The Doppler Effect
Michael Fowler 3/29/07
Introduction
(Flashlet here)
The Doppler effect is the perceived change in frequency of sound emitted by a source moving relative to the observer: as a plane flies overhead, the note of the engine becomes noticeably lower, as does the siren noise from a fast-moving emergency vehicle as it passes. The effect was first noted by Christian Doppler in 1842. The effect is widely used to measure velocities, usually by reflection of a transmitted wave from the moving object, ultrasound for blood in arteries, radar for speeding cars and thunderstorms. The velocities of distant galaxies are measured using the Doppler effect (the red shift).
Sound Waves from a Source at Rest
To set up notation, a source at rest emitting a steady note generates circular wavecrests:
The circles are separated by one wavelength
and they travel outwards at the speed of sound v.
If the source has frequency f0,
the time interval 
between wave crests
leaving the source
As a fresh wave crest is emitted, the previous crest has
traveled a distance , so, since it’s moving at speed v,
and therefore
Sound Waves from a Moving Source
The Doppler effect arises because once a moving source emits a circular wave (and provided the source is moving at less than the speed of the wave) the circular wave crest emitted continues its outward expansion centered on where the source was when it was emitted, independent of any subsequent motion of the source.
Therefore, if the source is moving at a steady speed, the centers of the emitted circles of waves will be equally spaced along its path, indicating its recent history. In particular, if the source is moving steadily to the left, the wave crests will form a pattern:
Or, to be more realistic (from Wikipedia Commons):
It is evident that, as a result of the motion of the source, waves traveling to the left have a shorter wavelength than they had when the source was at rest. And it’s easy to understand why.
Denoting the steady source velocity by us, in the time 
between crests being
emitted the source will have moved to the left a distance 
At the same time, the
previously emitted crest will itself have moved to the left a distance 
Therefore, the actual
distance between crests emitted to the left will be
These waves, having left the source, are of course moving at
the speed of sound v relative to the air—the
motion of the source does not affect the speed of sound in air. Therefore, as these waves of wavelength 
arrive at an observer
placed to the left so the source is moving directly towards him, he will hear a
frequency 
That is, the observed frequency
(Note that for the common case 
, we can approximate,
.)
By an exactly parallel argument, for a source moving away from an observer at speed us, the frequency is lower by the corresponding factor:
Stationary Source, Moving Observer
Consider now an observer moving at speed uobs directly towards a stationary frequency f0 source. So,
she’s moving to meet the oncoming wave crests.
Remember, the wave crests are apart in the air, and
moving at v. Suppose her time between meeting successive
crests is 
. During this time,
she moves 
, the wave crest moves 
coming to meet her,
and between them they cover the distance between crests.
It is evident from the diagram that the time interval she will measure between meeting successive crests is
and therefore the sound frequency she measures is
Source and Observer Both Moving Towards Each Other
For this case, the arguments above can be combined to give:
Both motions increase the observed frequency. If either observer or source is moving in the opposite direction, the observed frequency is found by switching the sign of the corresponding u.
Doppler Effect for Light
The argument above for the Doppler frequency shift is accurate for sound waves and water waves, but fails for light and other electromagnetic waves, since their speed is not relative to an underlying medium, but to the observer. To derive the Doppler shift in this case requires special relativity. A derivation can be found in my Modern Physics notes.
The Doppler shift for light depends on the relative velocity u of source and observer:
for motion towards each other.
Other Possible Motions of Source and Observer
We’ve assumed above that the motions of source and observer are
all along the same straight line. But as
we hear the change in frequency of a jet engine passing overhead, the note
drops smoothly, because we’re off the straight line path of the plane. The actual note heard as a function of time
can be found from fairly simple geometric considerations to be 
, where 
is the angle between
the straight line path and a line from the source to the observer. This factor is incorporated in police speed
radar units. One interesting point: if 
. This seems very
reasonable, but is not the case for
light, where observed time dilation of the source gives a frequency shift. This was found unequivocally in a beautiful
series of experiments in the 1930’s (by Ives and Stillwell) attempting to
disprove special relativity.