Physics 325
Lecture 20
Relativistic Doppler Effect
We begin with an observer at 90
˚
.
Non-relativistically, as we just saw, this observer
sees no frequency shift.
However, relativity tells us that the clock on the moving train is
different than the clock with the stationary observer, so time-dilation slows the period of
emitted waves according to
τ
γτ
′
=
(20.1)
The wavelength and frequency shift accordingly:
f
f
τ
λ
γ
τ
λ
′
=
=
=
′
′
(20.2)
This is a purely relativistic effect.
Now the rest of the argument goes very much like the
train example above.
Let’s repeat it with a moving source emitting light waves
v
G
S’
S
The source is at rest in the S’ frame and is emitting light waves.
An observer measures
them from the frame S.
S’ moves with velocity
v
G
with respect to S.
If S’ were at rest
with respect to S, then S would see a wave train of length
c
∆
t
in time
∆
t
.
However, since
S’ is approaching S at speed
v
, the wave train gets shorter because the end of it is emitted
closer to S by a distance
v
∆
t
, therefore the length of the wave train observed by S is
x
c t
v t
∆
= ∆ − ∆
If the wave train consists of
n
full waves, we can write
(
)
n
c
v
λ
t
=
−
∆
(20.3)
and the frequency is
(
)
c
cn
f
c
v
t
λ
=
=
−
∆
(20.4)

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