Page:Elektrische und Optische Erscheinungen (Lorentz) 057.jpg

is the component of ${\mathfrak {p}}$ with respect to the connection line $Q_{0}P$.

The "observed" period of oscillation is thus

$T'={\frac {T}{1+{\frac {p_{r}}{V}}}}=T\left(1-{\frac {{\mathfrak {p}}_{r}}{V}}\right){,}$

what is in agreement with the known law of Doppler^[1].

1. The derivation given here can easily be generalized so that it can be applied to all similar cases, for example also to sounding bodies. An arbitrary body A move with constant velocity ${\mathfrak {p}}$ in a medium that either remains at rest, or comes into a stationary state of motion. In this latter case (which also encloses the former one) we find at any point P, which translates with the body A, always the same state of motion, and it can be said, that the whole figure representing the distribution of velocities in the vicinity of A, shares the translation ${\mathfrak {p}}$.
   Furthermore, imagine now that the parts of the body perform simple oscillations of period T and of constant amplitude. It seems clear without further ado, when a sufficiently long time has elapsed since the beginning of this motion, that in the just-mentioned point P, the deviation from equilibrium or rather from the stationary state of flow, must necessarily have the period T. If we now introduce the co-ordinates x, y, z with respect to a system of axes progressing with the body (relative coordinates), and if we restrict ourselves to a space, that is so far from A and so small that we can speak of plane waves in it, then the above deviation can be represented by expressions of the form

   $\phi =A\cos {\frac {2\pi }{T}}\left\{t-{\frac {a_{x}x+a_{y}y+a_{z}z}{V}}+p\right\}.$

   (45)

   Here, $a_{x}$, $a_{y}$, $a_{z}$ are the direction constants of the wave normal, while V is the velocity of propagation.

   If we now want to know, by which frequency φ (in a stationary point) its sign is varying, then we have to introduce coordinates $\mathbf {x}$, $\mathbf {y}$, $\mathbf {z}$ with respect to stationary axes. By using the relations (44), (45) transforms into

   $\phi =A\cos {\frac {2\pi }{T}}\left\{t+{\frac {{\mathfrak {p}}_{n}}{V}}t-{\frac {a_{x}\mathbf {x} +a_{y}\mathbf {y} +a_{z}\mathbf {z} }{V}}+p\right\}{,}$

   where

   ${\mathfrak {p}}_{n}=a_{x}{\mathfrak {p}}_{x}+a_{y}{\mathfrak {p}}_{y}+a_{z}{\mathfrak {p}}_{z}$

   are the components of ${\mathfrak {p}}$ with respect to the wave normal

   For the observed oscillation period we now obtain

   $T'={\frac {T}{1+{\frac {p_{n}}{V}}}}=T\left(1-{\frac {{\mathfrak {p}}_{n}}{V}}\right).$

   What we have already stated without proof, namely that the period T exists throughout in the medium, is nothing else than what Petzval, in his attacks against Doppler's theory, called the law of the immutability of the oscillation period (Wiener Sitz.-Ber., vol 8, p. 134, 1852). He only forgot to notice, that this law only would apply, if we consider the phenomena as a function of t and the relative coordinates.

   The proof of the theorem is, by the way, easy to give, when the oscillations are infinitely small, and when we have to do with homogeneous linear differential equations.

   As regards the acoustic phenomena, the problem was discussed in detail by Was (Het beginsel van Doppler in de geluidsleer, Leiden, Engels, 1881).