Mar 1, 1997

Under what conditions do beats and subharmonics occur?

It is common for people to talk about "subharmonics" and derive them as the result of two frequencies f1 - f2. This is incorrect unless the system under analysis is non-linear. I will make the distinction between subharmonics and beats, a distinction that is sometimes ignored.

Beats:

If the system is linear it will not experience subharmonics. It will however experience the beat effect. Let's first develop a useful trig identity:

     / m - n \    / m + n \
2 cos| ----- | sin| ----- |
     \   2   /    \   2   /

      /    / m \    / n \      / m \    / n \ \   /    / m \    / n \      / m \    / n \ \
  = 2 | cos| - | cos| - | + sin| - | sin| - | | * | cos| - | sin| - | + sin| - | cos| - | |
      \    \ 2 /    \ 2 /      \ 2 /    \ 2 / /   \    \ 2 /    \ 2 /      \ 2 /    \ 2 / /

      / sin(n)    2 / m \   sin(m)    2 / n \   sin(m)    2 / n \   sin(n)    2 / m \ \
  = 2 | ------ cos  | - | + ------ sin  | - | + ------ cos  | - | + ------ sin  | - | |
      \   2         \ 2 /     2         \ 2 /     2         \ 2 /     2         \ 2 / /

      / sin(n)   sin(m) \
  = 2 | ------ + ------ |
      \   2        2    /

  = sin(m) + sin(n) 

               / m - n \    / m + n \  
Therefore 2 cos| ----- | sin| ----- | = sin(m) + sin(n)
               \   2   /    \   2   /

  

Given two frequencies w1 and w2 and summing their result as in K sin(w1*t) + L sin(w2*t) we see, given M = (K+L)/2 and N = (K-L)/2:

K sin(w1*t) + L sin(w2*t)                                                         (1)

= M sin(w1*t) + N sin(w1*t) + M sin(w2*t) - N sin(w2*t)

        / (w1+w2)  \    / (w1-w2)  \         / (w1-w2)  \    / (w1+w2)  \
= 2N cos| ------- t| sin| ------- t| + 2M cos| ------- t| sin| ------- t|    (2)
        \    2     /    \    2     /         \    2     /    \    2     /

As we can see there are two ways of seeing this signal as (1) or as (2). In (2) the original frequencies are lost and all that is left are the beats. The average frequency (w1+w2)/2, beats with a frequency of (w1-w2)/2; which is half what most people expect. The average frequency is the only one that can be heard. The beat frequency is not heard as a tone, it is just the modulation of the average frequency.

During experiments I noticed that as the separation between frequencies increases the beat frequency increases. If the beat frequency is too high for your ear to detect then you will be oblivious to the beating phenomenon and hear only the average frequency. As the beat frequency goes even higher you begin to hear the two frequencies for what they really are: equation (1). There is always the possibility that the beat is in the range that you can switch perspectives and hear either the average frequency beating or the two distinct tones.

Subharmonics:

done later, I need to get a book.