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

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On the Principle of Doppler. By W. Voigt.

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

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

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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:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0.$

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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.