Question

A car traveling at \(25.0 \mathrm{~m} / \mathrm{s}\) honks its horn as it directly approaches the side of a large building. The horn produces a long sustained note of frequency \(f_{0}=230 .\) 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!)?

Short answer

The beat frequency heard by the driver is approximately 46.07 Hz.

Step-by-step solution

Write the Doppler effect formula for a moving source and stationary observer

The Doppler effect formula for a moving source and a stationary observer is given by: \(f_{\text{observed}} =\left(\dfrac{c_{\text{sound}}}{c_{\text{sound}}\pm v_{\text{source}}}\right) f_{0}\) Here, \(c_{\text{sound}}\) is the speed of sound in the medium (air), \(v_{\text{source}}\) is the speed of the source (car), and \(f_{0}\) is the original frequency of the horn. In this case, we will use a negative sign as the car is approaching the building and the observed frequency will be higher than the actual frequency of the horn.

Substitute the given values and solve for the observed frequency

The given values for the problem are: - \(v_{\text{source}} = 25.0\, \text{m/s}\) - \(f_{0} = 230\, \text{Hz}\) - \(c_{\text{sound}} = 343\, \text{m/s}\) Plug these values into the Doppler effect formula: \(f_{\text{observed}} =\left(\dfrac{343}{343 - 25}\right)230\) Now, calculate the observed frequency: \(f_{\text{observed}} \approx 276.07\, \text{Hz}\)

Calculate the beat frequency

Beat frequency is the difference between the original frequency and the observed frequency: \( f_{\text{beat}} = |f_{\text{observed}} - f_{0}|\) Now, plug the values into the formula: \( f_{\text{beat}} = |276.07 - 230|\) The beat frequency is: \( f_{\text{beat}} \approx 46.07\, \text{Hz}\) The driver hears a beat frequency of approximately 46.07 Hz.