Transverse Wave equation for a string of changing length?

[czechman45]

I'm trying to learn more about the physics of guitars. I followed through the derivation of the transverse wave equation and that makes sense, but it seems like several of the simplifying assumptions might not apply. There are a lot of approximations with small angles and small slopes. I don't know how small is considered 'small', but I'm willing to take those on faith. The one that I think might make a difference is the assumption that the string elements have no longitudinal motion. The length between the bridge and the nut on a guitar stays the same but the string is not fixed at these points. It runs over then and then secures to the guitar further along it's length. When the string is plucked the string has to either stretch and/or recruit some of the string from these other portions (the parts not originally between the bridge and nut). This is changing the length of the string. Has there been work in the wave equation to account for this? Does it make a difference?

Thanks!

 

[Simon Bridge]

A guitar string should really be modeled as a spring - so the tension is not a constant.
As you mentioned, to play different notes, you change the length of the string.
http://www.bsharp.org/physics/guitar
http://ffden-2.phys.uaf.edu/211.web.stuff/billington/main.htm
... a model that tries to cover everything that could possibly happen will be very complex.

 

[olivermsun]

I think the OP is saying that a) the motion of segments of the string is really not perpendicular to the rest position of the string, and b) the small angle approximation, leading to zero change in string length due to stretching is not strictly correct.

These are just traditional approximations in deriving the linear wave equation which make the system tractable. They do capture the behavior of the string very well, particularly for small transverse motions, and qualitatively under most realistic situations (where you haven't broken the guitar!). People have of course considered what happens when you choose NOT to make these approximations, but the first thing that happens is that the restoring force has a sinusoidal function instead of a constant (spring) coefficient times displacement. For evaluating this system, computers are very useful. :)

It is true however that in a real guitar, the stretching term does matter. You can hear an example of this when you pluck the string very hard and the pitch changes (same goes for a bowed instrument). The deviation from purely 1-d motion of each string "element" is less noticeable.