## Doppler Effect and Sonic Booms

The characteristic sound of a motorcycle buzzing by is an example of the **Doppler effect**. The high-pitch scream shifts dramatically to a lower-pitch roar as the motorcycle passes by a stationary observer. The closer the motorcycle brushes by, the more abrupt the shift. The faster the motorcycle moves, the greater the shift. We also hear this characteristic shift in frequency for passing race cars, airplanes, and trains. It is so familiar that it is used to imply motion and children often mimic it in play.

The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer. Although less familiar, this effect is easily noticed for a stationary source and moving observer. For example, if you ride a train past a stationary warning bell, you will hear the bell’s frequency shift from high to low as you pass by. The actual change in frequency due to relative motion of source and observer is called a **Doppler shift**. The Doppler effect and Doppler shift are named for the Austrian physicist and mathematician Christian Johann Doppler (1803–1853), who did experiments with both moving sources and moving observers. Doppler, for example, had musicians play on a moving open train car and also play standing next to the train tracks as a train passed by. Their music was observed both on and off the train, and changes in frequency were measured.

What causes the Doppler shift? this figure, this figure, and this figure compare sound waves emitted by stationary and moving sources in a stationary air mass. Each disturbance spreads out spherically from the point where the sound was emitted. If the source is stationary, then all of the spheres representing the air compressions in the sound wave centered on the same point, and the stationary observers on either side see the same wavelength and frequency as emitted by the source, as in this figure. If the source is moving, as in this figure, then the situation is different. Each compression of the air moves out in a sphere from the point where it was emitted, but the point of emission moves. This moving emission point causes the air compressions to be closer together on one side and farther apart on the other. Thus, the wavelength is shorter in the direction the source is moving (on the right in this figure), and longer in the opposite direction (on the left in this figure). Finally, if the observers move, as in this figure, the frequency at which they receive the compressions changes. The observer moving toward the source receives them at a higher frequency, and the person moving away from the source receives them at a lower frequency.

We know that wavelength and frequency are related by \({v}_{w}=\mathrm{f\lambda }\), where \({v}_{w}\) is the fixed speed of sound. The sound moves in a medium and has the same speed \({v}_{w}\) in that medium whether the source is moving or not. Thus \(f\) multiplied by \(\lambda \) is a constant. Because the observer on the right in this figure receives a shorter wavelength, the frequency she receives must be higher. Similarly, the observer on the left receives a longer wavelength, and hence he hears a lower frequency. The same thing happens in this figure. A higher frequency is received by the observer moving toward the source, and a lower frequency is received by an observer moving away from the source. In general, then, relative motion of source and observer toward one another increases the received frequency. Relative motion apart decreases frequency. The greater the relative speed is, the greater the effect.

### The Doppler Effect

The Doppler effect occurs not only for sound but for any wave when there is relative motion between the observer and the source. There are Doppler shifts in the frequency of sound, light, and water waves, for example. Doppler shifts can be used to determine velocity, such as when ultrasound is reflected from blood in a medical diagnostic. The recession of galaxies is determined by the shift in the frequencies of light received from them and has implied much about the origins of the universe. Modern physics has been profoundly affected by observations of Doppler shifts.

For a stationary observer and a moving source, the frequency *f*_{obs} received by the observer can be shown to be

\({f}_{\text{obs}}={f}_{s}(\cfrac{{v}_{w}}{{v}_{w}±{v}_{s}}),\)

where \({f}_{s}\) is the frequency of the source, \({v}_{s}\) is the speed of the source along a line joining the source and observer, and \({v}_{w}\) is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away from the observer, producing the appropriate shifts up and down in frequency. Note that the greater the speed of the source, the greater the effect. Similarly, for a stationary source and moving observer, the frequency received by the observer \({f}_{\text{obs}}\) is given by

\({f}_{\text{obs}}={f}_{s}(\cfrac{{v}_{w}±{v}_{\text{obs}}}{{v}_{w}}),\)

where \({v}_{\text{obs}}\) is the speed of the observer along a line joining the source and observer. Here the plus sign is for motion toward the source, and the minus is for motion away from the source.

### Example: Calculate Doppler Shift: A Train Horn

Suppose a train that has a 150-Hz horn is moving at 35.0 m/s in still air on a day when the speed of sound is 340 m/s.

(a) What frequencies are observed by a stationary person at the side of the tracks as the train approaches and after it passes?

(b) What frequency is observed by the train’s engineer traveling on the train?

**Strategy**

To find the observed frequency in (a), \({f}_{\text{obs}}={f}_{s}(\cfrac{{v}_{w}}{{v}_{w}±{v}_{s}}),\) must be used because the source is moving. The minus sign is used for the approaching train, and the plus sign for the receding train. In (b), there are two Doppler shifts—one for a moving source and the other for a moving observer.

**Solution for (a)**

(1) Enter known values into \({f}_{\text{obs}}={f}_{s}(\cfrac{{v}_{w}}{{v}_{w}\phantom{\rule{0.25em}{0ex}}–\phantom{\rule{0.25em}{0ex}}{v}_{s}}).\)

\({f}_{\text{obs}}={f}_{s}(\cfrac{{v}_{w}}{{v}_{w}-{v}_{s}})=(\text{150 Hz})(\cfrac{\text{340 m/s}}{\text{340 m/s – 35.0 m/s}})\)

(2) Calculate the frequency observed by a stationary person as the train approaches.

\({f}_{obs}=(\text{150 Hz})(\text{1.11})=\text{167 Hz}\)

(3) Use the same equation with the plus sign to find the frequency heard by a stationary person as the train recedes.

\({f}_{\text{obs}}={f}_{s}(\cfrac{{v}_{w}}{{v}_{w}+{v}_{s}})=(\text{150 Hz})(\cfrac{\text{340 m/s}}{\text{340 m/s}+\text{35.0 m/s}})\)

(4) Calculate the second frequency.

\({f}_{obs}=(\text{150 Hz})(0.907)=\text{136 Hz}\)

**Discussion on (a)**

The numbers calculated are valid when the train is far enough away that the motion is nearly along the line joining train and observer. In both cases, the shift is significant and easily noticed. Note that the shift is 17.0 Hz for motion toward and 14.0 Hz for motion away. The shifts are not symmetric.

**Solution for (b)**

(1) Identify knowns:

- It seems reasonable that the engineer would receive the same frequency as emitted by the horn, because the relative velocity between them is zero.
- Relative to the medium (air), the speeds are \({v}_{s}={v}_{\text{obs}}=35.0 m/s.\)
- The first Doppler shift is for the moving observer; the second is for the moving source.

(2) Use the following equation:

\({f}_{\text{obs}}=[{f}_{s}(\cfrac{{v}_{w}±{v}_{\text{obs}}}{{v}_{w}})](\cfrac{{v}_{w}}{{v}_{w}±{v}_{s}}).\)

The quantity in the square brackets is the Doppler-shifted frequency due to a moving observer. The factor on the right is the effect of the moving source.

(3) Because the train engineer is moving in the direction toward the horn, we must use the plus sign for \({v}_{obs};\) however, because the horn is also moving in the direction away from the engineer, we also use the plus sign for \({v}_{s}\). But the train is carrying both the engineer and the horn at the same velocity, so \({v}_{s}={v}_{obs}\). As a result, everything but \({f}_{s}\) cancels, yielding

\({f}_{\text{obs}}\text{=}{f}_{s}.\)

**Discussion for (b)**

We may expect that there is no change in frequency when source and observer move together because it fits your experience. For example, there is no Doppler shift in the frequency of conversations between driver and passenger on a motorcycle. People talking when a wind moves the air between them also observe no Doppler shift in their conversation. The crucial point is that source and observer are not moving relative to each other.