The Doppler Effect
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.
EXAMPLE