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## Transverse Wave equation for a string of changing length?

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!