Page:Ueber das Doppler'sche Princip.djvu/6

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we finally get

\begin{align}
\xi & =xq+(x\alpha_{1}+y\beta_{1}+z\gamma_{1})\alpha_{1}(1-q)-\varkappa\alpha_{1}t\\
\eta & =yq+(x\alpha_{1}+y\beta_{1}+z\gamma_{1})\beta_{1}(1-q)-\varkappa\beta_{1}t\\
\zeta & =zq+(x\alpha_{1}+y\beta_{1}+z\gamma_{1})\gamma_{1}(1-q)-\varkappa\gamma_{1}t\\
\tau & =t-\frac{\varkappa}{\omega^{2}}(x\alpha_{1}+y\beta_{1}+z\gamma_{1}).
\end{align} 12)

This is the general form (2) from which we started, but with constants entirely defined by \varkappa, \alpha_{1}, \beta_{1}, \gamma_{1}, it contains what is usually understood by the principle of Doppler, so far it is true.

If it is possible to neglect ϰ² next to ω², then q = 1 and we very simply obtain:

\begin{align}
\xi & =x-\varkappa\alpha_{1}t\\
\eta & =y-\varkappa\beta_{1}t\\
\zeta & =z-\varkappa\gamma_{1}t\\
\tau & =t-\frac{\varkappa}{\omega^{2}}(x\alpha_{1}+y\beta_{1}+z\gamma_{1}).
\end{align} 13)

The condition (1') is in this case:

0=\frac{\varkappa}{\omega^{2}}\frac{\partial}{\partial t}\left(U\alpha_{1}+V\beta_{1}+W\gamma_{1}\right) 13')

and with the assumed negligence it is only to the extent necessary to be fulfilled, that the term, which is multiplied in \frac{\varkappa}{\omega}, is of the first order.

If, besides the illuminating surface, the observer is also in motion, such as with the constant velocity ϰ' in a direction given by the direction cosines α', β ', γ', then the displacements u, v, w,, which are only related to a coordinate system X', Y', Z' moving with the observer, i.e., we must replace in (12) or (13) x by x'+\varkappa'\alpha't, y by y'+\varkappa'\beta't, z by z'+\varkappa'\gamma't.

With those findings we give some applications.

1) Let a plane parallel to the YZ-plane be set in vibrations in accordance with the law

\overline{W}=A\sin\frac{2\pi t}{T}{,}

then the motion propagated in positive X-axis is given by:

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