Problem — Fundamental and third harmonic of a string

Problem

A guitar string of length $L = 65\;\text{cm}$ has a transverse-wave propagation speed $v = 286\;\text{m/s}$. Calculate the fundamental frequency and that of the third harmonic.

Solution

Setup. String fixed at both ends: normal modes $f_n = n\,\dfrac{v}{2L}$, with $L = 0{,}65\;\text{m}$.

Fundamental ($n=1$):

$f_1 = \frac{v}{2L} = \frac{286}{2\cdot 0{,}65} = \frac{286}{1{,}30} = \ev{220\;\mathrm{Hz}}$

Third harmonic ($n=3$):

$f_3 = 3\,f_1 = 3\cdot 220 = \ev{660\;\mathrm{Hz}}$

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Topics: Onde Concepts: Onde stazionarie