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I have been looking through the physics.se and all over the internet for weeks now honestly, and I still don't understand how harmonics are formatted on the particle level. Yes, I know that only certain wavelengths fit an integer number of times into the length of the system and I understand how and why they would resonate if we drive the system with the frequency matching this wavelength. I see how the incident and reflected waves would have constructive interference and create nodes and antinodes.

However what I fail to understand is how a single pulse or a driving force at the fundamental frequency can possibly lead to formation of harmonics? The compressions at the fundamental frequency travel from one end to the other and back, only to be met with newly introduced disturbance wave. At what point and how do the particles of the wavefront separate into distinct wavefront of the harmonics?

EDIT: Are the harmonic frequencies introduced by the forcing vibration in the first place? For example by non-linearities in vibrations of lip-buzzing, reeds, mouthpieces and oral cavities - or are there other phenomena that are causing harmonics? Would a sine wave driving a trumpet at its natural frequency (lip buzzing at only one resonant frequency of the trumpet) create harmonics, and if yes how?

And again just to be clear, I am NOT asking how and why they resonate once they are created and what frequencies fit the Fourier series etc - but the principle of their creation in the first place.

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    $\begingroup$ I believe that you what you are asking is related to non-linear acoustics. In linear acoustics, there is no generation of frequencies that were not present in the input (or the forcing function) of a system. Thus, for example in pipes and/or musical instruments, the forcing function is wideband and only the "resonant frequencies" (apologies for abuse of the term but I believe it communicates the meaning) "surive". $\endgroup$
    – ZaellixA
    Oct 18, 2022 at 13:09
  • $\begingroup$ So if we take for example a trumpet - all of the (or most of) the harmonics are the consequence of instability/non-linearities in the buzzing of the lips? I mean that would explain it, but is that really the case? $\endgroup$
    – Dimitri
    Oct 18, 2022 at 14:42
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    $\begingroup$ Most probably not. To my (rather limited indeed) knowledge, the resulting spectrum of the trumpet (just using the same example) is the filtered (from the tube) response of the wideband excitation resulting from the lips (buzzing or whatever is that they do). So, consider a wideband (white- or pink-noise like) excitation which is filtered by the tube which is the body of the trumpet and only the supported frequencies (kinda the resonances) are what comes out, since the rest of the frequencies are not supported by the tube. $\endgroup$
    – ZaellixA
    Oct 18, 2022 at 17:22
  • $\begingroup$ But how does this wideband, pink-noise like disturbance get generated in the first place? The buzzing of the lips definitely has a pitch, it definitely is not white noise like. ( see youtube.com/watch?v=2q6tA_SkYTE for reference) It is not a bad idea that I record and check the spectrum of the buzzing. It might indeed be broadband enough to create the basis for resonance harmonic overtones. However I still don't understand how is it that you think this white-noise-like disturbance is generated? $\endgroup$
    – Dimitri
    Oct 18, 2022 at 20:00
  • $\begingroup$ To further complicate the situation, there also might be some effect of the coupling of the larynx, vocal tract and lips and the waves in the trumpet, which might result in resonance at the upper harmonic frequencies - which might not be available to hear when listening to the buzzing on it's own. ( newt.phys.unsw.edu.au/jw/brassacoustics.html#sound ) $\endgroup$
    – Dimitri
    Oct 18, 2022 at 20:07

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