18 The Doppler Effect
211
coordinates x
and t
) which has the velocity +v with respect to the reference system
o . The argumentation in this case begins just as it did in the first case. The emitter
S determines by counting the number of wave crests, an oscillation period T o of
the emitter’s membrane in the reference system
, as we have already described
in (234),
T o =
1
ν o
=
t
n o
.
(234)
The observer O with his clocks stationary in o once again finds out that the emitter
S’s clock goes behind according to Eq. (121). The emitter therefore does not produce
a wave crest in space every T o seconds, but, according to observer O’s clocks every
T seconds, as we already discovered in Eq. (235),
T =
T o
1 − v 2 /c
2
T
.
(235)
These wave crests move towards the observer with the velocity c T . The second wave
crest sent out after time T has to bridge the same distance R o as the first wave crest
and moves with the same velocity c T . Therefore, the second wave arrives exactly time
T after the first crest arrives, which enables the observer to calculate the frequency
ν according to
ν =
1
T
=
1 − v 2 /c
2
T
T o
and thus
ν = ν o
1 −
v 2
c
2
T
.
Internal observer
Transversal Dopplereffect
(240)
The internal observer therefore registers, in contrast to the outside observer, for the
second characteristic of the acoustic Doppler effect:
There is a transversal acoustic Doppler effect.
And now comes the most important thing. We compare our Eqs. (239) and (240) to
the corresponding Doppler effect equations for light waves, electromagnetic waves
generally. We can check up on these in every physics coursebook or monograph on
relativity, for example A. P. French [23], and find:
The equations formulated by the internal observer inside of a crystal for the acoustic Doppler
effect are absolutely identical to the complete description of the Doppler phenomena for
light, if one replaces only the transversal sound velocity c T with the speed of light c L . 4
4 The method, originating from W. Voigt, of calculating the Doppler effect from the relativistic
invariance of a harmonic wave’s phase (see also the corresponding explanation made by A. Pais
211
coordinates x
and t
) which has the velocity +v with respect to the reference system
o . The argumentation in this case begins just as it did in the first case. The emitter
S determines by counting the number of wave crests, an oscillation period T o of
the emitter’s membrane in the reference system
, as we have already described
in (234),
T o =
1
ν o
=
t
n o
.
(234)
The observer O with his clocks stationary in o once again finds out that the emitter
S’s clock goes behind according to Eq. (121). The emitter therefore does not produce
a wave crest in space every T o seconds, but, according to observer O’s clocks every
T seconds, as we already discovered in Eq. (235),
T =
T o
1 − v 2 /c
2
T
.
(235)
These wave crests move towards the observer with the velocity c T . The second wave
crest sent out after time T has to bridge the same distance R o as the first wave crest
and moves with the same velocity c T . Therefore, the second wave arrives exactly time
T after the first crest arrives, which enables the observer to calculate the frequency
ν according to
ν =
1
T
=
1 − v 2 /c
2
T
T o
and thus
ν = ν o
1 −
v 2
c
2
T
.
Internal observer
Transversal Dopplereffect
(240)
The internal observer therefore registers, in contrast to the outside observer, for the
second characteristic of the acoustic Doppler effect:
There is a transversal acoustic Doppler effect.
And now comes the most important thing. We compare our Eqs. (239) and (240) to
the corresponding Doppler effect equations for light waves, electromagnetic waves
generally. We can check up on these in every physics coursebook or monograph on
relativity, for example A. P. French [23], and find:
The equations formulated by the internal observer inside of a crystal for the acoustic Doppler
effect are absolutely identical to the complete description of the Doppler phenomena for
light, if one replaces only the transversal sound velocity c T with the speed of light c L . 4
4 The method, originating from W. Voigt, of calculating the Doppler effect from the relativistic
invariance of a harmonic wave’s phase (see also the corresponding explanation made by A. Pais