waves - Physics of a guitar

I understand that when you pluck a guitar string, then a bunch of harmonic frequencies are produced rather than just the frequency of the desired note.

If this is true, why does C2 sound so different from C1? I mean, C2 is a harmonic of C1, and should therefore be heard when C1 is played. Why are all these harmonics produced on top of the targeted note in the first place?

Also, what happens if you don't pluck at the center of a string?

Answer

Let's look at frequency instead of notes. Let's say the string has a natural frequency of $100 Hz$ and that harmonics are present when you pluck it. Then, the frequency content of the sound will be of the form:

$a_1 \cdot 100 Hz + a_2 \cdot 200 Hz + a_3 \cdot 300 Hz + ... $

Now, let's say you fret this string halfway such that the natural frequency becomes $200 Hz$. When plucked, the frequency content of the sound will be of the form:

$b_1 \cdot 200 Hz + b_2 \cdot 400 Hz + b_3 \cdot 600 Hz + ... $

See the difference? The 2nd sound is missing many frequencies that the first sound contained.

There are a number of reasons harmonics are produced. One is that when you pluck the string, the initial configuration is not a pure sinusoid but more like a sawtooth or triangle. It's easy to show mathematically that a sawtooth or triangle shape can be "built up" from the fundamental and harmonics. Only the the sinusoid has a single frequency.

Plucking at a different point changes the initial configuration and thus the frequency content.