74.

To begin with, we consider the change of color which a monochromatic ray suffers by reflection from the reflector, which is 5 moving with an infinitely small velocity. For this purpose we consider first the case of a ray which falls normally from below on the reflector and hence is reflected normally downward. Let the plane $A$ ([fig:5]Fig. 5) represent the position of the reflector at the time $t$, the plane $A^{\prime }$ the position at the time $t+\delta t$, where the distance $A{A}^{\prime }$ equals $v\delta t$, $v$ denoting the velocity of the reflector. Let us now suppose a stationary plane $B$ to be placed parallel to $A$ at a suitable distance and let us denote by $\lambda$ the wave length of the ray incident on the reflector and by ${\lambda }^{\prime }$ the wave length of the ray reflected from it. Then at a time $t$ there are in the interval $AB$ in the vacuum containing the radiation ${\displaystyle \frac{AB}{\lambda }}$ waves of the incident and ${\displaystyle \frac{AB}{{\lambda }^{\prime }}}$ waves of the reflected ray, as can be seen, e.g., by thinking of the electric field-strength as being drawn at the different points of each of the two rays at the time $t$ in the form of a sine curve. Reckoning both incident and reflected ray there are at the time $t$ $AB(\frac{1}{\lambda }+\frac{1}{{\lambda }^{\prime }})$ waves in the interval between $A$ and $B$. Since this is a large number, it is immaterial whether the number is an integer or not.

Similarly at the time $t+\delta t$, when the reflector is at $A^{\prime }$, there are $A^{\prime }B(\frac{1}{\lambda }+\frac{1}{{\lambda }^{\prime }})$ waves in the interval between $A^{\prime }$ and $B$ all told.

The latter number will be smaller than the former, since in the shorter distance $A^{\prime }B$ there is room for fewer waves of both kinds than in the longer distance $AB$. The remaining waves must have been expelled in the time $\delta t$ from the space between the stationary plane $B$ and the moving reflector, and this must have taken place through the plane $B$ downward; for in no other way could a wave disappear from the space considered.

Now $\nu \delta t$ waves pass in the time $\delta t$ through the stationary plane $B$ in an upward direction and ${\nu }^{\prime }\delta t$ waves in a downward direction; hence we have for the difference $({\nu }^{\prime }-\nu )\delta t=(AB-A^{\prime }B)(\frac{1}{\lambda }+\frac{1}{{\lambda }^{\prime }})$ or, since $AB-A^{\prime }B=v\delta t,$ and

$$\begin{gathered} \lambda =\frac{c}{\nu }{\lambda }^{\prime }=\frac{c}{{\nu }^{\prime }} \\ {\nu }^{\prime }=\frac{c+v}{c-v}\nu \end{gathered}$$

or, since $v$ is infinitely small compared with $c$, ${\nu }^{\prime }=\nu (1+\frac{2v}{c}).$