Device that detect waves in a frequency range and beats

[Soren4]

Homework Statement

You have a device that can measure sound waves only if the frequency of the wave is in the range $0.8 kHz- 20 kHz$. You have a whistle that produces sound waves at $21.5kHz$. You ride a bike moving away from a wall, at the same time you blow the whistle and hold the device in your hand. What is the velocity you must have in order to make the device measure the minimum beat frequency that it can detect?

Homework Equations

Doppler effect and beats

The Attempt at a Solution

I premit I don't have any problem with the Doppler effect, the only problem here is to understand what to look for if I'm asked the "minimum beat frequency" that can be detected by the instrument. If $f_1$ is the frequency of the whistle and $f_2$ is the frequency modified by the reflection on the wall at a certain velocity, which condition between the following two is correct?

• $|f_2-f_1| > 0.8 kHz$
• $\frac{f_2+f_1}{2} <20kHz$

$|f_2-f_1|$ is the beat frequency, while $\frac{f_2+f_1}{2}$, as explaied at https://en.wikipedia.org/wiki/Beat_(acoustics) , is the frequency of the wave resulting from the interference of the two.

Now, in my view, what really counts is $\frac{f_2+f_1}{2}$, because that's the "real" frequency of the resulting wave, and I know that the device has a limitation on the frequency of the wave, but I'm not convinced about it. Which of the two would be correct, and (possibly) why?

Thanks a lot in advice

 

[TSny]

which condition between the following two is correct?

• $|f_2-f_1| > 0.8 kHz$
• $\frac{f_2+f_1}{2} <20kHz$

• Good question. The problem states at first that the device measures the frequency of the wave, but later it states that the device will measure the beat frequency. My guess is that you are to assume the device detects the beat frequency since that is what it says at the end of the problem statement. Try working it out for both conditions and see if one of the conditions leads to an unrealistic speed for a bike.