12.5 The Doppler effect for sound waves

You may have noticed a change in pitch of the note heard when an emergency vehicle passes you while sounding its siren. The pitch is higher as the vehicle approaches you, and lower as it moves away (recedes). This is an example of the Doppler effect; you can hear the same thing if a train passes at speed while sounding its whistle.
Figure 12.11 shows why this change in frequency is observed. It shows a source of sound emitting waves with a constant frequency $f_{\mathrm{S}}$, together with two observers $\mathrm{A}$ and $\mathrm{B}$.

•  If the source is stationary (Figure 12.11a), waves arrive at $\mathrm{A}$ and $\mathrm{B}$ at the same rate, and so both observers hear sounds of the same frequency $f_{\mathrm{s}}$.
•  If the source is moving towards $\mathrm{A}$ and away from B (Figure 12.11b), the situation is different. From the diagram, you can see that the waves are squashed together in the direction of A and spread apart in the direction of $B$.

Observer A will observe, or detect, waves whose wavelength is shortened. More wavelengths per second arrive at A, and so A observes a sound of higher frequency than $f_{\mathrm{s}}$. Similarly, the waves arriving at B have been stretched out and $\mathrm{B}$ will observe a frequency lower than $f_{\mathrm{s}}$.

Figure 12.11: Sound waves (green lines) emitted at constant frequency by a a stationary source, and $\mathbf{b}$ a source moving with speed $v_{\mathrm{s}}$. The separation between adjacent green lines is equal to one wavelength.

An equation for observed frequency

There are two different speeds involved in this situation. The source is moving with speed $v_{\mathrm{s}}$. The sound waves travel through the air with speed $v$, which is unaffected by the speed of the source. (Remember, the speed of a wave depends only on the medium it is travelling through.)
The frequency and wavelength observed by an observer will change according to the speed $v_{\mathrm{s}}$ at which the source is moving relative to the stationary observer. Figure 12.12 shows how we can calculate the observed wavelength $\lambda_0$ and the observed frequency $f_0$.

The wave sections shown in Figure 12.12 represent the $f_{\mathrm{s}}$ wavelengths emitted by the source in $1 \mathrm{~s}$. Provided the source is stationary (Figure 12.12a), the length of this section is equal to the wave speed $v$. The wavelength observed by the observer is simply:
$
\lambda_0=\frac{v}{f_{\mathrm{s}}}
$
The situation is different when the source is moving away (receding) from the observer (Figure 12.12b).

In $1 \mathrm{~s}$, the source moves a distance $v_{\mathrm{s}}$. Now the section of $f_{\mathrm{s}}$ wavelengths will have a length equal to $v+$ $v_{\mathrm{s}}$.

Figure 12.12: Sound waves, emitted at constant frequency by a a stationary source, and b a source moving with speed $v_{\mathrm{S}}$ away from the observer (that is, the person hearing the sound).
The observed wavelength is now given by:
$
\lambda_0=\frac{\left(v+v_{\mathrm{s}}\right)}{f_{\mathrm{s}}}
$
The observed frequency is given by:
$
f_0=\frac{v}{\lambda_0}=\frac{f_{\mathrm{s}} \times v}{\left(v+v_{\mathrm{s}}\right)}
$

where $f_0$ is the observed frequency, $f_{\mathrm{S}}$ is the frequency of the source, $v$ is the speed of the wave and $v_{\mathrm{S}}$ is the speed of the source relative to the observer.
This shows us how to calculate the observed frequency when the source is moving away from the observer. If the source is moving towards the observer, the section of $f_{\mathrm{s}}$ wavelengths will be compressed into a shorter length equal to $v-v_{\mathrm{S}}$, and the observed frequency will be given by:
$
f_0=\frac{v}{\lambda_0}=\frac{f_{\mathrm{s}} \times v}{\left(v-v_{\mathrm{s}}\right)}
$
We can combine these two equations to give a single equation for the Doppler shift in frequency due to a moving source:
observed frequency, $f_0=\frac{f_{\mathrm{s}} \times v}{\left(v \pm v_{\mathrm{s}}\right)}$

where the plus sign applies to a receding source and the minus sign to an approaching source. Note these important points:
The frequency fs of the source is not affected by the movement of the source.
The speed v of the waves as they travel through the air (or other medium) is also unaffected by the movement of the source.

Note that a Doppler effect can also be heard when an observer is moving relative to a stationary source, and also when both source and observer are moving. There is more about the Doppler effect and light in Chapter 31.