Beat Frequency from Reflected Sound by Approaching Car

To find the beat frequency detected at the stationary sound source, we apply the Doppler effect in two successive stages.

Step 1: Sound received by the approaching vehicle (Moving Observer) The source is stationary, and the car (observer) moves towards the source with a speed $u = 2\text{ ms}^{-1}$. The speed of sound in air is $v = 330\text{ ms}^{-1}$, and the emitted frequency is $f_0 = 492\text{ Hz}$.

The frequency of sound detected by the car, $f'$, is given by: $f' = f_0 \left( \frac{v + u}{v} \right)$

Step 2: Sound reflected by the vehicle back to the source (Moving Source) The car now acts as a moving source of frequency $f'$ moving towards the stationary observer (the original source) at speed $u$.

The frequency of the reflected sound received back at the original source, $f''$, is given by: $f'' = f' \left( \frac{v}{v - u} \right)$

Substituting the expression for $f'$ into the formula for $f''$: $f'' = f_0 \left( \frac{v + u}{v} \right) \left( \frac{v}{v - u} \right) = f_0 \left( \frac{v + u}{v - u} \right)$

Step 3: Calculating the Beat Frequency The beat frequency $f_{\text{beat}}$ is the absolute difference between the frequency of the reflected sound $f''$ and the emitted sound $f_0$: $f_{\text{beat}} = f'' - f_0$ $f_{\text{beat}} = f_0 \left( \frac{v + u}{v - u} - 1 \right) = f_0 \left( \frac{2u}{v - u} \right)$

Now, substituting the given values:

• $f_0 = 492\text{ Hz}$
• $v = 330\text{ ms}^{-1}$
• $u = 2\text{ ms}^{-1}$

$f_{\text{beat}} = 492 \times \left( \frac{2 \times 2}{330 - 2} \right)$ $f_{\text{beat}} = 492 \times \frac{4}{328}$

Simplifying the fraction: $\frac{492}{328} = \frac{3}{2} = 1.5$

Thus: $f_{\text{beat}} = 1.5 \times 4 = 6\text{ Hz}$

The beat frequency detected at the source is 6 Hz.