A car traveling at 22.1 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 = 235 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 340 m/s.

Question

A car traveling at 22.1 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 = 235 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 340 m/s.

Answer 1

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Answer 2

To find the beat frequency that the driver hears, we need to understand the concept of beat frequency and how it is related to the frequencies of the two sound waves.

First, let's understand what beat frequency is. When two sound waves of slightly different frequencies interfere with each other, they create a phenomenon called beats. The beat frequency is the difference between the frequencies of the two sound waves.

In this case, we have the original sound wave emitted by the car's horn with frequency f0 = 235 Hz, and the reflected sound wave off the building.

We know the speed of sound in air, which is 340 m/s. The car is approaching the building, so the frequency of the reflected sound wave (fr) can be calculated using the Doppler effect formula:

fr = f0 * (v + vr) / (v - vs)
Where,
fr = reflected frequency
f0 = original frequency (235 Hz)
v = speed of sound in air (340 m/s)
vr = velocity of the car relative to the building (speed towards the building)
vs = velocity of the car relative to the air (speed of the car)

In this case, the car is directly approaching the building, so the velocity of the car relative to the building (vr) is equal to the velocity of the car (vs). Therefore,

fr = f0 * (v + vr) / (v - vr)

We also know that the car is traveling at 22.1 m/s, so vr = -22.1 m/s (negative because the car is approaching the building).

Substituting the values,

fr = 235 * (340 + (-22.1)) / (340 - (-22.1))

Simplifying the equation,

fr = 235 * (340 - 22.1) / (340 + 22.1)

fr ≈ 229.41 Hz

Now that we have the frequency of the reflected sound wave, we can calculate the beat frequency. The beat frequency is the absolute difference between the two frequencies:

beat frequency = |fr - f0|

Substituting the values,

beat frequency = |229.41 Hz - 235 Hz|

beat frequency ≈ 5.59 Hz

So, the beat frequency that the driver hears is approximately 5.59 Hz.