Doppler Effect for Light — Question 3

Question 3

A moving detector counts phase cycles per unit proper time

A plane vacuum wave in S travels along +x with phase $\phi=kx-\omega t$ and frequency $f=600\,\mathrm{THz}$. A detector follows $x=vt$, with $v=0.60c$.

1. Find the detected angular frequency from $|d\phi/d\tau|$, using $dt/d\tau=\gamma$.

2. Find the measured frequency and wavelength, taking $c=3.00\times10^8\,\mathrm{m/s}$.

Question 3 - Solution

A moving detector counts phase cycles per unit proper time

(a) Method and calculation

$$\phi(t)=(kv-\omega)t,\qquad\omega=ck.$$$$\boxed{\omega_D=\gamma(\omega-kv)=\gamma(1-\beta)\omega}.$$

(b) Interpretation and further calculation

$$\boxed{f_D=1.25(0.40)(600)=300\,\mathrm{THz}}.$$$$\boxed{\lambda_D=c/f_D=1000\,\mathrm{nm}}.$$

Physical check

Counting cycles per laboratory time would omit the detector proper-time conversion.