# what beat frequencies will be heard from the superposition of 3 sources of unequal frequency?

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?

• In general with three or more frequencies all combinations of beats are possible and furthermore beats from one combination of wave inputs can interfere with beats from another. In other words beats could interfere with beats. But with 3 or more waves interfering the observed superposition begins to appear quite chaotic at least in the time domain. Best to view as as a spectrum in the frequency domain. Commented Jan 8, 2017 at 18:43
• Oh really? My book has n answer for this as 12. I've been wondering all the way like how did they get it? :/ Commented Jan 8, 2017 at 19:49
• Fundamentally beat frequencies come about from waves that interfere with one another. So the more fundamental wave frequencies you have, the more possibilities of interference. Commented Jan 8, 2017 at 19:52
• So you say there might be more than one beat frequency?! If so can we find those more than one beat frequency by these normal calculative methods? Commented Jan 8, 2017 at 19:53
• @sammy garbil edited the question Commented Jan 9, 2017 at 4:14

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:

• @John Rennie & al.: it is a bit frustrating to spend time answering a question, and find that the question has been put on hold later. But I believe that is the rule of the game.
– user130529
Commented Jan 9, 2017 at 10:39
• still I am flabbergasted on why this question went off topic :/ Commented Jan 9, 2017 at 16:27
• @user129402 : It is a good (useful) question. But the site has a very strict policy on "homework-like" questions such as this, regardless of their usefulness. You have shown effort and I think you are asking a conceptual question about what beats can be heard in such cases, rather than simply "show me how to do the calculation". In any case, your question may be a duplicate of Beat frequency for 3 waves which Floris answers by simply showing the waveform which results from the sum of 3 frequencies. Commented Jan 9, 2017 at 17:02
• @sammy gerbil. Thanks for the link but sorry I couldn't understand the programming or whatsoever stuff that the answer had :/ I M still confused on how to find beat frequency for the waves and that question appears too often in my assignment :/ some suggests using Fourier series. And found another question like that which goes on like this n+1, n, n-1 are the frequencies ,what will be their beat frequency? I thought it would be 1 since it's the GCD for those 3 factors but the answer contrary to mine says 2 which means they subtracted the lowest and highest frequencies?! I M yet to clarify this Commented Jan 9, 2017 at 18:19
• @user129402 : I added for you two figures illustrating the two cases whether the GCD exists or not.
– user130529
Commented Jan 9, 2017 at 20:02