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I have seen similar questions regarding this but containing answers that somewhat disagree with each other which makes it hard for me to understand this.

My question is mainly about when plucking a string instrument at the end side of it, not in the middle. From several slowmotion videos, including this one and this one, I can see that this creates a single propagating wave that reflects back and forth.

If there is no periodic plucking, there is no way that this single propagating wave would interfere with itself to produce standing waves and thus harmonics. And yet, I keep reading that plucking a string instrument (once) would produce (a mixture of) harmonics. How is this possible with a single propagating wave that can not interfere with itself? Are there other conditions, other than interfering propagating waves, that produce harmonics?

Phy
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    Are you familiar with Fourier analysis? The plucking pulse contains a wide mixture of frequencies, some of those are in harmony with the fundamental of the string. – PM 2Ring Sep 17 '19 at 22:04
  • @PM2Ring I have read a bit about it. Someone showed this picture in his answer: https://physics.stackexchange.com/a/111916/143509. Two questions came up because of this:

    1. Does that mean that these Fourier standing waves are actually by definition something else that are not caused by interfering propagating waves?

    2. Can those Fourier standing waves actually be seen during the propagation of a single propagating wave ?

     – Phy Sep 17 '19 at 22:13
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    Mathematically the harmonics are a multitude of solutions that all satisfy the wave equation (with boundaries). The superposition principle then demands the 'complete' solution is a sum of all these particular solutions (fundamental + harmonics). You can see the math in action here: http://www.sciencemadness.org/talk/viewthread.php?tid=65532&page=17#pid451284 – Gert Sep 17 '19 at 22:13
  • @Gert Thanks a lot for the link. One question though; does that mean that these Fourier standing waves are actually by definition something else that are not caused by interfering propagating waves? – Phy Sep 17 '19 at 22:24
  • interfering propagating waves They're not relevant. As said above and shown in the link, the fundamental and harmonics are simply the MANY (infinite, in fact) solutions of the wave equation. These are then summated to provide the complete solution. This is remarkably similar to (some) analytical solutions of the Schrodinger Equation, which also yields an infinite number of particular solutions, to be summated. I suppose you could claim this summation is your interference but personally I don't think it's useful to do so. – Gert Sep 17 '19 at 22:57
  • If you like I can formulate an answer according to these lines but it'll be tomorrow (late here). – Gert Sep 17 '19 at 23:05
  • @Gert I was confused because other sources say that harmonics are caused by interfering propagating waves while the Fourier analysis states that a single propagating wave itself consists of harmonics.

    That being said, you are saying that there are infinite possible solutions which would mean to me that there are an infinite number of harmonics possible for a certain propagating wave. Yet, when you lightly touch the string on a certain spot to create a node and then pluck the string, the loudness of the total sound would decrease because there are less harmonics that share this node. How?

     – Phy Sep 18 '19 at 17:41
  • That being said, you are saying that there are infinite possible solutions which would mean to me that there are an infinite number of harmonics possible for a certain propagating wave. Yes but there is such a thing called the amplitude spectrum. Look at this: http://www.sciencemadness.org/talk/viewthread.php?tid=65532&page=18. Where I wrote: "So the amplitudes diminish very quickly with increasing values of n" For harmonics, amplitude (volume) decreases quickly with increasing pitch. So in reality, only a few of the lowest harmonics influence the sound generated. – Gert Sep 18 '19 at 17:57
  • Also you keep using the term propagating wave but these here are standing waves. – Gert Sep 18 '19 at 18:00
  • @Gert Thanks for the explanation. So if I understand this correctly, the single wave that is bouncing back and forth is a result of many superpositioned standing waves? And as time goes on, many superpositioned high frequency standing waves dissipate such that, for example, only 1 standing wave is left over which will dominate the oscillation of the string, turning its movement into that particular standing wave instead of a wave that is moving back and forth? Is this correct? – Phy Sep 26 '19 at 17:20

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Anything happening on a guitar string can be always written as a superposition of standing waves of different harmonic frequencies. Even the initial motion, which is clearly bouncing back and forth, can be written in this way. (This should not be surprising. After all, standing waves themselves are written by combining solutions that only propagate in one direction. I'm just saying we can undo that.) So technically the answer to your question is just, yes, essentially by definition.

There's another sense in which the answer is yes. As the oscillation continues, higher frequency standing waves decay due to dissipation, leaving behind a simple combination of a few, low-frequency standing waves. So even though the solution is always formally a combination of standing waves, as time goes on, the guitar's behavior will look more and more like the examples of simple standing waves that you find in your textbooks. You can see this starting to happen towards the end of the first video you linked.

knzhou
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  • Thanks for the explanation. So if I understand this correctly, the single wave that is bouncing back and forth is a result of many superpositioned standing waves? And as time goes on, many superpositioned standing waves dissipate such that, for example, only 1 standing wave is left over which will dominate the oscillation of the string, turning its movement into that particular standing wave instead of a wave that is moving back and forth? Is this correct? – Phy Sep 26 '19 at 17:09
  • @JohnnyGui Yup, sounds right! – knzhou Sep 26 '19 at 17:46