Help Needed: Solving Homework Assignment on Beat Frequency

[Inertigratus]

So I had a homework assignment which I couldn't complete in time. One of the questions was what the title is about. I also asked for help earlier in the homework section.
See my post: https://www.physicsforums.com/showthread.php?t=487676.

Right now I just want to learn how this works for the coming test.
Perhaps the title in my other post is misleading as I don't really know if there is a "beat frequency" and/or if it has anything to do with the problem.

Any ideas on how to solve this?

Thanks!

 

[kuruman]

Any ideas on how to solve this?

Sure. Start with what you know about sound beats when you have two waves and then use superposition to add a third wave. The time dependence of the amplitude at a fixed location for two waves is
$$y(t)=\cos(2\pi f_1 t)+\cos(2\pi f_2 t)=2\cos \left(2\pi\frac{f_1+f_2}{2}t\right) \cos \left(2\pi\frac{f_1-f_2}{2}t \right)$$When the frequencies are close you can approximate $f_1\approx f_2=f_0$ and write
$$y(t)=2\cos (2\pi f_0) \cos \left(2\pi\frac{f_1-f_2}{2}t \right)$$When you have a third wave you take all the possible combinations and write $$y(t)=2\cos (2\pi f_0) \left[ \cos \left(2\pi\frac{f_1-f_2}{2}t \right) +\cos \left(2\pi\frac{f_2-f_3}{2}t \right) +\cos \left(2\pi\frac{f_3-f_1}{2}t \right) \right]$$ In this particular problem we have $f_1=200~ \mathrm{Hz}$, $f_2=201~ \mathrm{Hz}$ and $f_3=202~ \mathrm{Hz}$ which gives $$y(t)=2\cos (2\pi f_0) \left[ 2\cos (2\pi~\Delta f~t ) +\cos (2\pi \times 2\Delta f~t ) \right]$$where $\Delta f=0.5~\mathrm{Hz}$. You can use any number between 200 and 202 Hz for $f_0$. Of course, the intensity is the square of the amplitude.