Challenge — low E on a guitar string

Problem

A string should sound a low E$_2$ at $80\;\mathrm{Hz}$ in the fundamental. Wave speed on the string $v = 200\;\mathrm{m/s}$. Calculate the minimum length and comment on why in real guitars the low E is obtained with thicker strings, not longer ones.

Solution

Setup. For a string fixed at both ends the fundamental is $f_1 = v/2L$; inverting:

$L = \frac{v}{2 f_1}$

Substitution.

$L = \frac{200}{2\cdot 80} = \frac{200}{160} = 1{,}25\;\mathrm{m}$

$\ev{L = 1{,}25\;\mathrm{m},\ \text{too long for a guitar}}$

Comment. A guitar fretboard is about $0{,}65\;\mathrm{m}$ long: $1{,}25\;\mathrm{m}$ is nearly double, impractical. To lower the note without lengthening the string, the speed $v = \sqrt{T/\mu}$ is reduced: increasing the linear density $\mu$ (thicker strings, often wound with a metal winding) lowers $v$ and hence $f_1$, for the same length and similar tension. That’s why the bass strings of a guitar are denser (thicker), not longer.

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