Page:Ueber das Doppler'sche Princip.djvu/10

 The wave surfaces are thus spheres, but not around the illuminating point, but constructed around a center, which location is far off by the $$\frac{\varkappa}{\omega}$$ part of their radii to the opposite direction of motion.

Therefore, a stationary observer, since the perpendicular to the wave surface through the location of observation gives the direction in which the light source is to be perceived, would see the illuminating point at the location where it was at time $$\frac{r}{\omega}$$, in other words; he would observe, if his radius vector r includes the angle $$\phi$$ with the direction of motion, an "aberration" of the size $$\frac{r}{\omega}\ \sin\varphi$$ in the direction opposite to the motion of the point.

Concerning the propagated amplitudes (M) and (N), according to the above they have, at position $$x\ y\ z$$ at time t, those values as if the illuminating point permanently remained at the attained position at this time t, whereas the wave surface in $$x\ y\ z$$ has the form, as if the illuminating point would remain at the attained location at time $$\textstyle{t-\frac{r}{\omega}}$$. So, the wave area and amplitude are not connected in the sense of a stationary illuminating point, because the latter depends on the present position, the former depends on an abandoned position of the illuminating point.

Thus, the peculiar result is given that such a moving illuminating point of constant intensity, which at time t has the distance r from the observer, is seen by him in that position, which he attained at time $$\frac{r}{\omega}$$, but with the intensity that corresponds to the current (larger or smaller) distance.

The applicability of the above general considerations on the problems of optics is limited by the constraint (1'), which has lead to the formulas (10') and (13').

Such a limitation does not take place in the analogous problems of the acoustics of fluids. For the propagated dilation δ we have as the only condition

$\frac{\partial^{2}\delta}{\partial t^{2}}=\omega^{2}\Delta\delta.$|undefined 