Harmonics on vibrating string

Question

If we gently touch a string at a rational fraction of its length before plucking it, we get a pure harmonic.

I've always wondered why that is the case. Wikipedia has a picture illustrating it, but I don't understand why the other higher harmonics don't sound. For instance, if I touched it halfway through, I'd expect to get at least the harmonics that correspond to powers of two.

Mathematically, how do we account for the loss of degrees of freedom?

i.e. How can we model a string which vibrates according to the wave equation and is lightly being touched and then plucked?

Answer 1

I believe the answer is that you do get the higher harmonics. Consider that, if you didn't, a guitar would sound remarkably like a perfect synthetic sine wave.

Strings obey a wave equation, so we can approach the problem via superposition. Take the fourier transform of whatever plucking does to the string. Each frequency should get some amount of energy. Generally speaking lower frequencies will have more energy. Now, for each frequency, analyze it on its own. Frequencies which do not have a node at the string length will attenuate, leaving only a frequency distribution containing frequencies with nodes at the string length. This includes your harmonics. I believe the picture you drew only shows the lowest frequency that vibrates when you touch it in any place.

I believe, if you want to get just one harmonic, you have to carefully displace the entire string along that sine wave to ensure no harmonics show up. Displace it more haphazardly (such as "plucking"), and the harmonics appear.

In real life strings, I think there are two additional factors. First, your gentle touch is imperfect, attenuating higher frequencies more because the slightly soft region where you touch the string measures a larger fraction of a wavelength at higher frequencies. There is also resistance from the air (even the spherical cow deals with friction), and loading on the guitar, which shape the sound. If these effects attenuate a higher frequency more, the sound will approach a more pure tone.

The second effect is the human ear. We are so unbelievably used to hearing overtones that our brain actually discards them! The waves are reaching our ears, with all harmonics, but our brain just chooses not to hear all of them. This has two neat effects I know of:

• In the era where large subwoofers were hard to find, synthesizers often left the fundamental of a bass note out, synthesizing only the harmonics. The human ear would hear the harmonics, realize they match a harmonic pattern they are used to, and 'fill in' the fundamental, even though it was never actually played.
• Throat singing relies on creating shapes with one's mouth and throat which resonate at a given harmonic's frequency. When we hear this, our ear cancels out most of the harmonics, leaving only the fundamental. However, this carefully crafted resonance is so unusual that our brain assumes it must be a second source of sound, giving the illusion that two notes are being sung at the same time!