Question

A police siren contains two frequencies, producing a wavering sound (beats). Explain how the siren sound changes as a police car approaches, passes, and moves away from a pedestrian.

Short answer

Answer: As the police car approaches the pedestrian, the pitch of the siren gets higher, and the beat frequency speeds up due to the Doppler effect. When the car passes the pedestrian, the beat frequency remains relatively unchanged as the car is neither moving towards nor away from the pedestrian. As the car moves away, the pitch gets lower, and the beat frequency slows down, again due to the Doppler effect.

Step-by-step solution

Understanding Beats

Beats are produced when two sound waves having similar, but not equal, frequencies interfere with each other. As the two waves move in and out of phase, they produce a pattern of alternating constructive and destructive interference, creating a rhythmic sound with periodic variations in volume. The beat frequency is given by the difference between the frequencies of the two waves, i.e., Beat Frequency = |f1 - f2|.

Understanding Doppler Effect

The Doppler effect is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. In the case of the police siren, as the police car moves towards or away from the pedestrian, the perceived frequency (f') changes. The Doppler effect for sound waves can be given by the formula: f' = f * (v + vo) / (v + vs), where f' is the observed frequency, f is the source frequency, v is the speed of sound, vo is the observer's speed, and vs is the source's speed (the observer's and source's speed are positive if towards, and negative if away from each other).

As the Police Car Approaches the Pedestrian

When the police car approaches the pedestrian, the source (police car) is moving towards the observer (pedestrian). Therefore, vs is negative in the Doppler effect formula. This results in the observed frequency being greater than the source frequency, i.e., f' > f. Consequently, the apparent difference in frequency between the two siren frequencies will also increase, producing faster beating and a higher pitch.

As the Police Car Passes the Pedestrian

When the police car passes by the pedestrian, the source is momentarily neither moving towards nor away from the observer. In this case, the Doppler effect won't have a significant impact on the observed frequency. The pedestrian will hear the siren frequencies at their source value, i.e., f' = f, and therefore, the beat frequency will remain unchanged.

As the Police Car Moves Away from the Pedestrian

As the police car moves away from the pedestrian, the source is now moving away from the observer, making vs positive in the Doppler effect formula. This causes the observed frequency to be lesser than the source frequency, i.e., f' < f. As a result, the apparent difference in frequency between the two siren frequencies will decrease, producing slower beating and a lower pitch. Combining each of these steps, we can see that as the police car approaches the pedestrian, the pitch of the siren gets higher, and the beat frequency speeds up. While the car passes the pedestrian, the beat frequency remains relatively unchanged, and as the car moves away, the pitch gets lower, and the beat frequency slows down.