So far we have only discussed cases where the source of
waves is at rest. Often, waves are emitted by a source that moves
with respect to the medium that carries the waves, like when a speeding
cop car blares its siren to alert onlookers to stand aside. The
speed of the waves,
v, depends only
on the properties of the medium, like air temperature in the case
of sound waves, and not on the motion of the source: the waves will
travel at the speed of sound (
343 m/s) no matter how
fast the cop drives. However, the frequency and wavelength of the
waves
will depend on the motion of the wave’s source.
This change in frequency is called a
Doppler shift.Think
of the cop car’s siren, traveling at speed
, and emitting waves with frequency
f and
period
T =
1/
f.
The wave crests travel outward from the car in perfect circles (spheres
actually, but we’re only interested in the effects at ground level).
At time
T after the first wave crest
is emitted, the next one leaves the siren. By this time, the first
crest has advanced one wavelength,
, but the car has also traveled a distance of
. As a result, the two wave crests are closer
together than if the cop car had been stationary.
The shorter wavelength is called the Doppler-shifted wavelength,
given by the formula
. The Doppler-shifted
frequency is given by the formula:
Similarly, someone standing behind the speeding siren
will hear a sound with a longer wavelength,
, and a lower frequency,
.
You’ve probably noticed the Doppler effect with passing
sirens. It’s even noticeable with normal cars: the swish of a passing
car goes from a higher hissing sound to a lower hissing sound as
it speeds by. The Doppler effect has also been put to valuable use
in astronomy, measuring the speed with which different celestial
objects are moving away from the Earth.
As the car approaches, the sound waves will have shorter
wavelengths and higher frequencies, and as it goes by, the sound
waves will have longer wavelengths and lower frequencies. More precisely,
the frequency as the cop car approaches is:
