Addition of Harmonics in a string wave

[Jufa]

In basic optics, we are given the general solution of the wave equation (massless string of length L) as a linear combination of normal modes, that need to have some of the permitted frequencies due to boundary conditions. In laboratory, we observed that phenomenon. We generated a wave in a string that, as it reflexes at its end, ended up generating a standing wave. We did that for some number of harmonics.
My question (more than a question is a matter of discussion) is: We are often given the view of the harmonics as some sort of an orthogonal basis of the wave equation solutions space, in the sense that any of the solutions can be uniquely written as some linear combination of them and that the energy of the solution is equal to the sum of the energies of the harmonics of which it has been decomposed.
From this, I think there is a quite reasonable analogy between a standing wave and a free particle. Let me explain that. For a free particle, we have an infinite number of solutions for its movement. Every solution can be written as a straight movement with no acceleration and a certain velocity (let's fix the initial position, as we would fix where the standing wave was). I hope the analogy is clear until this moment. Now if I choose, for the particle, one solution as the movement along the "x" axis with a certain velocity and I do the same for the other orthogonal axis ("y" and "z") I obtain an orthogonal basis for the solutions in the same sense I obtained it for the standing wave (any solution will be a combination of this three and the energy resulting from the sum will be equal to the sum of energies. This is the analogy, and maybe the most remarkable differences are that for the particle we only have 3 degrees of freedom whereas that for the standing wave we have infinite. Now I would like to go further.
When speaking about the free particle I feel very comfortable talking about orthogonality of solutions because I associate it to the fact that I can "modify" the contribution of one element of the basis without disturbing the others. In other words, I can push the particle toward the "y" direction, which will increase its velocity in that direction, without modifying velocity in any of the other two directions. Now my intuition tells me that this sort of "non-disturbing modification" should have some analogy in the standing wave since it seems that analogies worked quite well between both systems until the moment. So I think that you should be able to modify the amplitude of one harmonic without disturbing others. Now is where I fail to figure how you can do that, how you "push" one single harmonic without interfering others. In the laboratory, we had a generator that made the string vibrate in a certain frequency. Once we reached a permitted frequency we could observe its corresponding normal mode but we never tried to create a superposition of them and, in fact, I have no clue of how could we make that.
Thank you for reading.

 

[Baluncore]

Once we reached a permitted frequency we could observe its corresponding normal mode but we never tried to create a superposition of them and, in fact, I have no clue of how could we make that.

So long as a transmission line, (the string), is linear, the supported traveling waves will all be independent and can exist simultaneously on the line. Since the end reflections lock the position of the standing wave pattern any waveform that can be described by a Fourier sum of the fundamental and harmonics can be supported on the line.

How to excite the line with multiple harmonics? Use a metal guitar string. Run a constant DC current through the string, (but not enough to heat it). Place a horseshoe shaped electromagnet with the magnetic field cutting the wire. Excite the electromagnet with a fundamental-frequency square-wave and notice the square standing wave that appears on the string. Switch the excitation to a triangle wave and observe that. Now add another electromagnet, rotated 90° to the first. Excite one magnet with the square and the other with the triangle, look from two directions to see the two separate independent waves, each composed of many independent harmonics, all of which are permitted resonant waves on the string.

Enjoy the orthogonality.

 

[sophiecentaur]

Excite the electromagnet with a fundamental-frequency square-wave and notice the square standing wave that appears on the string.

This assumes that the Force on the string (current in magnet) is the same as the Position of it. The Force determines the Acceleration and not the position. You would need to pre-distort the excitation waveform before you could get near the wanted wave shape on the string. That's a basic problem but you would also have a problem designing an electromagnet that would operate 'at a point' or have a field with a known distribution. Perhaps the 'perfect' electromagnet would be acceptable for a thought experiment by the the force / displacement problem would still exist.

 

[Baluncore]

I agree, the situation is analogous to a speaker voice coil.

 

[sophiecentaur]

I agree, the situation is analogous to a speaker voice coil.

If it were easy to do that sort of thing with a fixed string, I'm sure there would be dozens of YouTube movies showing it. (It would be very impressive). All I have ever seen is almost formless squiggles, sometimes with traveling waves, when a string has been plucked with a longitudinal component.