18 The Doppler Effect
201
therefore,
ν = ν o
1
1 − v/c T
.
Doppler effect
Moving emitter
(229)
This is the formula for the Doppler increase of frequency if the observer is at rest and
the emitter moves towards the observer with a velocity v. According to our definition
on the sign of velocity (see Fig. 18.2), v should be used in Eq. (229) positively if the
emitter moves towards the observer. The moment it passes by the observer we would
have to change the sign of the velocity v in Eq. (229) to −v. The emitter then moves
away from the observer with the velocity v, and the later registers a lowering of the
frequency.
For the case where the velocity v is very small compared to the sound velocity
c T , thus v/c T 1, we can replace Eq. (229) according to
1
1+x
≈ 1 − x for x 1
with
ν = ν o
1 +
v
c T
.
Moving emitter
Low velocity,v c T
(229a)
Equation (229a) can be applied very easily for our opening case with the fire engines.
The velocity of the fire engines is nowhere near that of sound, and we determine for
the jump in frequency that we register the value ν = 2 ν o v/c T . The fireman emits
a signal tone of the frequency ν o . Whilst he moves towards us, we register a higher
frequency according to (229a), and when the fire engine drives past us and moves
away, we register a lowered frequency, thus the factor 2 .
2
(II) We now let the emitter be at rest and the observer moves, with its receiving
apparatus
3 towards the emitter with the velocity v. Due to the fact that our observer
is to the right of the emitter, v is used with the opposite sign. The velocity is counted
greater than zero when the observer approaches the emitter, see Fig. 18.3. The latter
once again emits its eigen frequency ν o , which, because of the emitter’s condition of
rest in the medium, transmits to the medium itself. Therefore, a harmonic wave of
the frequency ν o and the wavelength λ o = c T /ν o moves through the medium, see
Fig. 18.3. The observer O is at the position x o + λ o = x o + c T /ν o , when the first
wave crest arrives. The next arriving wave crest then finds itself at x o . If it arrives
after time T , then O measured a frequency of ν = 1/T . This wave crest now moves
2 With a sound velocity of c T = 330 ms −1 and a velocity for the fire engine of v = 80 kph = 80 ×
1000 m
60×60 s = 22, 2 ms −1 (so that our approximation condition v/c T =
330
22,2 = 0, 067 1 is fulfilled)
we get if we have a signal tone of the frequency ν o = 440 Hz, a change in the frequency of ν =
54 Hz the moment the fire engine rushes past us, in other words the signal tone frequency suddenly
falls from around 467 Hz to around 413 Hz, which cannot be overheard.
3 The human ear is an excellent receiver for sounds in the frequency area between 50 Hz and 16,000
Hz, and readily responds to changes of frequency in this area. Our eye cannot copy this out of
purely physiological reasons. The eye cannot distinguish between pure spectral colours, which are
relevant for the Doppler effect, and mixed colours. It is therefore not alone the velocity of light
that is responsible for there not being a simple view of an optical Doppler effect.
201
therefore,
ν = ν o
1
1 − v/c T
.
Doppler effect
Moving emitter
(229)
This is the formula for the Doppler increase of frequency if the observer is at rest and
the emitter moves towards the observer with a velocity v. According to our definition
on the sign of velocity (see Fig. 18.2), v should be used in Eq. (229) positively if the
emitter moves towards the observer. The moment it passes by the observer we would
have to change the sign of the velocity v in Eq. (229) to −v. The emitter then moves
away from the observer with the velocity v, and the later registers a lowering of the
frequency.
For the case where the velocity v is very small compared to the sound velocity
c T , thus v/c T 1, we can replace Eq. (229) according to
1
1+x
≈ 1 − x for x 1
with
ν = ν o
1 +
v
c T
.
Moving emitter
Low velocity,v c T
(229a)
Equation (229a) can be applied very easily for our opening case with the fire engines.
The velocity of the fire engines is nowhere near that of sound, and we determine for
the jump in frequency that we register the value ν = 2 ν o v/c T . The fireman emits
a signal tone of the frequency ν o . Whilst he moves towards us, we register a higher
frequency according to (229a), and when the fire engine drives past us and moves
away, we register a lowered frequency, thus the factor 2 .
2
(II) We now let the emitter be at rest and the observer moves, with its receiving
apparatus
3 towards the emitter with the velocity v. Due to the fact that our observer
is to the right of the emitter, v is used with the opposite sign. The velocity is counted
greater than zero when the observer approaches the emitter, see Fig. 18.3. The latter
once again emits its eigen frequency ν o , which, because of the emitter’s condition of
rest in the medium, transmits to the medium itself. Therefore, a harmonic wave of
the frequency ν o and the wavelength λ o = c T /ν o moves through the medium, see
Fig. 18.3. The observer O is at the position x o + λ o = x o + c T /ν o , when the first
wave crest arrives. The next arriving wave crest then finds itself at x o . If it arrives
after time T , then O measured a frequency of ν = 1/T . This wave crest now moves
2 With a sound velocity of c T = 330 ms −1 and a velocity for the fire engine of v = 80 kph = 80 ×
1000 m
60×60 s = 22, 2 ms −1 (so that our approximation condition v/c T =
330
22,2 = 0, 067 1 is fulfilled)
we get if we have a signal tone of the frequency ν o = 440 Hz, a change in the frequency of ν =
54 Hz the moment the fire engine rushes past us, in other words the signal tone frequency suddenly
falls from around 467 Hz to around 413 Hz, which cannot be overheard.
3 The human ear is an excellent receiver for sounds in the frequency area between 50 Hz and 16,000
Hz, and readily responds to changes of frequency in this area. Our eye cannot copy this out of
purely physiological reasons. The eye cannot distinguish between pure spectral colours, which are
relevant for the Doppler effect, and mixed colours. It is therefore not alone the velocity of light
that is responsible for there not being a simple view of an optical Doppler effect.