B Moving to a higher harmonic in a standing wave

[sophiecentaur]

What you need to do is to plot out the function cos(ω₁t) + cos(ω₂t) + cos(ω₃t), that is the sum of three cos waves. Alternatively, there is an identity that expresses the expression in terms of half angles. There are dozend of links to that - here's one.
Plotting out functions is easily done with any half decent spreadsheet but you could probably do it with Mathmatica. You will mostly get a pretty messy looking waveform but choosing suitable values for the ωs could give a shape with identifiable features.
With not a lot of trouble, you can plot out any waveform that you can specify mathematically.
Edit: Sorry, I missed out the t's in the formula. I have just inserted them after second reading (after breakfast)

 

[Saw]

Thanks a lot, I did it in two sites, www.mathopenref.com and https://www.desmos.com/calculator

You can see the results in these links:

https://www.desmos.com/calculator/koofvpqxap
<A HREF="http://www.mathopenref.com/graphfun...(x*(a-b)/2)&ah=20&a=20&bh=20&b=19&ch=20&c=0.5">GFE</A>

To get the "envelope" drawn, I had to use an amplitude of 2 and a frequency difference of (f1-f2)/2, as the above Wiki equation somehow suggests.

Actually, one could say that there are two envelopes, upper and lower one, but logically I could only add one of them, because otherwise they would annul each other. So the final outcome is the addition of wave 1 at f1 =20 and A=1 plus wave 2 at f2 = 19 and A = 1 plus the envelope at frequency = (20-19)/2 and A = 2.

If we consider that, in a system like this (traveling waves), constructive interference is the most akin thing to resonance, well, here there are constructive and destructive interferences... I don't know how you would "rate" the operation in this respect. But the outcome is not messy, it looks aesthetically appealing to me. I await your comments and was also wondering: if I want to draw now a standing wave in one of this tools, how should I? I have looked at the applicable equation for standing waves and imagine that in the end I would hit on it but if you have some handy trick...

 

[sophiecentaur]

But the outcome is not messy, it looks aesthetically appealing to me

I agree, it does look pretty when the frequencies are harmonically related. Of course, if you were to use only one of the high frequency signals, the result would look pretty similar (with the right scaling factor). It's dominated by the low frequency waveform (that'll be to do with our psychology, I'd guess).

 

[Saw]

I agree, it does look pretty when the frequencies are harmonically related.

But apart from that, isn't this like "a sort of" resonance?

Look at this version (https://www.desmos.com/calculator/ojmpkxfrnt) where you add new waves at half the frequency difference. Continuing this way, the envelope would get steeper and steeper... and doesn't this put the system in higher risk of breaking up?

Think of the analogous case of the two-pendulum system. The coupled pendula oscillate at the average of the two original frequencies f1 and f2, corresponding to the two normal modes, the symmetric and the antisimmetric one. But they reach their maximal amplitudes at the rate of the frequency difference (or rather, it seems, half the frequency difference).

You can choose to put energy into the system at any of those frequencies, the higher or the lower one: every time that a pendulum turns round or only when it reaches its "maximal amplitude" (the envelope is a "wave of amplitudes", so it also has its own maxima).

Of course in the first case, the effect would be stronger and more rapidly devastating. That is resonance in strict sense. But then you would be expending more energy as well, so that is not strange.

Instead in the second case, the effect would be softer, because you spend less energy, but I believe that you would be "optmizing" it, you would always be doing positive, never negative work. So this would amount to a sort of resonance, less intense but equally economical. Or maybe not, I am not sure of that, what do you think?

 

[sophiecentaur]

But apart from that, isn't this like "a sort of" resonance?

I don't see what you want out of this. You have produced a pattern by adding three sinusoids. How is that 'like' anything? It's just some random maths. The three sinusoids are having no effect on each other at all; that's what superposition is about.
Why don't you just accept all this as a bit of maths and stop trying to imply there is more to it?

less intense but equally economical

This is just getting more and more fanciful.

 

[Saw]

This is just getting more and more fanciful.

No, not at all!

Why? Because I used the words "intense and economical"? I was trying to use a colorful expression, but regardless the choice of words, the idea in itself is spotless. I am realizing right now that what we call "resonance frequency" is the frequency at which we obtain two objectives: (i) ensure that all the energy that we inject into the system is absorbed (no negative work) and (ii) we inject the maximum amount of energy that is allowed without compromising objective (i). But we can also think of another concept, call it as you want, where you satisfy (i) but not (ii).

Just divide, for example, the resonance frequency by two. E.g., the father pushes the swing not every time that it arrives, but every other one. This will be less effective, it does not satisfy (ii), but I am sure now that it will satisfy (i) and the energy put into the swing will be fully absorbed and contribute to higher amplitude. (I leave damping aside.)

You have produced a pattern by adding three sinusoids. How is that 'like' anything? It's just some random maths. The three sinusoids are having no effect on each other at all; that's what superposition is about.
Why don't you just accept all this as a bit of maths

As commented before in my post #30, once that you have superposed two traveling waves in the conditions that we are considering, they will travel together for ever and they are as one single wave for all practical purposes. In fact, if I have stipulated that the first wave is the result of adding two constituent waves, it is only for didactic purposes, because it makes it clearer how it can be again decomposed into its two constituents, only for analysis purposes. But I could perfectly have stipulated that the wave in question were generated by my hand waving at a given frequency with varying amplitude. Then we would have a "real" single wave. And in that case, what would happen to your objection?

In fact, I am basculating now to the other strictly analogous beats example of the two-pendulum system. This is a single physical pehomenon. I displace one pendulum and set the system in motion. Full stop at the physical realm. But then we do an intellectual decomposition of the system purely within our heads. The two normal modes that I am mentioning, in the real experiment, have never existed. They only come out as an intellectual exercise for analysis purposes.

stop trying to imply there is more to it?

Well, I am not implying great things. What I am concluding so far is that one can "excite" an oscillating system (make it absorb all the energy provided to it) even if you don't do it at its resonant frequency but a lower one. Also that when a system can be decomposed (either due to physical origin or by math analysis) into two modes and its corresponding frequencies, a good exciting frequency is the difference between those two. But there may be many more shades, details to it and I just wanted to explore them, with PFers help if possible...

 

[sophiecentaur]

You are not exciting it with a lower frequency. You are exciting it with pulses at half the frequency and those pulses have a frequency component at the resonance frequency.
The theory is all very well established. There are no new frontiers here for you to conquer. Why not just learn the accepted theory and it will answer all your questions at the same time?

 

[Saw]

Oh, I am not pretending that I am conquering any frontier! If I give that wrong impression, I will struggle to avoid it. What I want is of course learning accepted theories, although as everybody I have my learning path and rhythm of progress.

For example, I started calling this situation a sort of resonance because that was the best expression that I could find with my limited knowledge. You said that I was being fanciful and then I explained my reasons, stating that if this should not be called resonance, it was at least an exciting frequency that allowed for full energy absorption. You say now that the right expression is "a frequency component at the resonance frequency". That is fine for me. It amounts to the same and is closer to my original expression.

And what about you? Do you now agree that you cannot dismiss the analysis just because it is based on an addition of waves? Do you agree that the system could perfectly have a single physical origin and the analysis would still be valid? And do you then agree that the so called frequency difference, which is obtained by decomposing the system into two modes and subtracting their respective frequencies, is a component of the resonant frequency of the system?

And note again: if this is true, I am not saying that I am innovating anything, I am just asking if this is accepted theory, which is what I want to learn.

 

[sophiecentaur]

You seem to be making far too much of this. All your have shown with that graph is the way the displacement changes with time with three sinusoids in a linear system. The way it looks doesn't imply that there is any resonance (I.e. no build up of energy in an oscillator at a particular frequency until a maximum steady level is reached)
A system with many modes could be excited that way and could have a displacement / time graph like that. But, you need to consider what resonance actually entails. The energy input system and the Q would be highly relevant. You would need to adjust the three inputs so that the Qs of the modes were just right to bring about that situation (even briefly). I have frequently suggested to you that you should actually learn more about resonance before trying to use it in a personal explanation of a bit of book work. There's a lot to be said for going along with convention - at least at this elementary level. It makes it possible to read sources and to understand the terms they use.

 

[Saw]

Noted. I will do that!

 

[sophiecentaur]

Noted. I will do that!

Good man!