A car traveling at 24.3 m/s honks its horn as it directly approaches the side of a large building. The horn produces a long sustained note of frequency f0 = 213 Hz. The sound is reflected off the building back to the car's driver. The sound wave from the original note and that reflected off the building combine to create a beat frequency. What is the beat frequency that the driver hears (which tells him that he had better hit the brakes!)? Assume the speed of sound in air is 342 m/s.
Calculate the Doppler-shifted frequency of the received echo, fr.
Let V be the car speed and a the speed of sound.
fr = fo*[(V+a)/(V-a)] = 1.153*fo = 246 Hz
Subtract fo from that for the beat frequency, 33 Hz
To find the beat frequency that the driver hears, we need to determine the difference between the frequency of the reflected sound wave and the frequency of the original sound wave.
The speed of sound in air is given as 342 m/s, and the car is traveling at 24.3 m/s towards the building. Therefore, the speed of the sound wave relative to the building is the difference between the speed of sound and the speed of the car:
Relative speed of sound = Speed of sound - Speed of car
Relative speed of sound = 342 m/s - 24.3 m/s
Relative speed of sound = 317.7 m/s
The driver hears a beat frequency because the reflected sound wave has a slightly different frequency from the original sound wave. The frequency difference can be determined using the Doppler effect formula:
Δf = (v / c)∙f0
where:
Δf = Frequency difference (beat frequency)
v = Relative speed of sound
c = Speed of sound
f0 = Frequency of the original sound wave
Plugging in the values:
Δf = (317.7 m/s / 342 m/s)∙213 Hz
Δf ≈ 196 Hz
Therefore, the beat frequency that the driver hears is approximately 196 Hz.