Page:Elektrische und Optische Erscheinungen (Lorentz) 056.jpg

In an area that isn't too extended, we may also regard $b_{x}$, $b_{y}$, $b_{z}$ as constant, and thus regard the motion as a system of plane waves. The direction constants $b'_{x}$, $b'_{y}$, $b'_{z}$ of the wave normal are obviously to be determined from the condition

$b'_{x}:b'_{y}:b'_{z}=\left(b_{x}+{\frac {{\mathfrak {p}}_{x}}{V}}\right):\left(b_{y}+{\frac {{\mathfrak {p}}_{y}}{V}}\right):\left(b_{z}+{\frac {{\mathfrak {p}}_{z}}{V}}\right)$

(43)

For ${\mathfrak {p}}=0$, $b'_{x}$, $b'_{y}$, $b'_{z}$ fall into $b_{x}$, $b_{y}$, $b_{z}$, and the waves are perpendicular to $Q_{0}P$. This is not the case if the light source is moving. From (43) follows, that the waves are perpendicular to the line that connects P with that point at which the light source was at the moment, when the light was sent that reaches P at time t.

The law of Doppler.

§ 37. In a point that moves together with the luminous molecule — and thus also for an observer who shares the translation — the values of ${\mathfrak {d}}_{x},...{\mathfrak {H}}_{x},...$ are changing, as we have seen (§ 30), as often in unit time as it corresponds to the actual period of oscillation T of the ions.

We can also examine, with which frequency these values in a stationary point are changing their sign. This frequency causes the oscillation period for a stationary observer. The question can be solved immediately, if instead of x, y, z we introduce new coordinates $\mathbf {x}$, $\mathbf {y}$, $\mathbf {z}$, which refer to a stationary system of axes. If the two systems have the same directions of axes and the same origin at time t = 0, then

$x=\mathbf {x} -{\mathfrak {p}}_{x}t,\ y=\mathbf {y} -{\mathfrak {p}}_{y}t,\ z=\mathbf {z} -{\mathfrak {p}}_{z}t{,}$

(44)

and by (42) for ${\mathfrak {d}}_{x},...{\mathfrak {H}}_{x},...$ we obtain expressions of the form

$A\cos {\frac {2\pi }{T}}\left\{t+{\frac {{\mathfrak {p}}_{r}}{V}}t-\left({\frac {b_{x}}{V}}+{\frac {{\mathfrak {p}}_{x}}{V^{2}}}\right)\mathbf {x} -{\text{etc.}}...+C\right\}{,}$

where

${\mathfrak {p}}_{r}=b_{x}{\mathfrak {p}}_{x}+b_{y}{\mathfrak {p}}_{y}+b_{z}{\mathfrak {p}}_{z}$