Basically we have the function
$$f(t)=\sin(2π\nu_1 t)+\sin(2π\nu_2 t)+\sin(2π\nu_3 t)$$
Let $T$ be the fundamental period of $f(t)$.
Beat frequency is defined as the number of peaks in intensity per unit time for the resultant wave ....
Which means $$f'(t)=0 = 2π\nu_1 \cos(2π\nu_1 t) +2π\nu_2 \cos(2π\nu_2 t) +2π\nu_3 \cos(2π\nu_3 t)$$
This equation is also periodic with period $T$.
So beat frequency can be defined as the number of roots of $f'(t)=0$ where $t \in [0,T)$ divided by T.
Is there any simple way of calculating this? (I don't need a formula ... A simple algorithm will also do)
Does it help if all three frequencies are natural numbers (because that's usually the case when dealing with tuning fork related problems)?
I tried considering three vectors of lengths $\nu_1, \nu_2,\nu_3$. The vectors are rotating with angular speeds $2π\nu_1,2π\nu_2,2π\nu_3$.
If the frequencies are natural numbers I think $T=1/\gcd (\nu_1,\nu_2,\nu_3)$
Now we need to find the number of times the cos component of the vector sum of the rotating vectors gives zero.
My teacher had a weird but simple way of doing this ... He said that all 3 vectors must either be parallel, or 2 vectors parallel and 1 antiparallel, or the 3 vectors should form a triangle.
I didn't completely understand the method or the reasoning behind it.
Does such a method indeed exist?