Our final topic for this chapter is an interesting effect that occurs when you are in motion relative to a wave source. It is called the Doppler effect. You’ve likely noticed that the pitch of an ambulance’s siren drops as it goes past you. Why?

FIGURE 16.30a shows a source of sound waves moving away from Pablo and toward Nancy at a steady speed $v_{\text{s}}$. The subscript s indicates that this is the speed of the source, not the speed of the waves. The source is emitting sound waves of frequency $f_0$ as it travels. The figure is a motion diagram showing the position of the source at times $t=0$, T, 2T, and 3T, where $T=1\text{/}{f}_0$ is the period of the waves.

FIGURE 16.30 A motion diagram showing the wave fronts emitted by a source as it moves to the right at speed $v_{\text{s}}$.

Nancy measures the frequency of the wave emitted by the approaching source to be $f_{+}$. At the same time, Pablo measures the frequency of the wave emitted by the receding source to be $f_{-}$. Our task is to relate $f_{+}$ and $f_{-}$ to the source frequency $f_0$ and speed $v_{\text{s}}$.

After a wave crest leaves the source, its motion is governed by the properties of the medium. That is, the motion of the source cannot affect a wave that has already been emitted. Thus each circular wave front in FIGURE 16.30b is centered on the point from which it was emitted. The wave crest from point 3 was emitted just as this figure was made, but it hasn’t yet had time to travel any distance.

The wave crests are bunched up in the direction the source is moving, stretched out behind it. The distance between one crest and the next is one wavelength, so the wavelength ${\text{λ}}_{+}$ Nancy measures is less than the wavelength ${\text{λ}}_0=v\text{/}{f}_0$ that would be emitted if the source were at rest. Similarly, ${\text{λ}}_{-}$ behind the source is larger than ${\text{λ}}_0$.

These crests move through the medium at the wave speed v. Consequently, the frequency $f_{+}=v\text{/}{\text{λ}}_{+}$ detected by the observer whom the source is approaching is higher than the frequency $f_0$ emitted by the source. Similarly, $f_{-}=v\text{/}{\text{λ}}_{-}$ detected behind the source is lower than frequency $f_0$. This change of frequency when a source moves relative to an observer is called the Doppler effect.

The distance labeled d in Figure 16.30b is the difference between how far the wave has moved and how far the source has moved at time $t=3T$. These distances are

The distance d spans three wavelengths; thus the wavelength of the wave emitted by an approaching source is

You can see that our arbitrary choice of three periods was not relevant because the 3 cancels. The frequency detected in Nancy’s direction is

where $f_0=1\text{/}T$ is the frequency of the source and is the frequency you would detect if the source were at rest. We’ll find it convenient to write the detected frequency as

Proof of the second version, for the frequency $f_{-}$ of a receding source, is similar. You can see that $f_{+}>f_0$ in front of the source, because the denominator is less than 1, and $f_{-}<f_0$ behind the source.