Page:Ueber das Doppler'sche Princip.djvu/1

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On the Principle of Doppler.
W. Voigt.

It is known that the differential equations for the oscillations of an elastic medium are:

\frac{\partial^{2}u}{dt^{2}}=\omega^{2}\Delta u

\frac{\partial^{2}v}{dt^{2}}=\omega^{2}\Delta v

\frac{\partial^{2}w}{dt^{2}}=\omega^{2}\Delta w


where ω is the propagation velocity of the oscillations - or more precisely the propagation velocity of plane waves with constant amplitude. It is presupposed that u, v, w fulfill the relation:

\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0. 1')

Now let u = U, v = V, w = W be solutions of these equations, which on a given surface f(\bar{x},\bar{y},\bar{z})=0 adopt given values \bar{U}, \bar{V}, \bar{W} which depend on time, then we can say that these functions U, V, W represent the law by which the surface f = 0 is illuminating.