The sound from the whistle of a truck moving at 17.0 m/s toward a cliff reflects from the cliff and is heard by the driver of the truck.When it is parked at the side of the road,the whistle from the truck produces sound of frequency 2.267 kHz.As observed by the driver of the truck,what is the change in frequency the sound from the whistle due to the reflection? The speed of sound is 343 m/s.
A)236 Hz
B)228 Hz
C)207 Hz
D)118 Hz
E)472 Hz

Question

Quizplus · Accepted Answer

The change in frequency due to the Doppler effect can be calculated using the formula for the observed frequency during both the approach of the truck towards the cliff and the sound reflecting back. The formula for the Doppler effect when the source is moving towards a stationary observer (or reflector in this case) is $f' = f \left(\frac{v + v_o}{v - v_s}\right)$, where $f'$ is the observed frequency, $f$ is the source frequency, $v$ is the speed of sound, $v_o$ is the observer's speed towards the source (0 in this case since the cliff doesn't move), and $v_s$ is the speed of the source towards the observer. For the truck moving towards the cliff, the observed frequency by the cliff (acting as an observer) is:$$f' = 2267 \left(\frac{343 + 0}{343 - 17}\right) = 2267 \left(\frac{343}{326}\right)$$When the sound reflects off the cliff, the truck (now the observer) is moving towards the source of the sound (the reflected sound), so the formula is applied again with the truck as the observer:$$f'' = f' \left(\frac{343 + 17}{343}\right)$$Substituting $f'$ from the first calculation into the second:$$f'' = 2267 \left(\frac{343}{326}\right) \left(\frac{360}{343}\right)$$The change in frequency ($\Delta f$) is the difference between $f''$ and the original frequency $f$:$$\Delta f = f'' - f$$Plugging in the numbers:$$\Delta f = 2267 \left(\frac{343}{326}\right) \left(\frac{360}{343}\right) - 2267$$This calculation simplifies to show the proportional increase in frequency due to the Doppler effect twice (once for the sound moving towards the cliff and once for the sound reflecting back to the moving truck). The actual calculation will show that the change in frequency is approximately 236 Hz, taking into account the effect of the truck moving towards the cliff and the reflected sound moving towards the moving truck.

The Sound from the Whistle of a Truck Moving at 17.0