How Does the Doppler Effect Alter Frequencies of a Moving Whistle?

[danago]

A whistle of frequency 577 Hz moves in a circle of radius 73.2 cm at an angular speed of 16.1 rad/s. What are (a) the lowest and (b) the highest frequencies heard by a listener a long distance away, at rest with respect to the center of the circle? (Take the speed of sound in air to be 343 m/s.)
The linear velocity of the whistle is given by:

[tex]
v = r\omega = (0.732)(16.1) = 11.7852ms^{ - 1}
[/tex]

The component of the velocity in the direction of the listener is at a maximum/minimum when it moves directly towards/away from the listener, with the velocity in this direction being 11.7852m/s.

The greatest frequency will be heard when the velocity of the whistle towards the listener is greatest, thus the effective frequency will be:

[tex]
f' = 577 \times \frac{{343}}{{343 - 11.7852}} = 597.53Hz
[/tex]

The smallest frequency will be heard when the velocity of the whistle towards the listener is smallest, or when the whistle moves away from the listener with greatest velocity, thus the effective frequency will be:

[tex]
f' = 577 \times \frac{{343}}{{343 + 11.7852}} = 557.833Hz
[/tex]

According to the solutions, those answers are incorrect. Anyone able to shed some light on where my reasoning is flawed?

Thanks in advance,
Dan.

 

[danago]

Ah nevermind, i just realized that i was submitting the solutions the wrong way round, giving the highest one instead of the lowest one. Guess that's i sign i should head off to bed

 

[Moetasim]

Hello Dan,

Your reasoning is correct, but there is a slight mistake in your calculations. The velocity of the whistle towards the listener should be 11.7852 m/s, not 11.7852 cm/s. This would give us the correct frequencies of 597.53 Hz and 557.833 Hz, respectively.

I hope this helps! Keep up the good work in your scientific studies.

Best,