Are Time Periods of Combined Vibrations Always Commensurable?

[NRa]

Hi. I have been reading about the superposition of simple harmonic vibrations of different frequencies and what entails to make the their combination periodic. This is quoted from the book Vibrations and Waves by A.P. French: "The condition for any true periodicity in the combined motion is that the periods of the component motions be commensurable-i.e. there exists two integers n₁ and n₂ such that
T = n₁T₁ = n₂T₂
The period of the combined motion is then the value of T as obtained above, using the smallest integral values of n₁ and n₂..."
This is quite understandable. However, when it comes to the beat phenomena we can't find out the time period of the combined motion through this. For example if we have frequencies 255 Hz and 257 Hz, the time period of the superposed motion is 1/256 s which isn't something you would arrive at using T = n₁T₁ = n₂T₂. I think i need a little bit guidance here to help me through because even though on the surface it seems easy to understand the beat phenomenon given the equation for the superposed, equal amplitude vibrations, however i can't see how the two time periods of the combining waves are commensurable? It is, it seems, a necessary condition to be fulfilled for periodicity after all.

 

[nasu]

256 as frequency is just a good approximation, in the case o beats. Very good, as long as the two frequencies are close enough.
But it is not the "real" frequency. The amplitude of each maximum changes a little after each period of 1/256 seconds.

 

[NRa]

256 as frequency is just a good approximation, in the case o beats. Very good, as long as the two frequencies are close enough.
But it is not the "real" frequency. The amplitude of each maximum changes a little after each period of 1/256 seconds.

Thank you for the reply. Just to be clear, the reason we can't say that the time periods of the two combining SHMs that are giving us a beat here, are commensurable because the amplitude is not constant? It's being modulated at 2 Hz and therefore we can't use T = n1T1 = n2T2 here?

 

[nasu]

Yes, you can and you need to if you want to find the "real" period. In the example given, with frequencies of 255 and 257 Hz, the period is T=1s. (n1=255, n2=257)
The frequency will be 1 Hz. So you see, the 256 is not the "real" frequency. If you look at a plot of the sum you will understand better.
You asked why this does not work for beats and I tried to say that it does, the other way, with half the difference of frequencies gives something that is approximately a frequency. Not in the sense that is close to the real frequency (1 Hz) but in the sense that the signal almost repeat itself. The repetition is not exactly "true", as the amplitude of each peak changes a little for each 1/256 s and only after 1 s it gets back exactly to what it was.

 

[NRa]

First of all, thank you for your quick replies. They are very helpful.
In the effort to hammer my understanding of this concept down to the last bit:

Yes, you can and you need to if you want to find the "real" period. In the example given, with frequencies of 255 and 257 Hz, the period is T=1s. (n1=255, n2=257)

two similar frequencies will give us a beat whose frequency will be the average of the two; it's amplitude will vary at a rate of half the difference of the two frequencies. What we will hear, in case of sounds waves, will be the intensity varying at twice this frequency. This was all nice and clear.

Now to put rest to the doubt of commensurable time periods: the beat waveform has a varying amplitude. If the superposing frequencies were 255 and 257 Hz and time period as you said, and as i had earlier,though unsure, arrived at, is 1 second than that means that a point on the waveform having a particular displacement and velocity at a certain instant will repeat these values exactly 1 second later. It's not easy to visualize this since the amplitude is varying unless you see a video, really.