# Translation:On the Principle of Doppler

On the Principle of Doppler  (1887)
by Woldemar Voigt, translated from German by Wikisource
In German: Ueber das Doppler’sche Princip, Göttinger Nachrichten, 1887 (2): 41-51. Session from January 8, 1887. Published March 10, 1887

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.

If we substitute in ${\displaystyle U,V,W,}$ respectively,

 ${\displaystyle {\begin{array}{l}x{\text{ by }}\xi =xm_{1}+yn_{1}+zp_{1}-\alpha t\\y{\text{ by }}\eta =xm_{2}+yn_{2}+zp_{2}-\beta t\\z{\text{ by }}\xi =xm_{3}+yn_{3}+zp_{3}-\gamma t\\t{\text{ by }}\tau =t-(ax+by+cz)\\\end{array}}}$ 2)

and describe the resulting functions, respectively, with (U), (V), (W), then by u = (U), v = (V), w = (W) it is possible to comply with (1). [AU 1]

For example, we obtain for the first of them:

${\displaystyle {\frac {\partial ^{2}(U)}{\partial \tau ^{2}}}\left(1-\omega ^{2}\left(a^{2}+b^{2}+c^{2}\right)\right)=\omega ^{2}\left\{{\frac {\partial ^{2}(U)}{\partial \xi ^{2}}}\left(m_{1}^{2}+n_{1}^{2}+p_{1}^{2}-{\frac {\alpha ^{2}}{\omega ^{2}}}\right)\right.}$
${\displaystyle +{\frac {\partial ^{2}(U)}{\partial \eta ^{2}}}\left(m_{2}^{2}+n_{2}^{2}+p_{2}^{2}-{\frac {\beta ^{2}}{\omega ^{2}}}\right)+{\frac {\partial ^{2}(U)}{\partial \zeta ^{2}}}\left(m_{3}^{2}+n_{3}^{2}+3_{3}^{2}-{\frac {\gamma ^{2}}{\omega ^{2}}}\right)}$
${\displaystyle +2{\frac {\partial ^{2}(U)}{\partial \eta \ \partial \zeta }}\left(m_{2}m_{3}+n_{2}n_{3}+p_{2}p_{3}-{\frac {\beta \gamma }{\omega ^{2}}}\right)}$
${\displaystyle +2{\frac {\partial ^{2}(U)}{\partial \zeta \ \partial \xi }}\left(m_{3}m_{1}+n_{3}n_{1}+p_{3}p_{1}-{\frac {\gamma \alpha }{\omega ^{2}}}\right)}$
${\displaystyle +2{\frac {\partial ^{2}(U)}{\partial \xi \ \partial \eta }}\left(m_{1}m_{2}+n_{1}n_{2}+p_{1}p_{2}-{\frac {\alpha \beta }{\omega ^{2}}}\right)}$
${\displaystyle -2{\frac {\partial ^{2}(U)}{\partial \tau \ \partial \xi }}\left(am_{1}+bn_{1}+cp_{1}-{\frac {\alpha }{\omega ^{2}}}\right)}$
${\displaystyle -2{\frac {\partial ^{2}(U)}{\partial \tau \ \partial \eta }}\left(am_{2}+bn_{2}+cp_{2}-{\frac {\beta }{\omega ^{2}}}\right)}$
${\displaystyle \left.-2{\frac {\partial ^{2}(U)}{\partial \tau \ \partial \zeta }}\left(am_{3}+bn_{3}+cp_{3}-{\frac {\gamma }{\omega ^{2}}}\right)\right\}}$

and this is fulfilled, because it indeed has to be:

${\displaystyle {\frac {\partial ^{2}(U)}{\partial \tau ^{2}}}=\omega ^{2}\left({\frac {\partial ^{2}(U)}{\partial \xi ^{2}}}+{\frac {\partial ^{2}(U)}{\partial \eta ^{2}}}+{\frac {\partial ^{2}(U)}{\partial \zeta ^{2}}}\right),}$

if there exist the following new equations:

 ${\displaystyle {\begin{array}{rl}1-\omega ^{2}(a^{2}+b^{2}+c^{2})&=m_{1}^{2}+n_{1}^{2}+p_{1}^{2}-{\frac {\alpha ^{2}}{\omega ^{2}}}\\&=m_{2}^{2}+n_{2}^{2}+p_{2}^{2}-{\frac {\beta ^{2}}{\omega ^{2}}}\\&=m_{3}^{2}+n_{3}^{2}+p_{3}^{2}-{\frac {\gamma ^{2}}{\omega ^{2}}}\end{array}}}$ 3)
 ${\displaystyle {\frac {\beta \gamma }{\omega ^{2}}}=m_{2}m_{3}+n_{2}n_{3}+p_{2}p_{3}}$ ${\displaystyle {\frac {\gamma \alpha }{\omega ^{2}}}=m_{3}m_{1}+n_{3}n_{1}+p_{3}p_{1}}$ ${\displaystyle {\frac {\alpha \beta }{\omega ^{2}}}=m_{1}m_{2}+n_{1}n_{2}+p_{1}p_{2}}$ 4)
 ${\displaystyle {\frac {\alpha }{\omega ^{2}}}=am_{1}+bn_{1}+cp_{1}}$ ${\displaystyle {\frac {\beta }{\omega ^{2}}}=am_{2}+bn_{2}+cp_{2}}$ ${\displaystyle {\frac {\gamma }{\omega ^{2}}}=am_{3}+bn_{3}+cp_{3}}$ 5)

If we take ${\displaystyle \alpha \beta \gamma }$ as given, then we have 12 constants available, so we can arbitrarily use three of them.

The solution is most comfortable when we use a temporary co-ordinate system X1, Y1, Z1, for which β and γ disappear in equations (2), α is equal to ϰ, that is, a co-ordinate system whose X1-axis falls in the direction, of which the direction cosine is proportional to X, Y, Z with α, β, γ.

Furthermore, it should be set

${\displaystyle {\begin{array}{clcclcclcclc}m_{h}^{2}+n_{h}^{2}+p_{h}^{2}&=&q_{h}^{2},&m_{h}/q_{h}&=&\mu _{h},&n_{h}/q_{h}&=&\nu _{h},&p_{h}/q_{h}&=&\pi _{h}\\\\a^{2}+b^{2}+c^{2}&=&d^{2},&a/d&=&\mu ,&b/d&=&\nu ,&c/d&=&\pi ,\end{array}}}$

then μ, ν, π are the direction cosines of 4 directions, which we will denote by δ1, δ2, δ3 and δ, against the system X1, Y1, Z1.

By these introductions our equations (3), (4) and (5) will be:

 ${\displaystyle 1-\omega ^{2}d^{2}=q_{1}^{2}-{\frac {\varkappa ^{2}}{\omega ^{2}}}=q_{2}^{2}=q_{3}^{2}}$ 3')

${\displaystyle \mu _{2}\mu _{3}+\nu _{2}\nu _{3}+\pi _{2}\pi _{3}=\mu _{3}\mu _{1}+\nu _{3}\nu _{1}+\pi _{3}\pi _{1}=\mu _{1}\mu _{2}+\nu _{1}\nu _{2}+\pi _{1}\pi _{2}=0}$

 ${\displaystyle {\text{that is }}\cos(\delta _{2},\delta _{3})=\cos(\delta _{3},\delta _{1})=\cos(\delta _{1},\delta _{2})=0}$ 4')

${\displaystyle \mu \mu _{1}+\nu \nu _{1}+\pi \pi _{1}={\frac {\varkappa }{\omega ^{2}q_{1}d}},\ \mu \mu _{2}+\nu \nu _{2}+\pi \pi _{2}+\mu \mu _{3}+\nu \nu _{3}+\pi \pi _{3}=0}$

 ${\displaystyle {\text{that is }}\cos(\delta ,\delta _{1})={\frac {\varkappa }{\omega ^{2}q_{1}d}},\ \cos(\delta ,\delta _{2})=\cos(\delta ,\delta _{3})=0.}$ 5')

According to (4'), the three directions δ1, δ2, δ3 are perpendicular to each other, according to (5') ${\displaystyle \delta _{1}}$ falls into δ, then it must be:

 ${\displaystyle \mu =\mu _{1},\ \nu =\nu _{1},\ \pi =\pi _{1}\ {\text{ and }}\ {\frac {\varkappa }{\omega ^{2}q_{1}d}}=1.}$ 6)
Substituting this in (3'), q1, q2, q3 are determined.

At first we obtain, since only positive signs are meaningful:

${\displaystyle q_{1}=1{\text{ or }}{\frac {\varkappa }{\omega }}}$ ${\displaystyle d={\frac {\varkappa }{\omega ^{2}}}{\text{ or }}{\frac {1}{\omega }}.}$

I will only use the first solution, the second is of no interest;[1] it follows from it:

 ${\displaystyle d={\frac {\varkappa }{\omega ^{2}}},\ q_{1}=1,\ q_{2}=q_{3}={\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}=q.}$ 7)

Consequently, we can write equations (2):

 {\displaystyle {\begin{aligned}\xi _{1}&=x_{1}\mu _{1}+y_{1}\nu _{1}+z_{1}\pi _{1}-\varkappa t&&=a_{1}-\varkappa t\\\eta _{1}&=\left(x_{1}\mu _{2}+y_{1}\nu _{2}+z_{1}\pi _{2}\right)q&&=b_{1}q\\\zeta _{1}&=\left(x_{1}\mu _{3}+y_{1}\nu _{3}+z_{1}\pi _{3}\right)q&&=c_{1}q\\\tau &=t-{\frac {\varkappa }{\omega ^{2}}}(\mu _{1}x+\nu _{1}y+\pi _{1}z)&&=t-{\frac {\varkappa a_{1}}{\omega ^{2}}}{,}\end{aligned}}} 8)

where for μh, νh, πh no more other conditions apply than those which result from their meaning as direction cosines of three successive perpendicular but otherwise quite arbitrary directions.

Therefore, the aggregates designated by ${\displaystyle a_{1}\ b_{1}\ c_{1}}$ can be considered as the coordinates of the point ${\displaystyle x_{1}\ y_{1}\ z_{1}}$ in relation to a coordinate system, which falls into the direction ${\displaystyle \delta _{1}\ \delta _{2}\ \delta _{3}}$.

Any such system μh, νh, πh gives a solution (U), (V), (W) from given U, V, W. If U, V, W adopt on a surface f(x, y, z) = 0 the given values ${\displaystyle {\overline {U}}}$, ${\displaystyle {\overline {V}}}$, ${\displaystyle {\overline {W}}}$, so (U), (V), (W) from those derivable ${\displaystyle ({\overline {U}}),({\overline {V}}),({\overline {W}})}$ to the surface ${\displaystyle (f)=f({\overline {\xi _{1}}},{\overline {\eta _{1}}},{\overline {\zeta _{1}}})=0}$, which because of the values of ξ1, η1, ζ1 has the property to move with uniform velocity ϰ parallel to a direction δ1 or A given by direction cosines ϰ. The solutions (U), (V), (W) give thus the laws by which certain surfaces in progressive motion are shining, if they only comply with the condition

${\displaystyle {\frac {(\partial U)}{\partial x}}+{\frac {\partial (V)}{\partial y}}+{\frac {\partial (W)}{\partial z}}=0}$
The two surfaces f = 0 and (f) = 0 have identical forms only if q = 1, i.e. ϰ is so small against ω, that ϰ² can be neglected with respect to ω². If this is the case, then they differ only by their position against the coordinate axes. By appropriate use of the arbitrary constants and the functions U, V, W we can obtain vivid special cases. By coordinate transformation we are lead to a (at least formally) general case, in which the shift of the surface is not parallel to the A-axis parallel, but directed in an arbitrary way.

We follow the special case, in which the three directions δ1, δ2, δ3 fall into the coordinate axes X1, Y1, Z1, that is

 ${\displaystyle \mu _{1}=\nu _{2}=\pi _{3}=1{,}}$ ${\displaystyle \mu _{2}=\mu _{3}=\nu _{1}=\nu _{2}=\pi _{1}=\pi _{3}=0}$ 9)

Then it is given, in a very simple and natural way, and formally identical with (8):

 {\displaystyle {\begin{aligned}\xi _{1}&=x_{1}-\varkappa t\\\eta _{1}&=y_{1}q\\\zeta _{1}&=z_{1}q\\\tau &=t-{\frac {\varkappa x_{1}}{\omega ^{2}}}{,}{\text{ where }}q={\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\end{aligned}}}[AU 2] 10)

The condition (1') is in this case

${\displaystyle (1-q){\frac {\partial (U)}{\partial \xi }}={\frac {\varkappa }{\omega ^{2}}}{\frac {\partial (U)}{\partial \tau }}}$

which can easily be exchanged with

 ${\displaystyle (1-q){\frac {\partial U}{\partial x}}={\frac {\varkappa }{\omega ^{2}}}{\frac {\partial U}{\partial t}}.}$ 10')

This states, that in U the arguments x and t only may occur in connection with ${\displaystyle (1-q)t+{\frac {\varkappa x}{\omega ^{2}}}}$, or not at all. The latter is the case if U = 0, that is, when the propagated vibrations are everywhere normal to the direction of translation of the illuminating surface.

If we pass from the assumed special co-ordinate system X1, Y1, Z1 to the general X, Y, Z, which is connected with the preceding by the relations

 ${\displaystyle x_{1}=x\alpha _{1}+y\beta _{1}+z\gamma _{1}}$ ${\displaystyle y_{1}=x\alpha _{2}+y\beta _{2}+z\gamma _{2}}$ ${\displaystyle z_{1}=x\alpha _{3}+y\beta _{3}+z\gamma _{3},}$ 11)
we finally get
 {\displaystyle {\begin{aligned}\xi &=xq+(x\alpha _{1}+y\beta _{1}+z\gamma _{1})\alpha _{1}(1-q)-\varkappa \alpha _{1}t\\\eta &=yq+(x\alpha _{1}+y\beta _{1}+z\gamma _{1})\beta _{1}(1-q)-\varkappa \beta _{1}t\\\zeta &=zq+(x\alpha _{1}+y\beta _{1}+z\gamma _{1})\gamma _{1}(1-q)-\varkappa \gamma _{1}t\\\tau &=t-{\frac {\varkappa }{\omega ^{2}}}(x\alpha _{1}+y\beta _{1}+z\gamma _{1}).\end{aligned}}} 12)

This is the general form (2) from which we started, but with constants entirely defined by ${\displaystyle \varkappa }$, \${\displaystyle alpha_{1}}$, ${\displaystyle \beta _{1}}$, ${\displaystyle \gamma _{1}}$, it contains what is usually understood by the principle of Doppler, so far it is true.

If it is possible to neglect ϰ² next to ω², then q = 1 and we very simply obtain:

 {\displaystyle {\begin{aligned}\xi &=x-\varkappa \alpha _{1}t\\\eta &=y-\varkappa \beta _{1}t\\\zeta &=z-\varkappa \gamma _{1}t\\\tau &=t-{\frac {\varkappa }{\omega ^{2}}}(x\alpha _{1}+y\beta _{1}+z\gamma _{1}).\end{aligned}}} 13)

The condition (1') is in this case:

 ${\displaystyle 0={\frac {\varkappa }{\omega ^{2}}}{\frac {\partial }{\partial t}}\left(U\alpha _{1}+V\beta _{1}+W\gamma _{1}\right)}$ 13')

and with the assumed negligence it is only to the extent necessary to be fulfilled, that the term, which is multiplied in ${\displaystyle {\frac {\varkappa }{\omega }}}$, is of the first order.

If, besides the illuminating surface, the observer is also in motion, such as with the constant velocity ϰ' in a direction given by the direction cosines α', β ', γ', then the displacements u, v, w,, which are only related to a coordinate system X', Y', Z' moving with the observer, i.e., we must replace in (12) or (13) ${\displaystyle x}$ by ${\displaystyle x'+\varkappa '\alpha 't}$, ${\displaystyle y}$ by ${\displaystyle y'+\varkappa '\beta 't}$, ${\displaystyle z}$ by ${\displaystyle z'+\varkappa '\gamma 't}$.

With those findings we give some applications.

1) Let a plane parallel to the YZ-plane be set in vibrations in accordance with the law

${\displaystyle {\overline {W}}=A\sin {\frac {2\pi t}{T}}{,}}$

then the motion propagated in positive X-axis is given by:

${\displaystyle W=A\sin {\frac {2\pi }{T}}\left(t-{\frac {x}{\omega }}\right).}$
Herein, we make the substitution according to (10), than we have

${\displaystyle (W){=}A\sin {\frac {2\pi }{T}}\left(1-{\frac {\varkappa }{\omega }}\right)\left(t-{\frac {x}{\omega }}\right).}$

This gives for x = ϰt:

 ${\displaystyle ({\overline {W}})=A\sin {\frac {2\pi t}{T}}\left(1-{\frac {\varkappa ^{2}}{\omega ^{2}}}\right)=A\sin {\frac {2\pi t}{T'}}{,}}$ 14')

thus we have an illuminating plane (moving parallel to the X-axes), which oscillates with a wave-length ${\displaystyle \textstyle {T'=T/\left(1-{\frac {\varkappa ^{2}}{\omega ^{2}}}\right)}}$ (only different of the second order of T). The propagated oscillation can be written:

 ${\displaystyle (W)=A\sin {\frac {2\pi }{T'\left(1-{\frac {\varkappa }{\omega }}\right)}}\left(t-{\frac {x}{\omega }}\right).}$ 14)

Thus we get, within the propagated wave, a reduced period of oscillation in the relation of ${\displaystyle \left(1-{\frac {\varkappa }{\omega }}\right)/1}$.

Is the observer is in motion as well, then:

${\displaystyle (W'){=}A\sin {\frac {2\pi }{T'\left(1-{\frac {\varkappa }{\omega }}\right)}}\left(t-{\frac {x'+\varkappa 't}{\omega }}\right)}$

${\displaystyle {=}A\sin 2\pi \left(t{\frac {(\omega +\varkappa '}{T'(\omega -\varkappa )}}-{\frac {x'}{T'(\omega -\varkappa )}}\right).}$

This formula gives the principle of Doppler for plane waves. But it is in no way universal, but essentially presupposes a plane wave with constant amplitude throughout.

2) The same plane is to be set in oscillation by the law:

${\displaystyle {\overline {W}}{=}Ae^{(\mu y+\nu z){\frac {2\pi }{T\omega }}}\sin {\frac {2\pi t}{T}}}$

- as it similar occurs when a wave with initially constant amplitude travels through a prism of an absorbing substance - then for the propagated wave it is given:

${\displaystyle W{=}Ae^{\frac {2\pi (\mu y+\nu z)}{T\omega }}\sin {\frac {2\pi }{T}}\left(t-{\frac {x\sigma }{\omega }}\right){\text{ where }}\sigma {=}{\sqrt {1+\mu ^{2}+\nu ^{2}}}.}$

If we substitute according to (10), it is, if ${\displaystyle \textstyle {{\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}=q}}$ is set:

${\displaystyle (W){=}Ae^{\frac {2\pi (\mu y+\nu z)q}{T\omega }}\sin {\frac {2\pi }{T}}\left[t\left(1+{\frac {\varkappa \sigma }{\omega }}\right)-x\left({\frac {\sigma }{\omega }}+{\frac {\varkappa }{\omega ^{2}}}\right)\right].}$

This gives for x = ϰt, if we write ${\displaystyle {\frac {\mu }{q}}=\mu '}$, ${\displaystyle {\frac {\nu }{q}}=\nu '}$:

${\displaystyle ({\overline {W}}){=}Ae^{\frac {2\pi (\mu 'y+\nu 'z)}{\omega T'}}\sin {\frac {2\pi t}{T}}{\text{, where }}T'{=}{\frac {T}{1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}},}$

thus we have an oscillating and at the same time propagating plane; however, the propagated displacement reads:

 ${\displaystyle (W)=Ae^{\frac {2\pi (\mu 'y+\nu 'z)}{\omega T'}}\sin {\frac {2\pi t}{T}}\left(t{\frac {1+{\frac {\varkappa \sigma }{\omega }}}{1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}-x{\frac {{\frac {\sigma }{\omega }}+{\frac {\varkappa }{\omega ^{2}}}}{1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\right){,}}$ 15)

where we now have ${\displaystyle \sigma ={\sqrt {1+(\mu ^{'2}+\nu ^{'2})q^{2}}}}$.

We notice that different laws as the Doppler principle are given, even if we limit ourselves to the first approximation, and ϰ²ω² is neglected compared to 1.

3) If the illuminating surface is a very small[AU 3] sphere of radius R, which oscillates according to the law for the rotation angle

${\displaystyle {\overline {\psi }}{=}A\sin {\frac {2\pi t}{T}}}$

around the X-axis, then, at the distance ${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}$ from the center of the sphere, the propagated rotations ψ are given by[2][AU 4]

 ${\displaystyle \psi ={\frac {R^{3}A}{r^{3}}}\left[\sin {\frac {2\pi }{T}}\left(t-{\frac {r-R}{\omega }}\right)+{\frac {2\pi (r-R)}{T\omega }}\cos {\frac {2\pi }{T}}\left(t-{\frac {r-R}{\omega }}\right)\right]}$ ${\displaystyle ={\frac {R^{3}A}{r^{3}}}{\sqrt {1+\left({\frac {2\pi (r-R)}{T\omega }}\right)^{2}}}\cos {\frac {2\pi }{T}}\left(t-{\frac {r-R}{\omega }}-\eta \right){,}}$ 16)

where

${\displaystyle {\frac {2\pi (r-R)}{T\omega }}{=}\operatorname {ctg} {\frac {2\pi \eta }{T}}}$

is set. It is therefore ${\displaystyle \textstyle {\eta ={\frac {T}{4}}}}$ for r = R and η = 0m if r is very great compared with the wavelength Tω.

The propagated displacements follow from ψ by:

${\displaystyle U{=}0,\ V=-\psi z,\ W=\psi y;}$

we briefly set

${\displaystyle U{=}0,\ V=MC,\ W=NC.}$

Substituting herein for x, y, z, the values ξ, η, ζ according to (10), then the periodic part C is given by:

 ${\displaystyle (C)=\cos {\frac {2\pi }{T}}\left(t-{\frac {\varkappa x}{\omega ^{2}}}-{\frac {1}{\omega }}\left({\sqrt {(x-\varkappa t)^{2}+y^{2}+z^{2}}}-R\right)-(\eta )\right){,}}$ 17)

if ${\displaystyle \operatorname {cotg} {\frac {2\pi (\eta )}{T}}{=}{\frac {2\pi }{T\omega }}\left({\sqrt {(x-\varkappa t)^{2}+y^{2}+z^{2}}}-R\right)}$

For ${\displaystyle (x-\varkappa t)^{2}+y^{2}+z^{2}=R^{2}}$, i.e., on the surface of a sphere which is displaced parallel to the X-axis, this becomes

${\displaystyle ({\overline {C}}){=}\sin {\frac {2\pi }{T}}\left(t\left(1-{\frac {\varkappa ^{2}}{\omega ^{2}}}\right)-{\frac {\varkappa }{\omega ^{2}}}{\sqrt {R^{2}-y^{2}-z^{2}}}\right),}$

thus, since under the assumption ${\displaystyle \textstyle {\frac {\varkappa ^{2}}{\omega ^{2}}}}$ and ${\displaystyle \textstyle {\frac {\varkappa R}{\omega ^{2}}}}$ of second-order it follows:

${\displaystyle ({\overline {C}}){=}\sin {\frac {2\pi t}{T}}.}$

(M) and (N) have the same value, as if the little sphere would oscillate as a state of equilibrium around the attained position ${\displaystyle x_{0}=\varkappa t}$ at time t. Therefore, by (U), (V), (W) we get the motion that was submitted by a rotating "illuminating point" with translational speed ${\displaystyle \varkappa }$ parallel to the direction of the rotation axis.

The propagated wave surfaces are assessed according to the value (17) for (C), which can be written by introduction of relative coordinates against the moving luminous point ${\displaystyle \xi =x-\varkappa t}$, ${\displaystyle y=\eta }$, ${\displaystyle z=\zeta }$ (neglecting ${\displaystyle \textstyle {\frac {\varkappa ^{2}}{\omega ^{2}}}}$ against 1) and for r which is great against ${\displaystyle T\omega }$:

${\displaystyle (C){=}\cos {\frac {2\pi }{T}}\left(t-{\frac {\varkappa \xi }{\omega ^{2}}}-{\frac {1}{\omega }}\left({\sqrt {\xi ^{2}+\eta ^{2}+\zeta ^{2}}}-R\right)\right).}$

The wave surfaces are thus spheres, but not around the illuminating point, but constructed around a center, which location is far off by the ${\displaystyle {\frac {\varkappa }{\omega }}}$ part of their radii to the opposite direction of motion.

Therefore, a stationary observer, since the perpendicular to the wave surface through the location of observation gives the direction in which the light source is to be perceived, would see the illuminating point at the location where it was at time ${\displaystyle {\frac {r}{\omega }}}$, in other words; he would observe, if his radius vector r includes the angle ${\displaystyle \phi }$ with the direction of motion, an "aberration" of the size ${\displaystyle {\frac {r}{\omega }}\ \sin \varphi }$ in the direction opposite to the motion of the point.

Concerning the propagated amplitudes (M) and (N), according to the above they have, at position ${\displaystyle x\ y\ z}$ at time t, those values as if the illuminating point permanently remained at the attained position at this time t, whereas the wave surface in ${\displaystyle x\ y\ z}$ has the form, as if the illuminating point would remain at the attained location at time ${\displaystyle \textstyle {t-{\frac {r}{\omega }}}}$. So, the wave area and amplitude are not connected in the sense of a stationary illuminating point, because the latter depends on the present position, the former depends on an abandoned position of the illuminating point.

Thus, the peculiar result is given that such a moving illuminating point of constant intensity, which at time t has the distance r from the observer, is seen by him in that position, which he attained at time ${\displaystyle {\frac {r}{\omega }}}$, but with the intensity that corresponds to the current (larger or smaller) distance.[AU 5]

The applicability of the above general considerations on the problems of optics is limited by the constraint (1'), which has lead to the formulas (10') and (13').

Such a limitation does not take place in the analogous problems of the acoustics of fluids. For the propagated dilation δ we have as the only condition

${\displaystyle {\frac {\partial ^{2}\delta }{\partial t^{2}}}{=}\omega ^{2}\Delta \delta .}$

The introduction of the substitutions (10), (12) or (13) always gives, if δ is given by the constraints along a given surface as an arbitrary function of time, the transition from the effect of a stationary source to the effect when it is in translational motion.

If, for example, we have ${\displaystyle {\overline {\delta }}=f(t)}$ on a very small sphere of radius R, then the propagated dilation is given by:

${\displaystyle \delta {=}{\frac {R}{r}}f\left(t-{\frac {r-R}{\omega }}\right).}$

}

The substitution (10) gives the influence of a translation of a "sounding" sphere parallel to the X-axis. The discussion of the result is equivalent to that employed under 3).

1. It follows from it ${\displaystyle q_{2}=q_{3}=0}$, as well as ${\displaystyle m_{2}\ n_{2}\ p_{2}}$, ${\displaystyle m_{3}\ n_{3}\ p_{3}}$ and therefore ${\displaystyle \zeta =\eta =0}$
2. W. Voigt, Crelles Journ. Vol. 89, 298.

## Notes by the author (1915)

From the reprint in: Physikalische Zeitschrift (1915), 16, 381-385 Online

The anniversary of the principle of relativity causes the editors to present an almost forgotten predecessor to the readers of the Physikalische Ztschrift. Indeed, in this note from the Nachr. d. Kgl. Ges. d. Wiss. zu Göttingen, meeting of 3. January 1887, the fundamental transformation of the optical differential equation was clearly formulated. Only the supplements (indicated by brackets) were added to this reprint by the author.

1. Due to the same order of all parts of equations (1), the right-hand sides of the substitution formulas (2) can be multiplied by a common factor, without changing the results.
2. This is, except the factor q which is irrelevant for the application, exactly the Lorentz transformation of the year 1904.
3. This will be made more precise, so that the radius should be small compared to the wave-length. Yet the formulas (16) and (17) don't require this assumption:
4. There one also finds the laws for the emission of a linearly oscillating sphere, which allows the same way of use.
5. The subsequent continuation of such problems didn't happen at that time, due to the physical difficulty of a complete realization of the presuppositions of this work - the translation of oscillating surfaces or bodies within the medium that transmits the oscillations - in the field of elasticity. This problem doesn't exist in the field of Lorentz's electrodynamics, in which the aether is extended through all ponderable bodies, and the latter is admitted to be freely movable within the first; therefore, the application of the method explained above is much more suitable for it. Indeed, the sphere problems explained above by an example, have the closest relation to the problem of the oscillating and simultaneously progressive electron, that became important many years after the preceding work.
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