Page:Ueber das Doppler'sche Princip.djvu/8

If we substitute according to (10), it is, if ${\displaystyle \textstyle {{\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}=q}}$ is set:

${\displaystyle (W)=Ae^{\frac {2\pi (\mu y+\nu z)q}{T\omega }}\sin {\frac {2\pi }{T}}\left[t\left(1+{\frac {\varkappa \sigma }{\omega }}\right)-x\left({\frac {\sigma }{\omega }}+{\frac {\varkappa }{\omega ^{2}}}\right)\right].}$

This gives for x = ϰt, if we write ${\displaystyle {\frac {\mu }{q}}=\mu '}$, ${\displaystyle {\frac {\nu }{q}}=\nu '}$:

${\displaystyle ({\overline {W}})=Ae^{\frac {2\pi (\mu 'y+\nu 'z)}{\omega T'}}\sin {\frac {2\pi t}{T}}{\text{, where }}T'={\frac {T}{1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}},}$

thus we have an oscillating and at the same time propagating plane; however, the propagated displacement reads:

 ${\displaystyle (W)=Ae^{\frac {2\pi (\mu 'y+\nu 'z)}{\omega T'}}\sin {\frac {2\pi t}{T}}\left(t{\frac {1+{\frac {\varkappa \sigma }{\omega }}}{1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}-x{\frac {{\frac {\sigma }{\omega }}+{\frac {\varkappa }{\omega ^{2}}}}{1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\right){,}}$ 15)

where we now have ${\displaystyle \sigma ={\sqrt {1+(\mu ^{'2}+\nu ^{'2})q^{2}}}}$.

We notice that different laws as the Doppler principle are given, even if we limit ourselves to the first approximation, and ϰ²ω² is neglected compared to 1.

3) If the illuminating surface is a very small[AU 1] sphere of radius R, which oscillates according to the law for the rotation angle

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

around the X-axis, then, at the distance ${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}$ from the center of the sphere, the propagated rotations ψ are given by[1][AU 2]

 ${\displaystyle \psi ={\frac {R^{3}A}{r^{3}}}\left[\sin {\frac {2\pi }{T}}\left(t-{\frac {r-R}{\omega }}\right)+{\frac {2\pi (r-R)}{T\omega }}\cos {\frac {2\pi }{T}}\left(t-{\frac {r-R}{\omega }}\right)\right]}$ ${\displaystyle ={\frac {R^{3}A}{r^{3}}}{\sqrt {1+\left({\frac {2\pi (r-R)}{T\omega }}\right)^{2}}}\cos {\frac {2\pi }{T}}\left(t-{\frac {r-R}{\omega }}-\eta \right){,}}$ 16)

where

${\displaystyle {\frac {2\pi (r-R)}{T\omega }}=\operatorname {ctg} {\frac {2\pi \eta }{T}}}$
1. W. Voigt, Crelles Journ. Vol. 89, 298.
1. This will be made more precise, so that the radius should be small compared to the wave-length. Yet the formulas (16) and (17) don't require this assumption:
2. There one also finds the laws for the emission of a linearly oscillating sphere, which allows the same way of use.