Natural Harmonics on a String

Question

Consider the Dirichlet boundary value problem of a guitar string stretched between two fixed points which is made to oscillate by pinching and releasing the string.

It can be shown in quite straightforward fashion that the solution is given by a Fourier series

$u(x,t) = \bigg( A_n \cos \frac{n \pi c t}{l} + B_n \sin \frac{n \pi c t}{l} \bigg) \sin \frac{n \pi x}{l}.$

The coefficients $A_n$ and $B_n$ can be related to the initial/boundary conditions. This basically implies that the oscillation of the string would be seen as something quite messy, since there is the dominant overtone which is superposed with various other harmonics.

Is there some way to choose the initial conditions so that something more simple is obtained, for example, so that only the low order terms are kept?

Also, is there a way to set the problem up so that only gets the overtone and the other harmonics are left out of the situation, to achieve

$x(t) = A \cos (2 \pi f_0 t) ,$

where $A$ is the amplitude and $f_0$ is the natural frequency.

In the guitar playing world, I think this is called a natural harmonic, which is achieved by lightly touching a string at certain points, but I am not sure how this fits in with the mathematical description above.

I appreciate this may sound like a music question, but the answer must have a physical basis.

Answer 1

Is there some way to choose the initial conditions so that something more simple is obtained, for example, so that only the low order terms are kept?

All terms die down in time after the string is plucked, and after some time, most higher harmonics are not audible, and only some lower ones are.

However, which harmonics are most audible then, and thus the timbre the tone has, depends on the initial conditions. One can get a symmetric standing wave oscillation, and thus presumably more intensity in low harmonics, by plucking the string in the middle of its free segment. The resulting tone has a different timbre (less intensity at the higher harmonics) compared to what results usually, when plucking the string near the bridge.

is there a way to set the problem up so that only gets the overtone and the other harmonics are left out of the situation, to achieve

$$ x(t) = A \cos (2 \pi f_0 t) , $$

The other harmonics are the overtones (frequencies that are multiples of the fundamental frequency). What you're after is a standing wave oscillation at the fundamental (=natural =base) frequency. Such oscillation is hard to set up by picking the string, because one would have to pick the string with very many picks at all its points in between the fret and bridge, so that the initial condition would not be two linear segments meeting at the pick, as usual, but a string bent into sine wave arc, touching all the picks.

One could maybe excite a wave that is very close to such an ideal standing wave, by a speaker emitting the proper sound wave of the same frequency, directed at the string, or if the string is metallic, by electric/magnetic field oscillation at the same frequency.

Using string harmonics technique (flageolet) does not produce a standing wave of one frequency exactly, there are still higher harmonics because the technique just makes sure the wave has a node in an unusual place where it can remain for the rest of the oscillation. This seems to boost some low frequencies and suppress the higher ones, but it does not remove all the harmonics.