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

$(W)=A\sin {\frac {2\pi }{T}}\left(1-{\frac {\varkappa }{\omega }}\right)\left(t-{\frac {x}{\omega }}\right).$
This gives for *x = ϰt*:

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

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

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

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

Is the observer is in motion as well, then:

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

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

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