Page:VaricakRel1912.djvu/19

8. The principle of Doppler.

From equation (43) we can see the frequency that an observer moving along the X-axis with velocity v will find. The light ray encloses the angle $\varphi$ with the X-axis (measured in System S). Considered as a parallel angle, this angle corresponds to the length $u_{1}$, and therefore we have the relation

(49)	$\cos \varphi =\operatorname {th} \,u_{1}$

and (43) goes over into

(50)	$\nu '=\nu \left(\operatorname {ch} \,u-\operatorname {th} \,u_{1}\operatorname {sh} \,u\right)$

from which we easily obtain

(51)	${\frac {\nu '}{\nu }}={\frac {\operatorname {ch} \,\left(u_{1}-u\right)}{\operatorname {ch} \,u_{1}}}$

We take two equidistant lines (Fig. 10) with the X-axis as their common center line and with the parameters $OA=u_{1},\ OA_{1}=u_{1}-u.$. The arcs of those two distance lines limited by the Y-axis and the perpendicular to the X-axis in point C, are

$AB=OC\ \operatorname {ch} \,u_{1},\ A_{1}B_{1}=OC\ \operatorname {ch} \,\left(u_{1}-u\right)$

Thus it is

(52)	${\frac {\nu '}{\nu }}={\frac {A_{1}B_{1}}{AB}}$

The ratio of the frequencies $\nu$ and $\nu '$ can be represented in the general case as the ratio of the arcs of two distance lines between common perpendiculars.

Now, let us assume that light propagates in the direction of the X-axis. The parallel angle $\varphi =0$ corresponds to the length $u_{1}=\infty$, whose hyperbolic tangens is one. From (43) we obtain in this case

$\nu '=\nu \left(\operatorname {ch} \,u-\operatorname {sh} \,u\right)$

or

(53)	$\nu '=\nu \cdot e^{-u}$

For $u_{1}=\infty$ those distance lines go over into the limiting circles with the common axes. Thus formula (53) agrees with Fig. 11. The ratio of the frequencies can also in this case be represented as the ratio of two limiting arcs between two common axes.