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

 we finally get

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:

The condition (1') is in this case:

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:

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