Trying to understanding resonance, harmonics wrt standing waves

Question

I keep getting confused about this and I'm not sure if it's because I keep thinking of guitar harmonics, so I thought I'd just try to explain: please tell me if my understanding is correct or not. A string on a guitar has 2 nodes. One at nut and one at tuning peg. When string is plucked it will vibrate at its fundamental frequency, which means the middle of the standing wave has maximum displacement from its equilibrium position.

But I know that there's a harmonic on a guitar at 12th fret. But when I pluck a harmonic on 12th fret what exactly does this represent wrt the standing wave? By placing my finger at 12th fret, am I imposing a node at that location? If so, then what is difference if I actually press down fully at 12th fret.

Also, am I correct in saying that a string that has nodes at both ends is the same as a closed pipe. A string that is attached at one end but other end free (an antinode) is the same as an open pipe.

The fundamental frequency for string with nodes at either end is the same as the 1st harmonic. I don't understand why this wouldn't be the 0th harmonic. Its length will correspond to one-half of a wavelength because it must attach to nodes at either end... but if we add one node in middle of string, then the length of string will equate to one complete wavelength, which corresponds to 1st harmonic. How wrong or right is this?

Answer 1

The string does not vibrate only in the first harmonic. It vibrates in a combination of several harmonics, the first one is just the more intense.

The second harmonic has a node exactly at the 12th fret. When you touch the string at this point you kill the first harmonic since your finger stops this harmonic of vibrating. In fact you kill any other harmonic which has an antinode at that point (the odd ones). The second harmonic (and the even ones) survive because they have a node at that point. That point would not vibrate anyway.

if I am not wrong you can make another harmonic (in the sense musicians refer to harmonic) at the 17th fret. What you do is to touch the string at this point which is a node for the third harmonic. Then you kill all the harmonics which do not have a node at that point. That includes the first and the second.

Note that since the intensity decrease with harmonic number the harmonic (musician's sense) in the 17th fret is less intense that the one in the 12th fret, which is itself less intense that the free string vibrating.

The string and tube behaviors according to nodes/antinodes are the same as long as you are referring to nodes/antinodes of displacement (and not pressure) in the case of tubes.

Aside note: Guitar players normally refer to "harmonic" to the act of extracting special sounds when slightly touching the strings in point near the 12th, 17th,... frets. For physicists, harmonic means the normal modes of vibrating strings.