Section 4: The Doppler Effect

Lesson 1: The Doppler Effect of Sound Waves

The Doppler Effect

To understand the Doppler Effect, let's imagine a person standing by the side of a road, listening to an ambulance approaching and then driving away, as seen in Figure 1. As the ambulance gets closer, the sound waves it produces get compressed or squished together. This causes the sound waves to reach the person's ear more frequently, so they perceive a higher pitch or frequency. It's similar to when a musical note sounds higher when played on a smaller instrument, such as a piccolo. Now, when the ambulance passes by and moves away, the sound waves get stretched out. They take longer to reach the person's ear, resulting in a lower pitch or frequency. This is similar to when a musical note sounds lower when played on a larger instrument, like a tuba.

Figure 1: As the ambulance travels to the right, there is a Doppler shift of the sound waves. This causes an observer behind the ambulance to observe a lower frequency and an observer ahead of the ambulance to observe a higher frequency (University of Saskatchewan).

This change in pitch or frequency is what we call the Doppler Effect. It happens not only with sound waves but also with other types of waves, like light or radio waves.

For a stationary observer and a moving source of sound, the frequency (f_obs) of sound perceived by the observer is

$$f_{obs}=f_s\frac{v_w}{v_w \pm v_s},$$

where f_s is the frequency of sound from a source in Hertz (Hz), v_s is the speed of the source in m/s along a line joining the source and observer, and v_w is the speed of sound in m/s. The minus sign is used for motion toward the observer, while the plus sign is for motion away from the observer.

Note that the greater the speed of the source, the greater the Doppler shift. Similarly, for a stationary source and moving observer, the frequency perceived by the observer (f_obs) is given by

$$f_{obs}=f_s\frac{v_w \pm v_{obs}}{v_w},$$

where v_obs is the speed of the observer in m/s along a line joining the source and observer. Here the plus sign is for motion toward the source, and the minus sign is for motion away from the source.

Lesson 2: Sonic Booms and Mach Speed

Breaking the Sound Barrier

What happens to the sound produced by a moving source, such as a jet airplane, that approaches or even exceeds the speed of sound? Suppose a jet airplane is coming nearly straight at you, emitting a sound of frequency f_s. The greater the plane’s speed, v_obs, the greater the Doppler shift and the greater the value of the observed frequency f_obs. Now, as v_obs approaches the speed of sound, v_s, f_obs approaches infinity, because the denominator in the equation for the Doppler shift approaches zero.

Figure 6: An F/A-18C Hornet breaking the sound barrier (US Navy)

An aircraft creates two sonic booms, one from its nose and one from its tail. During television coverage of the space shuttle landings, two distinct booms could often be heard. These were separated by exactly the time it would take the shuttle to pass by a point. Observers on the ground often do not observe the aircraft creating the sonic boom, because it has passed by before the shock wave reaches them. If the aircraft flies close by at low altitudes, pressures in the sonic boom can be destructive enough to break windows. Because of this, supersonic flights are banned over populated areas of the United States.

$$M=\frac{v_{obj}}{v_s}$$