The Vibrating String

Main idea

A string fixed at both ends vibrates in standing waves: each mode $\sin\frac{n\pi x}{L}$ swings back and forth at a frequency proportional to $n$ . Those frequencies are whole-number multiples of the lowest one, the harmonics, and that's why plucked strings sound musical.

u(x, t) = \sum_{n=1}^{\infty}\left({\htmlData{sym=0}{A_n}}\cos\frac{n\pi ct}{L} + {\htmlData{sym=1}{B_n}}\sin\frac{n\pi ct}{L}\right)\sin\frac{n\pi x}{L}

What the symbols mean

$A_n$
Shape coefficients
The sine coefficients of the initial shape f.
$B_n$
Velocity coefficients
From the initial velocity g: B_n·(nπc/L) are g's sine coefficients.

{\htmlData{sym=0}{f_n}} = \frac{nc}{2L}

What the symbols mean

$f_n$
Frequency of mode n
In vibrations per second: n times the fundamental c/(2L).

Explanation

A standing wave is simply a wave that wiggles in place instead of traveling, with some points, called nodes, that never move at all. It's called "standing" because the pattern stays put while the string swings up and down through it. A plucked string's motion is a mix of these, each swinging at its own steady rate, and that mix is what gives an instrument its sound. The formula has a few pieces: $\sin\frac{n\pi x}{L}$ (the shape of wiggle $n$ , with $n$ humps), $A_n$ and $B_n$ (how much of that wiggle the starting shape and the starting push put in), and the cosine and sine of $\frac{n\pi ct}{L}$ (the back-and-forth swinging in time). Add the wiggles up and you get the string. It looks long, but it's the heat-equation recipe with swinging in place of fading.

Where it comes from. Separating $u_{tt} = c^2u_{xx}$ with fixed ends gives the same eigenfunctions as the heat equation, but now $T'' = -c^2\lambda_nT$ : oscillation instead of decay. That one change comes from the extra time derivative. $T'' = -c^2\lambda_nT$ is the mass-on-a-spring equation, so each mode is its own little oscillator, with angular frequency $\frac{n\pi c}{L}$ . Divide by $2\pi$ and you get the frequency $f_n = \frac{nc}{2L}$ above.

Matching initial conditions: $u(x, 0) = \sum A_n\sin\frac{n\pi x}{L} = f(x)$ and $u_t(x, 0) = \sum B_n\frac{n\pi c}{L}\sin\frac{n\pi x}{L} = g(x)$ . Two conditions, two sets of coefficients: the starting shape fixes the $A_n$ , and the starting velocity fixes the $B_n$ .

Nodes. Mode $n$ has $n - 1$ points inside the string that never move. Touching a guitar string lightly at its midpoint kills the odd modes and you hear the octave (mode 2).

Music. Frequencies $f_1, 2f_1, 3f_1, \dots$ are harmonics, and their mix gives an instrument its timbre. Pitch rises with tension and falls with length and density, $f_1 = \frac{1}{2L}\sqrt{T/\rho}$ .

There's no decay in this model; real strings lose energy to the air and the instrument's body.

See it

t

0.00

A string plucked at its midpoint, with c = L = 1. Drag t and watch the kink: it splits into two corners that run out to the ends, flip upside down when they bounce off, and come back. At t = 1 the string is the starting triangle turned upside down, and at t = 2 it's right back where it began.

Worked examples

Example 1

A guitar string is 0.65 m long with wave speed 286 m/s. Find its fundamental frequency.

1
The fundamental is the frequency of mode 1, so $f_1 = \frac{c}{2L} = \frac{286}{1.3}$ .

$220$ Hz (the note A3)

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

$L = 0.65$ m, $c = 286$ m/s. Find the fundamental frequency in Hz.

Problem 2+5 XP

What is the frequency of the third harmonic?

Problem 3+5 XP

Solve $u_{tt} = 4u_{xx}$ on $[0, 1]$ , fixed ends, $u(x, 0) = \sin 2\pi x$ , $u_t(x, 0) = 0$ .

Problem 4+5 XP

Solve $u_{tt} = u_{xx}$ on $[0, 1]$ , fixed ends, $u(x, 0) = 0$ , $u_t(x, 0) = \sin\pi x$ .

Problem 5+5 XP

To raise a string's pitch by an octave (double the frequency) using tension alone, multiply the tension by:

Problem 6+5 XP

A violin's A string has a vibrating length of 0.33 m and is tuned to 440 Hz. What is the wave speed on the string, in m/s?

Problem 7+5 XP

Pressing a string against a fret shortens its vibrating length. To raise the pitch by a perfect fifth (frequency × 3/2), what fraction of the string's length must be cut off?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A string is plucked at its exact middle, so its initial shape is a triangle, with coefficients $B_n = \frac{8h}{n^2\pi^2}\sin\frac{n\pi}{2}$ . How does the amplitude of the third harmonic compare with the fundamental's (the ratio $|B_3|/|B_1|$ )?

Where it's used

Instrument makers tune strings by tension and length; engineers check bridges and cables for resonance with the same mode analysis.

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The Vibrating String

Main idea

u(x, t) = \sum_{n=1}^{\infty}\left({\htmlData{sym=0}{A_n}}\cos\frac{n\pi ct}{L} + {\htmlData{sym=1}{B_n}}\sin\frac{n\pi ct}{L}\right)\sin\frac{n\pi x}{L}

What the symbols mean

$A_n$
Shape coefficients
The sine coefficients of the initial shape f.
$B_n$
Velocity coefficients
From the initial velocity g: B_n·(nπc/L) are g's sine coefficients.

{\htmlData{sym=0}{f_n}} = \frac{nc}{2L}

What the symbols mean

$f_n$
Frequency of mode n
In vibrations per second: n times the fundamental c/(2L).

Explanation

Nodes. Mode $n$ has $n - 1$ points inside the string that never move. Touching a guitar string lightly at its midpoint kills the odd modes and you hear the octave (mode 2).

There's no decay in this model; real strings lose energy to the air and the instrument's body.

See it

t

0.00

Worked examples

Example 1

A guitar string is 0.65 m long with wave speed 286 m/s. Find its fundamental frequency.

1
The fundamental is the frequency of mode 1, so $f_1 = \frac{c}{2L} = \frac{286}{1.3}$ .

$220$ Hz (the note A3)

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

$L = 0.65$ m, $c = 286$ m/s. Find the fundamental frequency in Hz.

Problem 2+5 XP

What is the frequency of the third harmonic?

Problem 3+5 XP

Solve $u_{tt} = 4u_{xx}$ on $[0, 1]$ , fixed ends, $u(x, 0) = \sin 2\pi x$ , $u_t(x, 0) = 0$ .

Problem 4+5 XP

Solve $u_{tt} = u_{xx}$ on $[0, 1]$ , fixed ends, $u(x, 0) = 0$ , $u_t(x, 0) = \sin\pi x$ .

Problem 5+5 XP

To raise a string's pitch by an octave (double the frequency) using tension alone, multiply the tension by:

Problem 6+5 XP

A violin's A string has a vibrating length of 0.33 m and is tuned to 440 Hz. What is the wave speed on the string, in m/s?

Problem 7+5 XP

Pressing a string against a fret shortens its vibrating length. To raise the pitch by a perfect fifth (frequency × 3/2), what fraction of the string's length must be cut off?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.