistic speeds, the period $\tau$ is subject to time dilation. It relates to the period $\tau_o$ measured in the rest frame of the source via

$\begin{displaymath} \tau = \gamma \tau_o \end{displaymath}$ (5.43)

where

$\begin{displaymath} \gamma = \frac{1}{\sqrt{1 - \beta^2}} \end{displaymath}$ (5.44)
Combining Eq.'s 5.41 and 5.43, and noting that $\omega = \frac{2\pi}{\tau}$ , we find

$\begin{displaymath} \omega' = \frac{\sqrt{1 - \beta^2}} {1 - \beta \cos \theta} \omega_o \end{displaymath}$ (5.45)
As in the acoustic case, $\theta$ is determined by the positions of source and observer relative to $\vec{v}_s$ at the time the observed wave fronts were emitted.

Next: 6. Diffraction Up: 5. Electromagnetic Waves Previous: 5.5 Reflection, Refraction, and