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

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