Wiki shows the fundamental and first overtone frequencies as follows. I'm trying to relate these to a vibrating guitar string and interference. I'm assuming these animated images represent the string after interference of two separate waves in opposing directions, but am having trouble visualizing this using the upper left image. Can anyone explain the phenomenon?
If you look at the red wave, the section between $x=-1$ and $x=-0.5$, that part is the upper left image.
The second diagram at the top is like the red wave between $x=-1$ and $x=0$, and the third diagram is like the red wave between $x=-1$ and $x=0.5$ etc...
It happens like this because for the fundamental mode (the simplest vibration that is fixed at both ends), half a wavelength fits between the fixed points. For the first overtone (the second simplest vibration that is fixed at both ends), a full wavelength fits between the fixed points, and so on.
If you want to understand the vibration of guitar strings, I suggest you forget about the coloured image and just focus on the black and white images.
The key point about a guitar string is that it is (relatively) fixed at each end, so it needs to vibrate in a pattern in which the amplitude of vibration is close to zero at each end.
The black and white images show some of the simplest ways in which a string under tension will vibrate. If you pluck a guitar string at its exact mid-point, it will vibrate in a way that looks much like the very top left image- the amplitude of vibration will be a maximum at the middle of the string and taper down to zero at each end. So, apart from the ends, there is no point on the string that is not moving from side to side.
If instead you hold one finger against the exact midpoint of the string, and pluck the string three quarters of the way down, it will vibrate much like the top right image. At the centre of the string, where your finger has prevented it from being displaced, there will be stationary point, and the string will vibrate in two halves either side of it. Since the two halves are half the length of the full string, they vibrate at twice the frequency, so you will hear a note an octave higher if you pluck the string in that way.
If now you place a finger a third of the way along the string, and pluck it five-sixths of the way along, you can get the string to vibrate in the pattern shown in the third of the black and white images. There will be four points (including the fixed ends) at which the string does not vibrate at all, between which three segments of string will vibrate from side to side at a higher frequency again. In theory you can induce the string to vibrate at higher and higher frequencies by constraining more and more evenly-spaced points along its length, which effectively splits it up into an increasing number of increasingly small segments that vibrate at higher frequencies.
More generally, however, if you don't take care how you pluck the string, it will vibrate in a way that is a mixture of all those so-called fundamental modes of vibration. They will be superimposed to yield a complicated vibrational pattern in which all of the component parts are present to some extent.
The exact mix of the component parts depends on how and where you pluck the sting. If you pluck it gently close to the centre, the first fundamental mode of vibration will strongly dominate the mix, and you will hear a deeper tone than if you pluck the string very close to the end, where you will be forcing it to vibrate in a way that includes more of the higher frequency overtones.
