Page:SahaElectrodynamics.djvu/23

where

$\omega '=\omega \beta \left(1-{\frac {lv}{c}}\right),\ l'={\frac {l-{\frac {v}{c}}}{1-{\frac {lv}{c}}}},\ m'={\frac {m}{\beta \left(1-{\frac {lv}{c}}\right)}},\ n'={\frac {n}{\beta \left(1-{\frac {lv}{c}}\right)}}$.

From the equation for $\omega '$ it follows :— If an observer moves with the velocity v relative to an infinitely distant source of light emitting waves of frequency $\nu$, in such a manner that the line joining the source of light and the observer makes an angle of $\Phi$ with the velocity of the observer referred to a system of co-ordinates which is stationary with regard to the source, then the frequency $\nu '$ which is perceived by the observer is represented by the formula

$\nu '=\nu {\frac {1-\cos \Phi {\frac {v}{c}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$.

This is Doppler's principle for any velocity. If $\Phi =0$ then the equation takes the simple form

$\nu '=\nu \left({\frac {1-{\frac {v}{c}}}{1+{\frac {v}{c}}}}\right)^{\frac {1}{2}}$.

We see that —contrary to the usual conception— $\nu =\infty$, for $v=-c$.

If $\Phi '$=angle between the wave-normal (direction of the ray) in the moving system, and the line of motion of the observer, the equation for l' takes the form

$\cos \Phi '={\frac {\cos \Phi -{\frac {v}{c}}}{1-{\frac {v}{c}}\cos \Phi }}$.