How does making harmonics work on a guitar?

Question

If you excite the string without fretting it anywhere, you will get a fundamental harmonic.
If you fret it at the halfway length, you will get the second harmonic (higher frequency).
That seems to make sense, because you will "shorten" the length of the string, therefore you will increase the frequency.
However, for the next harmonic, you will fret at the 1/3 mark (and at the 1/4 mark for the next). Therefore, you will be exciting a longer length of the string than the case of the second harmonic. But still, the frequency keeps increasing. Why does that happen?
If you excite a longer length, wouldn't it make sense for the frequency to decrease? What am I missing?
Maybe it has something to do with the extra tension our fingers make on the string at different parts of it? But I don't know how that would work.
If possible, I'd like a layman friendly answer.

user137289 · Accepted Answer

Disclaimer: I am not a guitar player.
Plucking a guitar string does not only produce the fundamental frequency but also many overtones. We perceive the pitch as that of the fundamental.
Putting one's finger in the middle will quickly dampen the fundamental and the odd overtones. The even harmonics are hardly affected because those have a node in the middle. One hears a pitch that is one octave higher.
Similar for putting one's finger at one third of the string. It dampens the vibrational modes that do not have a node at that position.

Quantumwhisp · Answer

If you press at the 1/3-mark, you create a "node" of the possible vibrations at the 1/3-mark. This is an additional boundary-condition, that all the waves that occur on the string have to respect. This boundary-condition only allows waves with a frequency 3 times the fundamental to exist.
To go into more detail: The node at the 1/3 mark splits the string into two parts with 1/3 length and 2/3 length of the original string.
Since the wave travels along the complete length of the string, it is important to look at both parts. Yes, the 2/3 length part is longer than the 1/2 length part, which would give you the 2nd harmonic. But the 1/3 part is shorter. And the wave has to be able to exist on both parts of the string, not only on the long one.

How does making harmonics work on a guitar?

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