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I just learned standing waves in class. And I know that fundamental frequency is the minimum frequency with which a standing can exist on a string. Then we talked about the role of standing wave in music.

I don't very understand how do standing waves produce a note, since there is no net energy transfer through the wave. Also, what are the roles of different harmonics in the music? Can anyone help me out?

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The standing wave on the string creates a travelling pressure wave in the air around the string, i.e. sound. (In a real musical instrument, the mechanism of sound production is a little different, but the concept remains the same. See @Hilmar 's comment on this answer.) This carries away energy and damps the oscillation on the string. Hence, the standing wave will decay; to maintain the standing wave on the string, you must drive the oscillation, through a method like pulling a bow across the string or repeatedly plucking the string.

The principle of superposition tells us that if two waves $\psi_1$ and $\psi_2$ are allowed on the string (satisfies boundary conditions), any linear combination of them $c_1\psi_1+c_2\psi_2$ will also be allowed on the string. So, in real life, the standing wave on a string can be decomposed as a harmonic series, i.e. it is a linear combination of the pure harmonics. Typically, the fundamental frequency determines the pitch, and the smaller contributions at higher harmonics determine the timbre (tone quality).

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    $\begingroup$ The sound radiation by the string itself is often negligible. That’s why most musical instruments have bodies that do the actual radiation. The body is a mechanical transformer: it has a much larger surface area than the string itself. The damping comes mostly from the load that the body presents to the sting and partially from losses in the string itself. $\endgroup$
    – Hilmar
    Nov 10, 2021 at 21:10
  • $\begingroup$ @Hilmar Thanks for pointing that out! I've edited my answer to mention this. $\endgroup$
    – DanDan0101
    Nov 10, 2021 at 21:15
  • $\begingroup$ Re, "what are the roles of different harmonics in the music?" That question goes deep, and it goes deep in two different directions; In human physiology, the harmonic content of a "note" underlies its timbre. Timbre is the musicians name for how we can tell, just by hearing it, whether the note was played on a trumpet, played on a flute, on a violin, etc. The other direction is music theory, in which harmonics underlie the structure of musical scales and musical chords. Whichever way you go though, the question probably is off-topic for a physics forum. $\endgroup$
    – Solomon Slow
    Nov 10, 2021 at 23:53

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