1. The problem statement, all variables and given/known data
3 tuning forks of frequencies 200, 203, 207 Hz are sounded together. Find the beat frequency.
2. Relevant equations
Beat frequency= n1-n2 (n=frequency).
3. The attempt at a solution
I know that beat frequency is the difference in the frequencies of two superposing notes. But here 3 wave frequencies are given. The differences are 3, 4 and 7 Hz.
4. Conceptual doubt
Which of these 3 beat frequencies will actually be heard by the ear? All 3, the lowest (3Hz), or some combination of the 3?
Best Answer
The question does not specify what is the beat frequency in the case of more than 2 frequencies, so I will sketch what one could expect it to be. Each pair of frequencies produces the following beats: 3 Hz for the (200, 203) pair, 4 Hz for the (203, 207) pair and 7 Hz for the (200, 207) pair. The resulting sound combines those three beats. What could be called the beat frequency? The most natural choice is to pick the frequency which would be the fundamental frequency of those three beats, that is, the greatest common divisor of 3, 4, 7, which is 1. And indeed, the sum of the three beats is a more complex beat, periodic with period 1 Hz, hence the definition makes sense.
So the answer to your question is: the beat frequency is 1 Hz. Here is how it looks, the 1 Hz periodic pattern is visible:
Note that the beat frequency as defined above does not necessarily exist: if for example you have three frequences $f$, $f+3$, and $f+2\pi$, then the differences $3$, $2\pi$ and $2\pi-3$ have no greatest common divisor (there are no integers $p$, $q$, $r$ and no positive real number $d$ such that $3 = pd$, $2\pi = qd$ and $2\pi-3 = rd$). In the latter case, the resulting sound is not periodic.
Here is how it looks when there is no GCD, and hence no periodic pattern: