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

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

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

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

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

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

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

15)

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

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

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

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

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

16)

where

${\frac {2\pi (r-R)}{T\omega }}{=}\operatorname {ctg} {\frac {2\pi \eta }{T}}$

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.