Polarization of a Guitar String

[DannyCov]

Hi everyone,

When a guitar string is played its motion of displacement can be considered in two planes horizontal and vertically. With consideration of the nut and bridge of a guitar (ends) I've been told their their terminating conditions be different?
How does the motion cause a difference

Anyone have any ideas? This is puzzling me...

 

[lpfr]

I have asked myself the same question, and I do not have an answer for sure.
In the way guitar strings are exited, they are of course polarized. And mostly parallel to the soundboard. But this do not impairs the acoustic coupling.
Moreover, if you look the vibrating string, you will see that, in most of the cases, it is elliptically polarized and that the ellipse turns. Maybe that the parallel (to the soundboard) mode and the perpendicular mode do not have the same frequencies. This can be due to the fact that the zero of amplitude lies a little outside the bridge because the bridge vibrates. And the compliance of the bridge is not the same in vertical an parallel directions. Maybe that for the same reason the two modes are coupled.
But as I said, I'm not sure at all.

 

[AlephZero]

Yes, there are two modes with different frequencies and different damping factors, because the compliance is different in the two directions, and the sound radiation from the guitar body is greater for the perpendicular mode.

The "classical" guitar sound has notes that start loud (mostly perpendicular mode) but this dies away quickly and the softer parallel mode lasts longer. "strumming" the strings parallel to the body (with fingers or plectrum) produces a very different sound from plucking them in the perpendicular direction.

The "rotating ellipse" appearance does not depend on coupling between the modes, it is the same effect as Lissajous figures when the X and Y deflections of an oscilloscope have close but not identical frequences.

The sound radiated by the two modes IS coupled, in a complicated way because of the differnent damping rates and the slightly different frequencies. The different frequences produce beats. Typically there is only one clearly audible "beat" before the perpendicular mode dies away. The result is a loud start to the note, a rapid dip in sound output, the sound level then increases as the parallel mode becomes dominant, and then decays slowly.

Though the "rotation ellipse" doesn't depend on coupling between the modes, there is some coupling because the vibrating system is intrinsically nonlinear. When the string is displaced its length increases slightly and therefore its tension also increases. This explains why the pitch of the note falls as the amplitude decays (the pitch change is easily measurable). The changes in tension give some weak coupling between the modes.

Incidentally this weak coupling is more important in instruments with several strings per note, like the lute (2 strings per note) or piano (2 for mid range notes, 3 for high notes). The 3 piano strings start off vibrating more or less in phase perpendicular to the soundboard, and finish up vibration with 120 degree phase angles parallel to it. The transition between the two different vibration configurations is what makes a piano sound like a piano.