Page:Ueber das Doppler'sche Princip.djvu/7

Herein, we make the substitution according to (10), than we have

${\displaystyle (W)=A\sin {\frac {2\pi }{T}}\left(1-{\frac {\varkappa }{\omega }}\right)\left(t-{\frac {x}{\omega }}\right).}$

This gives for x = ϰt:

 ${\displaystyle ({\overline {W}})=A\sin {\frac {2\pi t}{T}}\left(1-{\frac {\varkappa ^{2}}{\omega ^{2}}}\right)=A\sin {\frac {2\pi t}{T'}}{,}}$ 14')

thus we have an illuminating plane (moving parallel to the X-axes), which oscillates with a wave-length ${\displaystyle \textstyle {T'=T/\left(1-{\frac {\varkappa ^{2}}{\omega ^{2}}}\right)}}$ (only different of the second order of T). The propagated oscillation can be written:

 ${\displaystyle (W)=A\sin {\frac {2\pi }{T'\left(1-{\frac {\varkappa }{\omega }}\right)}}\left(t-{\frac {x}{\omega }}\right).}$ 14)

Thus we get, within the propagated wave, a reduced period of oscillation in the relation of ${\displaystyle \left(1-{\frac {\varkappa }{\omega }}\right)/1}$.

Is the observer is in motion as well, then:

${\displaystyle (W')=A\sin {\frac {2\pi }{T'\left(1-{\frac {\varkappa }{\omega }}\right)}}\left(t-{\frac {x'+\varkappa 't}{\omega }}\right)}$ ${\displaystyle =A\sin 2\pi \left(t{\frac {(\omega +\varkappa '}{T'(\omega -\varkappa )}}-{\frac {x'}{T'(\omega -\varkappa )}}\right).}$

This formula gives the principle of Doppler for plane waves. But it is in no way universal, but essentially presupposes a plane wave with constant amplitude throughout.

2) The same plane is to be set in oscillation by the law:

${\displaystyle {\overline {W}}=Ae^{(\mu y+\nu z){\frac {2\pi }{T\omega }}}\sin {\frac {2\pi t}{T}}}$

- as it similar occurs when a wave with initially constant amplitude travels through a prism of an absorbing substance - then for the propagated wave it is given:

${\displaystyle W=Ae^{\frac {2\pi (\mu y+\nu z)}{T\omega }}\sin {\frac {2\pi }{T}}\left(t-{\frac {x\sigma }{\omega }}\right){\text{ where }}\sigma ={\sqrt {1+\mu ^{2}+\nu ^{2}}}.}$