Page:TolmanPostulate.djvu/6

true of any periodic disturbance. Let c be the velocity with which the disturbance travels, in this case that of light, λ₀ and n₀ the observed wave-length and frequency (of some particular line in the spectrum), when the source is at rest with respect to the observer, and λ and n the same quantities after the source has been set in motion towards the observer with the velocity v. Let us first consider—

Case I.— The velocity of light is independent of the velocity of the source. Then,

$\lambda =\lambda _{0}-{\frac {v}{c}}\lambda _{0}=\lambda _{0}\left(1-{\frac {v}{c}}\right)$

This is the ordinary formula for the Doppler effect and the derivation is simple, since it is evident that while an emitted wave-front is moving forward the distance λ₀, the source itself has moved forward the distance λ₀&midpoint;v/c, and the next wave-front to leave the source will have gained this distance over the earlier one. The frequency of the light will be equal to the velocity divided by the wave-length.

$n={\frac {c}{\lambda }}={\frac {c}{\lambda _{0}}}\left({\frac {c}{c-v}}\right)=n_{0}{\frac {c}{c-v}}=n_{0}\left(1+{\frac {v}{c}}+{\frac {v^{2}}{c^{2}}}+\cdot \cdot \cdot \right)$.

Case II. — The velocity of light and that of the source are additive, the velocity with which light passes the observer is v+c.

λ=λ₀.

This result is evident on inspection, since, under the conditions assumed, the velocity of light relative to the source is always the same, and the source and its surrounding disturbances move together as a whole, suffering no permanent change in configuration when the velocity of the source is changed. In detail, however, we see that, with respect to the observer, an emitted wave front moves forward the distance λ₀·(c+v)/c during the interval of time which elapses before the next wave front leaves the source, and during that time the source has moved forward the distance

${\frac {v}{c+v}}\lambda _{0}{\frac {c+v}{c}}=\lambda _{0}{\frac {v}{c}}$,