Moving to a higher harmonic in a standing wave

[Saw]

Imagine that you have plucked a string and it is vibrating as a standing wave at its fundamental tone (frequency f1). You leave it there and later on come back with the intention of bringing it up to the second tone (frequency f2). What should you do? It seems obvious: apply a stimulous oscilating at such second frequency (f2). But what happens then with the energy of the first vibration oscilating at f1? Is it necessarily wasted? Is there no way to benefit from such energy? Can't you build on the existing vibration and apply a stimulous of frequency (f2 - f1)? For example, if I arrive with my arm vibrating at (f2 - f1) and then grasp the string that was already vibrating at f1, will it somehow be raised to vibration level f2?

 

[DrClaude]

If f2 is the frequency of the second harmonic and f1 the fundamental frequency, then what is the value of f2-f1?

 

[Saw]

Well, the difference between f2 and f1... Say f1 = 3 and f2 = 6, then (f2 - f1) would be 6 - 3 = 3.

 

[DrClaude]

Exactly, so any excitation at f2-f1 is simply excitation of the fundamental.

 

[Saw]

Ah..., so frequencies would add up? I had thought that that way I would only increase the amplitude of the original wave. And If instead of moving to an overtone, I just wanted to increase the amplitude of the fundamental one, how should I physically proceed?

 

[marcusl]

When you study vibrations mathematically, you will learn that each mode is independent (their wavefunctions are "orthogonal"). Thus you can put energy into one, the other, both or neither, but what you do in one won't affect the other.

This is true for an ideal string, which is a linear system. Energy converts between modes only in non-linear systems, or in systems specially prepared with some kind of coupling (cross-moding in waveguides, e.g.).

 

[Saw]

each mode is independent (their wavefunctions are "orthogonal"). Thus you can put energy into one, the other, both or neither, but what you do in one won't affect the other.

Following this guidance, I have read the wiki on normal modes (https://en.wikipedia.org/wiki/Normal_mode), which says: "The most general motion of a system is a superposition of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are orthogonal to each other."

This is true for an ideal string, which is a linear system.

So I gather that the answer to my question would be negative at least for such linear systems: there is no way that I could take advantage of the energy of a given mode so as to "excite" it to a higher mode by supplying only the corresponding energy "difference".

If so, when we talk about the excitation of a mode, what do we mean: just setting such mode in motion through a driving force with the right frequency? Or maybe also an increase in the amplitude of the standing wave at such frequency/mode?

Energy converts between modes only in non-linear systems, or in systems specially prepared with some kind of coupling (cross-moding in waveguides, e.g.).

Any example of this you could guide me to?

 

[marcusl]

So I gather that the answer to my question would be negative at least for such linear systems: there is no way that I could take advantage of the energy of a given mode so as to "excite" it to a higher mode by supplying only the corresponding energy "difference".

That is correct.

If so, when we talk about the excitation of a mode, what do we mean: just setting such mode in motion through a driving force with the right frequency? Or maybe also an increase in the amplitude of the standing wave at such frequency/mode?

These are different ways of describing the same thing, or rather, the second follows from the first.

Any example of this you could guide me to?

There is a vast literature--just Google "waveguide mode conversion" and "waveguide mode coupling". Here's an open-source (i.e., free) article on RF waveguides, as one example:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=0ahUKEwjGqd_2jZzQAhVJr1QKHV-LBUkQFgg8MAQ&url=http://www.tuks.nl/pdf/Reference_Material/Nagelberg%20Shefer%20-%20Mode%20Conversion%20in%20Circular%20Waveguides.pdf&usg=AFQjCNFEHpGhq7-AivbgP4fTiKihpi_neQ&sig2=Z30fehN6FuQb__IfRhRslA&bvm=bv.138169073,d.cGw

You'll also see a huge literature on mode conversion in optical fiber guides, relating to the internet and transfer of voice and data across the globe.

 

[Saw]

Thanks a lot, I have looked at that literature, though so far it seems only "inspiring" as it's too advanced for me to follow now. Looking for a simpler model I have thought that I could focus on harmonic oscilators, which are after all a simplified version of standing waves. In this sense, when talking about standing waves, in lecture 49 section 4 (http://www.feynmanlectures.caltech.edu/I_49.html) Feynman discusses the system of two "coupled pendulums" (two pendulums joined by a spring). I think he is saying that the system may have two modes, one where the spring is unstretched and the two pendulums oscilate as if without it, and another where the spring "contributes a restoring force and raises the frequency". I still don't understand how that can physically be done, how this little toy operates but is it relevant for my question? Should I study this system as an example of "coupled" modes or are they also "orthogonal"?

 

[Saw]

I have seen the system mentioned by Feynman in an animation in this site:
http://physics-animations.com/Physics/English/link_txt.htm

I realize now that what is "coupled" is the componentes of the system. But the system has two modes with different frequencies which however would be "normal"...

Anyhow, this book, Optical Guided Waves and Devices, R.R.A.Syms and J.R.Cozens (http://www.gbv.de/dms/ilmenau/toc/110355911.PDF) has given me an orientation. If I understand it well, I don't need to look for examples of coupled modes from the origin. The example can be initially normal modes, which however become coupled due to some influence.

The relevant text is in chapter 11 and it refers to the cases of stimulated and spontaneous emission in these terms:

"we must remember that the photon is essentially a wave train of electric field with some characteristic frequency, and that the electron states E1 and E2 are simply normal modes of the unperturbed atom. If these modes are perturbed by interacting with an oscillating field, then mode coupling will occur strongly if the frequency of the field matches the difference in frequency between the two modes. This is clearly analogous to the mode coupling situations encountered in Chapter 10, where a spatially periodic perturbation was used to phase-match two propagating modes.

Since the new photon has been emitted in response to the stimulation of the driving field, the phases of the incident and stimulated photons must be linked. In fact, they are identical, so that the two photons are effectively joined together in a single wavetrain. We may describe this as a 'doubly occupied photon mode'. This effect, known as stimulated emission, is the key to the amplification of light in optoelectronic devices. Spontaneous emission can also be interpreted in terms of coupling between higher and lower states, except that in this case the stimulating perturbation is not derived from an external agency (an incident photon) but from fluctuations in the local field within the atom itself."

I think this more or less answers my question: what couples two modes and allows the frequency/energy transition is an influence filling the frequency gap between the two modes. I would just need to visualize how that happens...

 

[Saw]

I am now however coming back to "coupled oscillators" as a good ortientation to the question of the OP. I understand now how the two-pendulum system operates. Its energy is shifting from one pendulum to the other. One separates a first pendulum from equilibrium and releases it, so that it starts oscillating but is progressively damped by the other, which is "stealing" its energy until the first comes to an instantaneous halt and the second reaches maximal amplitude, only to suffer ultimateley the same fate and so on. This can be viewed as the superposition of the two above mentioned normal modes, which by the way have only slightly different frequencies. The same happens in acoustics when sound waves of close frequencies superimpose, first destructively, later on constructively. In the latter case it is said that the ear perceives such combined note at the average frequency of the two waves. In both cases (sound and pendula), the amplitude oscillates at a rate of change equal to the frequency difference between the two modes, which is called the "frequency beat".

The question would then be: what happens if I submit the coupled oscillators system to a driving force oscillating at the beat frequency? Will the amplitude oscillation increase in a typical phenomenon of resonance? And how could that physically happen, how would that driving force operate over the system?

 

[sophiecentaur]

Imagine that you have plucked a string and it is vibrating as a standing wave at its fundamental tone (frequency f1).

If you 'pluck' a string, it will take up many different modes because the modes relate to the Fourier transform of the shape of the string, which would be triangular. To get it to oscillate at any individual frequency, you can either start it off with a sinusoidal shape (difficult) or excite it with a single tone and with a low power signal - allowing it to build up in amplitude.

 

[Saw]

Thanks sophie, I assume that the string is already vibrating, for whatever reason, with one or several (superposed) modes (standing waves since the string is closed) at the fundamental tone (f1) or successive overtones (f2, f3). The question was whether it was possible, through a driving force with frequency (f2 - f1), excite mode 1 into mode 2 and so on. I have learned that in a linear system, where modes are normal, that is not possible: the new wave thus created will be again superposed (combined) with the others in terms of amplitude but it cannot alter the frequency of any existing waves.

So I have re-formulated the question in another way. We imagine that the superposed waves are just two, with different frequencies, f1 and f2. In the beats phenomenon, those frequencies would be very close and their difference (the "beats difference") very small. But I suppose that the question can still be formulated equally and the answer is the same, no matter the size of such "beats difference". The question is then: if you apply a driving force oscillating with the "beats frequency" (f2 - f1), what happens? I tend to believe that the answer is simply that you increase the amplitude of the envelope. This would be equivalent to (in the similar coupled oscillators example = two pendula joined by a spring) to giving a little push to pendulum 1 every time that a cycle is completed. Could anyone confirm that this is right?

 

[sophiecentaur]

their difference (the "beats difference") very small.

You mean that they are both very high overtones? 12th and 13th, perhaps? In this situation, you would be dealing with a pretty non-ideal situation. Any argument should also apply to a situation with the fundamental and the first overtone. (The idea of just "sliding" form one overtone to the next would still apply.)

 

[sophiecentaur]

excite mode 1 into mode 2 and so

Do you mean here that the energy at one frequency would turn up at another frequency by supplying energy at a third frequency? If the system is linear then how could this happen? If the exciter is at any frequency f' then it can't do anything but to cause the oscillator to move at f'. If there is a natural mode in the oscillator at f' then you can get a resonance at f'.
There is an important factor that is not often considered in this sort of discussion and that is the Q of the resonant response. There is always some resistance / friction in any real oscillator. If there were not, there would be a constant build up of energy, if the exciter is exactly at f, with no limit. In fact we always find that the loss will limit the total energy held in the oscillator when the loss equals the added power from the exciter. A real resonator will have a response curve and it will be possible to excite it at a frequency other than the centre response frequency. With a Q of 100, you get a significant response at +/- 1% of the centre frequency. See this link which relates to a simple RLC resonator. When you turn off an off-centre excitation, the 'off centre' oscillations will just die down but cannot change frequency.
All this relates to classical, linear oscillators. What goes on in a quantum system doesn't need to be the same at all - in QM, you have non linearity and an energy source 'within' the system. Are you trying to bring the two together? I don't think that's valid.

 

[Saw]

Do you mean here that the energy at one frequency would turn up at another frequency by supplying energy at a third frequency? If the system is linear then how could this happen? If the exciter is at any frequency f' then it can't do anything but to cause the oscillator to move at f'. If there is a natural mode in the oscillator at f' then you can get a resonance at f'.

Yes, I was told that above by marcusl and it is assumed in the way that I have re-formulated the question in post #13.

There is an important factor that is not often considered in this sort of discussion and that is the Q of the resonant response. There is always some resistance / friction in any real oscillator. If there were not, there would be a constant build up of energy, if the exciter is exactly at f, with no limit. In fact we always find that the loss will limit the total energy held in the oscillator when the loss equals the added power from the exciter. A real resonator will have a response curve and it will be possible to excite it at a frequency other than the centre response frequency. With a Q of 100, you get a significant response at +/- 1% of the centre frequency. See this link which relates to a simple RLC resonator. When you turn off an off-centre excitation, the 'off centre' oscillations will just die down but cannot change frequency.

Ok, although in principle this is not related to the question. We can assume that the oscillator is ideally undamped.

The question is only if impinging upon an oscillator (the two-pendulum system described above or a complex sound wave made up of two waves with different frequencies) with a force oscillating at the "beat difference" (difference between the frequencies of the two modes of the coupled oscillator or the two waves of the complex sound wave) is of any use. As commented, the answer seems obvious: you can increase the amplitude of such thing that is vibrating precisly at that "beat frequency", which is the amplitude of the envelope embracing the two coupled oscillators or the complex wave. You can modulate it, so to speak. I just wanted now confirmation that this is ok.

All this relates to classical, linear oscillators.

But even in that case, the nonlinear oscillation can be resolved into components, into harmonics by Fourier theorem, right?

What goes on in a quantum system doesn't need to be the same at all - in QM, you have non linearity and an energy source 'within' the system. Are you trying to bring the two together? I don't think that's valid.

No, I am just trying to conclude if you can do anyhthing with the above mentioned "frequency difference" in a classical system. It is true however that, in a QM system, a photon for example (external energy source) excites the atom also at a "frequency difference". But we had better not get distracted with the QM here...

 

[Saw]

But even in that case, the nonlinear oscillation can be resolved into components, into harmonics by Fourier theorem, right?

I think I mixed things there. Maybe I should have said the following: whilst a linear oscillation can be decomposed into harmonics, a nonlinear oscillation can be decomposed into an infinite series of sine and cosine functions? Or how else?

 

[DrClaude]

No, I am just trying to conclude if you can do anyhthing with the above mentioned "frequency difference" in a classical system. It is true however that, in a QM system, a photon for example (external energy source) excites the atom also at a "frequency difference". But we had better not get distracted with the QM here...

I think this is the crux of the "problem." If you take a quantum harmonic oscillator, it has a single frequency $\omega$. When you excited it, you are always exciting it at $\omega$, not at a "frequency difference." What is quantized is the energy, not the frequency.

 

[Saw]

If you take a quantum harmonic oscillator, it has a single frequency $\omega$. When you excited it, you are always exciting it at $\omega$, not at a "frequency difference." What is quantized is the energy, not the frequency.

Well, I didn't want to talk about the QM analog, but what you say...

My understanding is that both at classical and QM level you have simple oscillators with 1 degree of freedom (and one natural frequency), coupled oscillators with 2 or more degrees of freedom (and correlative frequencies) and waves with plenty of degrees of freedom (and frequencies).

Also that at quantum level, when an electron transition happens, because a photon is absorbed or released, the frequency of such photon (which is tracked by way of absorption or emission lines) multiplied by Planck's constant must correspond to the energy difference between energy levels. Mathematically, that looks identical to saying that such frequency must be equal to the frequency difference between frequency levels.

 

[DrClaude]

Also that at quantum level, when an electron transition happens, because a photon is absorbed or released, the frequency of such photon (which is tracked by way of absorption or emission lines) multiplied by Planck's constant must correspond to the energy difference between energy levels. Mathematically, that looks identical to saying that such frequency must be equal to the frequency difference between frequency levels.

That was my point: you are confusing the frequency of the quantum oscillator and that of the photon related to the transition. The transition appears at the frequency of the oscillator, not at a difference of frequencies in the oscillator. It is best to think of it in terms of energy even when considering the photon, and only convert (if needed) to the frequency of the photon afterwards.

 

[Saw]

you are confusing the frequency of the quantum oscillator and that of the photon.

That is precisely what I was trying to avoid... such confusion. There is on the one hand the frequency/ies of the oscillator/s and the frequency of "what happens". That is clear.

Our disagreement is that for you the frequency of "what happens" is the frequency of the oscillator.

The transition appears at the frequency of the oscillator, not at a difference of frequencies in the oscillator.

But if you look at the details, the surprise is that it is not. Though maybe I went astray somewhere. Let me explain my reasoning.

It is best to think of it in terms of energy even when considering the photon, and only convert (if needed) to the frequency of the photon afterwards.

Yes, I can do that:

(I am thinking of the hydrogen atom and simplifying with signs, for my own sake...)

Take an oscillator located at level n = 3 with E3 = hcR_c/3²
It suffers a "transition" down to level n = 2 with E2 = hcR_c/2²
The E difference is = hcR_c ( 1/2² - 1/3²)
But this is equal also to hf, being f the frequency of the photon.
So the f of the photon that will be emitted and detected is cR_c ( 1/2² - 1/3²)

But the funny thing is that if you think in terms of frequencies, you get the same result:

Take an oscillator located at level n = 3 with frequency f3 = cR_c/3²
It suffers a "transition" down to level n = 2 with frequency f2 = cR_c/2²
The frequency difference f2 - f3 is = cR_c ( 1/2² - 1/3²)
This is again the frequency of the photon causing the "transition".

Hence thinking in terms of frequencies does not look so wrong since it leads to a correct conclusion. And in the light of this the frequency at which the "phenomenon" appears is not the frequency of the oscillator but a frequency difference between oscillators' modes.

I can concede however that if you call such phenomenon an "oscillator transition", this sounds strange, because it is actually true (this is what I have actually learned through this thread) that you cannot excite or force an oscillator to transit to a higher frequency by applying a frequency difference, so it looks only logical to seek refuge in the idea that the influence/exciting source is oscillating at the frequency of such oscillator, thus increasing its amplitude. But if I didn't go astray above, that is not what happens in the phenomenon at hand.

Thus I turned to the mechanical realm looking for an example where a "frequency difference" was of any use. I have concluded that such thing can serve for increasing the amplitude not of an individual oscillator in a system but of the overall system. For example, in the two-pendulum case, an influence at the "beat frequency" would increase the overall amplitude of the system. Or in the complex sound wave, such influence would also enhance the amplitude difference between the beats. If you applied this construct to the QM system of an atom, you would infer that the photon does not provoke strictly speaking an electron transition but an overall increase of amplitude of the envelope (the atom) as such.

But please note: I am not at all proposing that as a theory, I am more prudent than that...! Forget the QM analog, I was just trying to understand if the above described mech systems work as I have explained before, which no one has answered so far!

 

[sophiecentaur]

That was my point: you are confusing the frequency of the quantum oscillator and that of the photon related to the transition. The transition appears at the frequency of the oscillator, not at a difference of frequencies in the oscillator. It is best to think of it in terms of energy even when considering the photon, and only convert (if needed) to the frequency of the photon afterwards.

The QW connection is even more dodgy if you take the Schroedinger idea of a wave solution because the De Broglie wavengths that you can assign to the bound electron are not 'harmonically related' and a foot in classical interpretation wouldn't include the possibility of the electron having two different wave functions at a time.
But, returning to the classical model, the suggestion is that the wave on the string could somehow 'bend' to another frequency, merely by nudging it with another input wave. That would involve discontinuity during the change and, as the string (etc.) has distinct permitted modes. How would it adopt an intermediate modes during the change?

We can assume that the oscillator is ideally undamped.

I don't think the result is very interesting or valid if you insist on no damping. You would be trying to drive the oscillator (which already carries energy on one mode) in such a way that there is no matching of that energy out of the system but that you can introduce more energy in the form of another frequency, that isn't a permitted mode. You could only hope to do that sort of thing if the oscillator had some response away from its normal modes - which requires damping. This would have a musical connection where you can 'bend a note' on a wind instrument from one mode to the next by 'insisting' with the embouchure. That relies entirely on having a pretty low system Q and lots of puff!

But even in that case, the nonlinear oscillation can be resolved into components, into harmonics by Fourier theorem, right?

The (mathematical) Harmonics of a single tone, subjected to a non linearity cannot be assumed to coincide with the Overtones that an oscillator can adopt. The difference can be slight or gross, depending on the exdample. I always quote the example of a quartz crystal oscillator. If you look on the side of it, there will usually be a stipulation that the frequency (mode) to use should be a certain overtone. (Stated frequency on the side). If you set it off in its fundamental and then select the 'nearest' harmonic frequency to the marking on the side, it will not be right.
When guitarists claim to be playing 'harmonics' by putting a finger lightly at points on the string, they are actually playing overtones. Guitar strings are actually not bad in that respect and the harmonics do lie pretty close to the overtones but, trying to synthesise, accurately, a guitar note by just assembling it with harmonics, is doomed to failure.

 

[Saw]

Sophie, let us leave aside the colateral matters (nonlinearity, damping, QM...) and focus on the question, as it is now formulated.

the suggestion is that the wave on the string could somehow 'bend' to another frequency, merely by nudging it with another input wave.

No, that is not the question any more! The original question was re-formulated quite a few posts ago. I will re-phrase it again for you, but better than before:

(The question can be posed based on the two coupled pendula system or based on a complex wave. I will use the second option which is simpler.)

Given two superposed sine waves of different frequencies f1 and f2, we get a complex wave that oscilates at the average frequency [(f1 + f2)/2] but whose amplitude varies at the frequency difference, which is the absolute value of (f1 - f2). This means that you hear a sound with a pitch at the average frequency but whose volume goes up and down with a rate of change equal to the frequency difference (which is for this reason called the "beat frequency").

Note that the envelope of this modulated wave, the green or red lines in the below image, are sine waves themselves, slower ones.

https://en.wikipedia.org/wiki/Envelope_(waves)#/media/File:Modulated_wave.png

The question is: what if you now superpose a third sound wave oscillating at the beat frequency (f1 - f2) and traveling in phase with such envelope? What is roughly speaking the result?

 

[sophiecentaur]

I can only answer with another question. What are you doing to one mode by trying to add energy to another orthogonal mode? By definition of what orthogonal means, the two modes cannot affect each other in a linear system. If you try to impress a periodic 'force' on the system that is not at one of the resonant frequencies you can get two results - if this exciter frequency is lower than the resonance frequencies then the whole system will move by the small displacement of the exciter, if the exciting frequency is higher then there will be no resulting movement. See the graph in this link which is only worth describing with a finite value of Q.

 

[sophiecentaur]

Note that the envelope of this modulated wave,

I just re-read this. It is NOT a modulated wave. It is just two independent frequencies which, when passed through an appropriate analysis, give a resultant that varies in amplitude at the difference frequency. The mathematical description is A+B with no hint of C X D.

 

[Saw]

What are you doing to one mode by trying to add energy to another orthogonal mode? By definition of what orthogonal means, the two modes cannot affect each other in a linear system.

No... I am not trying to do this. Don't be misled by the title of the thread. I am not trying to put energy into one mode and hopefully feel the effect in another. Since many posts ago, I re-formulated the question. I am just superposing waves. First, I combined two waves of different frequencies (f1 and f2) and I want to combine them with a third wave of another frequency (f1 - f2) and observe the outcome.

If you try to impress a periodic 'force' on the system that is not at one of the resonant frequencies you can get two results - if this exciter frequency is lower than the resonance frequencies then the whole system will move by the small displacement of the exciter, if the exciting frequency is higher then there will be no resulting movement. See the graph in this link which is only worth describing with a finite value of Q.

That is another thing, whether the third wave will find "resonance" in the system over which it will be superposed. Maybe not, but before deciding, please consider these comments on the description of the system.

The mathematical description is A+B with no hint of C X D.

The combined wave is a “beat wave”. Look at its math description in Wikipedia:

$$\cos (2\pi {f_1}t) + \cos (2\pi {f_2}t) = 2\cos \left( {2\pi \frac{{{f_1} + {f_2}}}{2}t} \right)\cos \left( {2\pi \frac{{{f_1} - {f_2}}}{2}t} \right)$$

The sum of waves A (f1) and B (f2) is equivalent to the product of two other waves: C = a (fast) combined “beat wave” oscillating at the average frequency [f2 + f1)/2] and D = a (slow) wave oscillating at (f1 – f2), the rate of change of the amplitude of the said “beat wave”.

It is NOT a modulated wave.

According to Wiki, “It can be said that the lower frequency cosine term is an envelope for the higher frequency one, ie that its amplitude is modulated”

 

[sophiecentaur]

You are just showing a mathematical identity there. Neither side of the equation corresponds to Modulation. The Wiki statement is just sloppy. Look somewhere more reputable for the distinction between addition and modulation. Modulation involves altering some characteristic of a carrier wave (present or suppressed). It does not involve merely putting another carrier near it.
Wiki says "that its amplitude is modulated” but the amplitude of what? is modulated? There isn't and there never was a carrier wave involved and no information / energy was used to vary any quantity associated with a carrier. Those are the essential descriptions of Modulation.

"It can be said" proves nothing. All sorts of things "can be said" but it doesn't make them true. I think you need to accept that you are trying to prove, or show, that your alternative view is valid. Perhaps if you read around some links that a Google search of the term "Modulation" and "carrier wave" you would see where your view is out of step.

Being wrong is no problem if the outcome is better understanding of a very basic subject.

 

[Saw]

Well, if the term "modulation" should apply only when you modulate a carrier wave so as to code information to be transported somewhere else, like in AM radio, let it be so... But really the question of this thread is very simple: if on top of the beat wave, you superpose a new wave oscillating at f1 - f2, what does the outcome look like?

 

[sophiecentaur]

you superpose a new wave

I'm sorry but I don't think you actually know what you mean by your question.What do you mean by the term "superpose"? If you can specify mathematically what you mean then we can discuss what the result will be like.
Is it Addition? A+B
Is it multiplication? A X B
Is it AM? A(1+B)
Btw, the term Superposition is commonly used to describe the co existence of a number of independent fields or waves that do not affect each other. It assumes a linear transmission medium. If you want to do more than Addition then you need a non linear process.

 

[Saw]

I assume that the medium is linear and I mean superposition, i.e. the alebraic addition of the amplitudes. This means that the waves do interfere constructively or destructively as the case may be at each moment and build up a common pattern, a complex wave, even if of course this pattern would vanish the moment that the interference stopped and each wave continued its way unaffected. Nevertheless, in my question the two waves travel in the same direction and in a non-dispersive medium, so at the same speed despite their different frequencies; hence I gather that the interference would last for ever. For practical purposes, the system would be as if it were one single wave, although it can be at any time decomposed in its constituent waves for the sake of analysis.

In fact the analogous case of the coupled oscillators that I have also discussed is actually a single and indivisible phenomenon from a physical perspective though you decompose it in normal modes for the sake of analysis and such analysis is very similar to the one at hand.

And an interesting complication of the waves case would be imagining that at a given time the medium becomes closed and the waves become stationary, but let us leave that aside for the time being and focus on the above question.

I can roughly guess what the outcome would look like but how can I get the exact picture? Any software that does it?

Said that, I would like to ask a lot of questions about the other alternatives, like wave multiplication or AM, which I understand much less or nothing, but maybe better at another moment/thread, so as to settle first that simple question.

And thanks for your help and patience!