Figure 12.1: Doppler shift for a source moving at an angle relative to to the observer. Here the source is
moving along a straight line track with velocityv, and the observer is standing off to one side of the track.
For the purpose of computing the Doppler shift, the effective speed of the source is the component ofvin the
direction of the observer,vcos.
For example, suppose a fire engine emits a sound with a frequency of 2000 Hz, and is moving directly
toward you at 50 m/s. You are stationary. What frequency do you hear? In this casef D 2000 Hz,
vsndD 343 m/s,vsourceD 50 m/s, andvobsD 0. Since the fire engine is moving toward you, you choose the
top signs, so the frequency you hear isf^0 D.2000Hz/Œ.343C0/=.34350/D 2341 Hz.
Eq. (12.1) covers the case where the source and observer are movingdirectlytoward or away from each
other. But what if they are moving at some angle relative to each other, rather than directly toward or away
from each other? In that case, the velocitiesvobsandvsourcethat you use in Eq. (12.1) are thecomponentsof
the velocity along a line connecting the source and the observer. Fig. 12.1 shows an example: a source of
sound is moving along a straight track, and the observer is standing off to one side. At any point, thevsource
we use in Eq. (12.1) is the component of the source’s velocity along the line connecting the source to the
observer at that point, orvsourcecos. If we perform this calculation usingvsourceD 50 m/s andd D 10 m
for each point along the track, we get the plot shown in Fig. 12.2.
It is a common misconception that in the case of a moving source, the frequency increases as the object
moves toward the observer, and decreases as it moves away. As you can see from Fig. 12.2, this is not the
case: the frequency decreases monotonically.
12.1 Relativistic Doppler Effect
Light waves exhibit the Doppler effect just as sound waves do, but the analysis is different. We’ll examine
light waves in more detail later, but for now we can just note that light waves are a type of transverse wave that
can travel through a vacuum. In discussing the Doppler effect for sound, we specified the speeds of both the
source and the observer relative to the reference frame of theair. However, there is no such reference frame
for light waves. According to Einstein’s special theory of relativity, there is no “universal” reference frame
with respect to which we can measure speeds of bodies—and furthermore, the theory says that the speed of
light is constant, regardless of the speed of the person making the measurement. So in the case of light waves,
it makes no sense to talk about the speeds of the source or the observer with respect to some fixed reference
frame, since there is no such frame—we can only talk about the speeds of the source and observerrelative to
each other. This means that the Doppler shift equation for light has only two speeds in it: the speed of light
c, and the relative speed between the source and observer,v. The Doppler equation for light waves (called