The Sound of a Plucked String

Pluck a guitar string and its starting shape is a triangle. A sine series rewrites that triangle as a sum of the string's pure vibration modes. Where you pluck decides which harmonics are strong, which is why tone changes as you move your hand.

How it works

Fourier sine series

In one sentence

The odd $2L$ -periodic extension of $f$ on $[0, L]$ has the Fourier sine series $\sum b_n\sin(n\pi x/L)$ with $b_n = \frac2L\int_0^Lf(x)\sin(n\pi x/L)\,dx$ , converging to $f$ wherever $f$ is continuous.

General formula

{\htmlData{sym=3}{f(x)}} = \sum_{{\htmlData{sym=2}{n}}=1}^{\infty} {\htmlData{sym=0}{b_n}}\sin\frac{{\htmlData{sym=2}{n}}\pi x}{\htmlData{sym=1}{{L}}}, \qquad {\htmlData{sym=0}{b_n}} = \frac{2}{\htmlData{sym=1}{{L}}}\int_0^{\htmlData{sym=1}{L}} {\htmlData{sym=3}{f(x)}}\sin\frac{{\htmlData{sym=2}{n}}\pi x}{\htmlData{sym=1}{{L}}}\,dx

Each sine is one mode the string can vibrate in by itself. The coefficients say how much of each mode the pluck put in.

What the symbols mean

$b_n$
Harmonic strength
How much of the nth mode the pluck puts in.
$L$
String length
1 in the graph.
$n$
Harmonic number
1 is the fundamental, 2 the octave above, and so on.
$f(x)$
Pluck shape
The string's starting shape: a triangle.

The pluck shape is a triangle with its peak at the pluck point $p$ on a string of length $L = 1$ .
Each coefficient $b_n$ comes from orthogonality: multiply by $\sin(n\pi x/L)$ and integrate. Here $b_n = \frac{2\sin(n\pi p)}{n^2\pi^2p(1 - p)}$ .
For a triangle $b_n$ falls off like $1/n^2$ , so a few terms already look close.
Harmonics with a node at the pluck point get $b_n = 0$ : plucking at the middle removes every even harmonic.

Learn the math

Fourier Sine SeriesDiff Eq · Lesson 44, Boundary Value Problems and Fourier Series The Fourier Series ToolkitPDE · Lesson 3, Foundations The Vibrating StringPDE · Lesson 10, Separation of Variables