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

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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.