Fundamental and harmonics of a string

Problem

A string of length $L = 60\;\mathrm{cm}$, fixed at both ends, has wave speed $v = 360\;\mathrm{m/s}$. Calculate the fundamental frequency and that of the first two harmonics.

Solution

Setup. String fixed at both ends: modes with frequency

$f_n = n\,\frac{v}{2L}, \qquad n = 1, 2, 3, \dots$

with $L = 0{,}60\;\mathrm{m}$, $v = 360\;\mathrm{m/s}$.

Fundamental ($n=1$):

$f_1 = \frac{v}{2L} = \frac{360}{2\cdot 0{,}60} = \frac{360}{1{,}2} = 300\;\mathrm{Hz}$

First two harmonics.

$f_2 = 2 f_1 = 600\;\mathrm{Hz}, \qquad f_3 = 3 f_1 = 900\;\mathrm{Hz}$

$\ev{f_1 = 300\;\mathrm{Hz},\quad f_2 = 600\;\mathrm{Hz},\quad f_3 = 900\;\mathrm{Hz}}$

The harmonics of a string fixed at both ends are integer multiples of the fundamental: it is this complete harmonic series that gives the “musical” sound.

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Topics: Onde Concepts: Onde stazionarie Skills: Analisi dimensionale