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

On the Principle of Doppler.
By
W. Voigt.

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

 ${\displaystyle {\frac {\partial ^{2}u}{dt^{2}}}=\omega ^{2}\Delta u}$ ${\displaystyle {\frac {\partial ^{2}v}{dt^{2}}}=\omega ^{2}\Delta v}$ ${\displaystyle {\frac {\partial ^{2}w}{dt^{2}}}=\omega ^{2}\Delta w}$ 1)

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:

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

Now let u = U, v = V, w = W be solutions of these equations, which on a given surface ${\displaystyle f({\bar {x}},{\bar {y}},{\bar {z}})=0}$ adopt given values ${\displaystyle {\bar {U}}}$, ${\displaystyle {\bar {V}}}$, ${\displaystyle {\bar {W}}}$ 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.