I get more hits from Google searches about electric guitar physics than anything else. So that seems to be a subject in high demand. If you haven’t seen my first post on the topic check it out. Currently, I’d like to talk about why notes played closer to the bridge of a guitar sound “dirty”. If you’ve never witnessed this phenomenon, I’ll try to have some examples prepared.
Guitar strings, like any string fixed at both ends, have resonant modes corresponding to particular energies and frequencies. The simplest and lowest energy modes have the fewest crests, so the lowest energy mode for a string is when the center of the string moves up and down. It is the tendency for things to fall into their lowest energy states, so it is this mode which produces the note you hear when a string is plucked.
Higher energy modes displace the string from equilibrium in more locations than the middle. So if you pluck a guitar string farther from its center, you introduce more of these modes corresponding to higher frequencies. A listener will still identify the note being played as the lowest energy resonant frequency, but there will be other higher frequencies audible at lower relative volume.
I’ve modeled this behavior mathematically so we can see it graphically. Waves follow a differential equation called the wave equation, which is generally
. This equation has a well understood solution, all we need to do is provide initial and boundary conditions. I’ve imagined a string fixed at both ends with a length of one unit. I’ve also decided that we’ll just pull one point on the string some distance away from equilibrium.

Plucked in the center.

Plucked 0.25 from the nearest fixed end.

Plucked 0.05 from the nearest fixed end.
I’ve written all of this up in a Python 2.6 script using the SciPy library, which I’ll provide you with to mess around with on your own (I think it’s fun, anyway). It will take two command line arguments (both optional), the first specifies the place to pluck the string, the second how far to pull it. Defaults are 0.5 and 0.05.
I’ve produced three graphs which show the initial conditions, the displacement of the string as a function of time at a fixed point on the string (0.75, since this is a fairly reasonable place for a pickup to be along an electric guitar), and a Fourier transform of the “signal” in the previous cell for plucks in three different locations. The important thing here is the result of the FFT.
A Fourier transform is means of going from a description of something’s position in time to a description of the frequencies which describe that motion in time. What it shows us here is that the oscillations in the string have higher frequency components when the string is plucked closer to a fixed endpoint. The scale on the y-axis is arbitrary and is just a way of judging relative prevalence of each frequency on the x-axis (also arbitrary). As mentioned before, this is due to higher energy harmonic modes being stimulated when the string is plucked further from its center. This is why the same guitar can make so many different sounds from clean to twangy!