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

we finally get

 {\displaystyle {\begin{aligned}\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{aligned}}} 12)

This is the general form (2) from which we started, but with constants entirely defined by ${\displaystyle \varkappa }$, \${\displaystyle alpha_{1}}$, ${\displaystyle \beta _{1}}$, ${\displaystyle \gamma _{1}}$, 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:

 {\displaystyle {\begin{aligned}\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{aligned}}} 13)

The condition (1') is in this case:

 ${\displaystyle 0={\frac {\varkappa }{\omega ^{2}}}{\frac {\partial }{\partial t}}\left(U\alpha _{1}+V\beta _{1}+W\gamma _{1}\right)}$ 13')

and with the assumed negligence it is only to the extent necessary to be fulfilled, that the term, which is multiplied in ${\displaystyle {\frac {\varkappa }{\omega }}}$, 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) ${\displaystyle x}$ by ${\displaystyle x'+\varkappa '\alpha 't}$, ${\displaystyle y}$ by ${\displaystyle y'+\varkappa '\beta 't}$, ${\displaystyle z}$ by ${\displaystyle z'+\varkappa '\gamma 't}$.

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

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

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

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