75.

When the radiation does not fall on the reflector normally but at an acute angle of incidence $\theta$, it is possible to pursue a very similar line of reasoning, with the difference that then $A$, the point of intersection of a definite ray $BA$ with the reflector at the time $t$, has not the same position on the reflector as the point of intersection, $A^{\prime }$, of the same ray with the reflector at the time $t+\delta t$ ([fig:6]Fig. 6). The number of waves which lie in the interval $BA$ at the time $t$ is ${\displaystyle \frac{BA}{\lambda }}$. Similarly, at the time $t$ the number of waves in the interval $AC$ representing the distance of the point $A$

from a wave plane $C{C}^{\prime }$, belonging to the reflected ray and stationary in the vacuum, is ${\displaystyle \frac{AC}{{\lambda }^{\prime }}}$.

Hence there are, all told, at the time $t$ in the interval $BAC$ $\frac{BA}{\lambda }+\frac{AC}{{\lambda }^{\prime }}$ waves of the ray under consideration. We may further note that the angle of reflection ${\theta }^{\prime }$ is not exactly equal to the angle 6 of incidence, but is a little smaller as can be shown by a simple geometric consideration based on Huyghens' principle. The difference of $\theta$ and ${\theta }^{\prime }$, however, will be shown to be non-essential for our calculation.

Moreover there are at the time $t+\delta t$, when the reflector passes through $A^{\prime }$, $\frac{B{A}^{\prime }}{\lambda }+\frac{A^{\prime }{C}^{\prime }}{{\lambda }^{\prime }}$ waves in the distance $B{A}^{\prime }{C}^{\prime }$. The latter number is smaller than the former and the difference must equal the total number of waves which are expelled in the time $\delta t$ from the space which is bounded by the stationary planes $B{B}^{\prime }$ and $C{C}^{\prime }$.

Now $\nu \delta t$ waves enter into the space through the plane $B{B}^{\prime }$ in the time $\delta t$ and ${\nu }^{\prime }\delta t$ waves leave the space through the plane $C{C}^{\prime }$. Hence we have $({\nu }^{\prime }-\nu )\delta t=(\frac{BA}{\lambda }+\frac{AC}{{\lambda }^{\prime }})-(\frac{B{A}^{\prime }}{\lambda }+\frac{A^{\prime }{C}^{\prime }}{{\lambda }^{\prime }})$

but

$$\begin{gathered} BA-B{A}^{\prime }=A{A}^{\prime }=\frac{v\delta t}{\cos \theta } \\ AC-A^{\prime }{C}^{\prime }=A{A}^{\prime }\cos (\theta +{\theta }^{\prime }) \\ \lambda =\frac{c}{\nu },{\lambda }^{\prime }=\frac{c}{{\nu }^{\prime }}. \end{gathered}$$

Hence ${\nu }^{\prime }=\frac{c\cos \theta +v}{c\cos \theta -v\cos (\theta +{\theta }^{\prime })}\nu .$

This relation holds for any velocity $v$ of the moving reflector. Now, since in our case $v$ is infinitely small compared with $c$, we have the simpler expression ${\nu }^{\prime }=\nu (1+\frac{v}{c\cos \theta }[1+\cos (\theta +{\theta }^{\prime })]).$ The difference between the two angles $\theta$ and ${\theta }^{\prime }$ is in any case of the order of magnitude ${\displaystyle \frac{v}{c}}$; hence we may without appreciable error replace ${\theta }^{\prime }$ by $\theta$, thereby obtaining the following expression for the frequency of the reflected ray for oblique incidence ${\nu }^{\prime }=\nu (1+\frac{2v\cos \theta }{c}).\mathrm{\backslash Label}[eqn](83)\text{upshape (83)}$