206
18 The Doppler Effect
The second wave crest, sent on to the first wave crest after time T o , moves with
the same velocity c T as the first, and must according to (232) have the same distance
R o to cover to the observer. Thus, the second wave crest arrives exactly T o later at
the observer, so that the observer determines the same frequency ν o as the emitter
sent out,
ν = ν o .
Moving emitter
Transversal observation
(233)
This is our second characteristic for the acoustic Doppler effect:
There is no transversal acoustic Doppler shift.
All of this has been known for over 150 years. However, that what is going to happen
next is new and exciting. We once again move into the world of our infinite crystal, into
a crystal where our internal observers are operating with their natural measuring-rods
and clocks (and where these observers perhaps philosophically discuss the results
of their observations). We think about how the Doppler effect could be determined
and measured using only the natural measuring-rods and clocks that the internal
observers have at their disposal (see Günther [34]). Here, we once again assume that
the critical velocity c o of the sine-Gordon equation coincides with the transversal
sound velocity c T . In fact, the whole experimental situation is exactly as we have
previously described and examined.
An internal observer S sits on an emitter, a standard emitter that produces per
construction an oscillation of the frequency ν o . This observer S determines this
frequency by counting the number n o of oscillations produced by the emitter during
a certain time interval t o in the same manner as described by us in (224). However,
here we have to be a little more careful. Observer S’s internal standard emitter is
per definition situated in its own reference system. Therefore, time interval t o is
a numerical date that is only valid for this special reference system. We will have
to take this into account. The same applies to observer O on his receiving station.
He determines the frequency from the number n of arriving wave crests during a
certain time interval t. Here t is a statement of time that only makes sense in the
reference system in which observer O as well as his receiver are at rest.
To be consistent, we will only take those standard emitters into consideration
where the internal observer S measures one and the same frequency ν o , irrespective
of the reference system in which he and his standard emitter are at rest. Exactly this
is what is meant when we make the demand: A standard emitter shall produce per
construction an oscillation of the frequency ν o . Here, one can, for example, think of
breather-similar oscillation systems or other oscillating dislocation configurations.
This (only meaningful) definition of a standard emitter of constant frequency has
one important consequence. An internal observer’s standard emitter is not a standard emitter as defined by an outside observer. An oscillating breather (the internal
observer’s standard emitter) is not a standard emitter for the outside observer. For
the outside observer, the moving breather is with respect to the crystal of a different
oscillationary construction than the breather being at rest with respect to the crystal,
18 The Doppler Effect
The second wave crest, sent on to the first wave crest after time T o , moves with
the same velocity c T as the first, and must according to (232) have the same distance
R o to cover to the observer. Thus, the second wave crest arrives exactly T o later at
the observer, so that the observer determines the same frequency ν o as the emitter
sent out,
ν = ν o .
Moving emitter
Transversal observation
(233)
This is our second characteristic for the acoustic Doppler effect:
There is no transversal acoustic Doppler shift.
All of this has been known for over 150 years. However, that what is going to happen
next is new and exciting. We once again move into the world of our infinite crystal, into
a crystal where our internal observers are operating with their natural measuring-rods
and clocks (and where these observers perhaps philosophically discuss the results
of their observations). We think about how the Doppler effect could be determined
and measured using only the natural measuring-rods and clocks that the internal
observers have at their disposal (see Günther [34]). Here, we once again assume that
the critical velocity c o of the sine-Gordon equation coincides with the transversal
sound velocity c T . In fact, the whole experimental situation is exactly as we have
previously described and examined.
An internal observer S sits on an emitter, a standard emitter that produces per
construction an oscillation of the frequency ν o . This observer S determines this
frequency by counting the number n o of oscillations produced by the emitter during
a certain time interval t o in the same manner as described by us in (224). However,
here we have to be a little more careful. Observer S’s internal standard emitter is
per definition situated in its own reference system. Therefore, time interval t o is
a numerical date that is only valid for this special reference system. We will have
to take this into account. The same applies to observer O on his receiving station.
He determines the frequency from the number n of arriving wave crests during a
certain time interval t. Here t is a statement of time that only makes sense in the
reference system in which observer O as well as his receiver are at rest.
To be consistent, we will only take those standard emitters into consideration
where the internal observer S measures one and the same frequency ν o , irrespective
of the reference system in which he and his standard emitter are at rest. Exactly this
is what is meant when we make the demand: A standard emitter shall produce per
construction an oscillation of the frequency ν o . Here, one can, for example, think of
breather-similar oscillation systems or other oscillating dislocation configurations.
This (only meaningful) definition of a standard emitter of constant frequency has
one important consequence. An internal observer’s standard emitter is not a standard emitter as defined by an outside observer. An oscillating breather (the internal
observer’s standard emitter) is not a standard emitter for the outside observer. For
the outside observer, the moving breather is with respect to the crystal of a different
oscillationary construction than the breather being at rest with respect to the crystal,