18 The Doppler Effect
203
wave of the frequency ν 1 in this medium,
ν 1 = ν o
1
1 − v 1 /c T
=
ν o c T
c T − v 1
that has the wavelength λ 1 ,
λ 1 =
c T
ν 1
=
c T − v 1
ν o
.
As above, the observer O covers this distance, because of its absolute velocity v 2 with
respect to the medium, with the velocity c T + v 2 . The observer therefore measures
a frequency ν according to
ν =
c T + v 2
λ 1
= ν o
c T + v 2
c T − v 1
= ν o
c T + v − v 1
c T − v 1
thus,
ν = ν o
1 +
v
c T − v 1
.
(231)
The observer thus measures the frequency (231) if the emitter has the unknown
velocity v 1 with respect to the medium (and the observer the unknown velocity v 2 ).
Upholding the relative velocity v between emitter and observer, the emitter and
the observer should swap positions. We could also imagine that both emitter and
observer possess identical transmitting installations as well as receiving systems.
They would then only have to swap functions; the former receiver emits and the
former emitter receives. This results in the emitter being to the right-hand side of the
receiver. For the relative velocity v = v 1 + v 2 is still valid. However, now v 1 = q
is the absolute velocity of the observer with respect to the medium, and the emitter
has with respect to the medium the absolute velocity v 2 , which we once again count
positive, if the emitter moves towards the observer. In order to measure the receiver’s
frequency ¯
ν , the above calculation is reused, with v 1 and v 2 in swapped positions
with an unchanged relative velocity v = v 1 + v 2 , thus
¯
ν = ν o
1 +
v
c T − v 2
,
(231a)
thus
¯
ν = ν o
1 +
v
c T + v 1 − v
.
(231b)
The observer thus measures the frequency (231b) if the emitter has the unknown
velocity v 2 with respect to the medium (and the observer the unknown velocity v 1 ).
We add up both frequencies (231) and (231b), write the variable q for the unknown
velocity v 1 and look in the developed function f (q) ,
f (q) = ν + ¯
ν = 2ν o + ν o
v
c T − q
+
v
c T + q − v
,