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

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


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\sin2\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}}.