ApliantWhy two patterns for one pair (n, m)?
On a square, the modes (n, m) and (m, n) have the same frequency. When two modes share a frequency, any mix of them is also a mode. The plate picks two special mixes: the sum (+) and the difference (−). On a real plate the two mixes have slightly different frequencies, so you can find each one. Here we split them by ±3.5% to copy this.
Look at (0, 2): the + mix draws a diamond, the − mix draws a cross. Chladni drew both in 1787.
Reading a tile
The bright lines are the nodal lines: the places that do not move, where the sand stays. The numbers count the lines: on the square, n and m are the number of half-waves along each side; on the circle, the first number counts the diameters and the second the circles.
The ratio under each tile is its frequency divided by the lowest one. More lines means a higher note.
Overtones: string, drum, timpani
Each line is one frequency, divided by the lowest one. Gray lines mark the whole numbers 1, 2, 3… where a string puts its overtones.
Modes in this sound
Bessel functions
A round drum's modes are Jn(jn,k r)·cos(nθ). The skin is fixed at the rim (r = 1), so jn,k must be a zero of Jn (the dots). The frequencies are proportional to these zeros.
Check the numbers
This page computes Jn with a backward recurrence (Miller's method) and finds each zero by bisection. Here they are next to published values.
Why does a drum sound less "musical" than a string?
A string's overtones are 2, 3, 4… times its lowest frequency. Our ears fuse them into one clear note. A round drum's overtones are 1.59, 2.14, 2.30, 2.65… times the lowest. They do not line up, so we hear a thud with a vague pitch.
Timpani makers cheat. The air in the kettle and the weight of the air on the skin shift the modes until the ones with nodal diameters sit close to 1 : 1.5 : 2 : 2.5 : 3. Those are the ratios of the harmonics 2, 3, 4, 5, 6 of one note, so a timpani plays a clear note. Press 🎵 to hear this idealized version.
A string tied at both ends
Grab the string with the mouse and let go. Or press "Pluck it". The motion is shown in slow motion.
The harmonics, one by one
The motion above is the sum of these standing waves. Pink dots are nodes (never move). Harmonic n vibrates n times faster than the first.
The recipe: how much of each harmonic
Writing a shape as a sum of sine waves is a Fourier series. Play with it in Fourier Draw.
Where you pluck changes the sound
A harmonic with a node where you pluck cannot start moving. Pluck at the middle and all even harmonics are missing: the sound is soft and hollow. Pluck near the end and many harmonics join: the sound is bright and twangy. Guitarists use this all the time.
A light touch does the opposite: it stops every harmonic that wants to move at that point. Touch at 1/2 and only the even harmonics survive. You hear a note one octave higher: a guitar harmonic. Chladni did the same with his fingers on his plates to force nodal lines where he wanted them.
Draw a drum
Its first 16 modes
From the lowest note to the highest. The number is the frequency divided by the lowest one. Click a mode to see it big.
Can one hear the shape of a drum?
Mark Kac asked this in 1966: if you know every frequency of a drum, do you know its shape? In 1992 Gordon, Webb and Wolpert found the answer: no. Drums A and B below have different shapes but exactly the same frequencies. Each is made of 7 copies of the same triangle.
Check: the square
For a square of side L the exact answer is known: λ = π²(p² + q²)/L². Here is the computer next to it.
How the computer finds the modes
The shape is cut into small triangles (4 per grid square). On each triangle the height of the skin is a flat piece. Then the equation −Δu = λu becomes a big matrix problem Au = λu, with one unknown per corner inside the shape (thousands of them).
We want the smallest λ. The trick: factor A once (Cholesky), then solve with it again and again. Each solve makes the low modes stand out more (inverse iteration, on 24 vectors at once). It runs in the background, in a Web Worker, so the page does not freeze.
The frequency of mode k is proportional to √λk. The modes are the eigenvectors of the Laplacian: see the Eigenvectors lab.
From ripples to modes
This is the wave equation ∂²u/∂t² = c²Δu, solved step by step on a grid of 300 × 180 points. Each point moves toward the average of its neighbors.
In a closed box, waves bounce back and forth. Most frequencies cancel themselves after many bounces. Only some frequencies come back in step with themselves and grow: the modes of the box. That is resonance, and it is what happens on a Chladni plate, much faster.
Sound is vibration
A plate, a drum skin or a string moves back and forth and pushes the air. The number of back-and-forths per second is the frequency, in hertz (Hz). Higher frequency, higher note: 110 Hz is a low A, 440 Hz is the A that orchestras tune to.
Standing waves
A wave runs along a string, hits the end and comes back. The two waves add up. For some frequencies the sum does not travel: it just goes up and down in place. That is a standing wave. Its still points are nodes; the points that move the most are antinodes.
Resonance
Push a swing at the right moment and it goes higher and higher. A plate is the same: shaken at one of its own frequencies it answers strongly; a bit off, it hardly moves. The peaks of the resonance curve are those frequencies. Q says how sharp the peaks are: a bell has a high Q, a pillow a very low one.
Sand finds the nodal lines
Where the plate moves, the sand is thrown up and lands a bit to the side. Where the plate is still, nothing throws it. So each grain wanders until it reaches a still place, and stays. After a few seconds the sand draws the nodal lines. In this page each grain gets a random kick proportional to how much the plate moves under it: |u(x, y)|.
Modes are eigenvectors
For a drum skin, each mode is a shape u that the Laplacian (the "curvature" of the skin) only stretches: −Δu = λu. These u are the eigenvectors of the Laplacian and the λ its eigenvalues. Frequency is proportional to √λ. Any motion of the drum is a mix of its modes. See the Eigenvectors lab.
The model on the Plate tab (an approximation)
A metal plate is stiff: it bends, it does not stretch like a skin. Its exact modes have no simple formula (Walther Ritz computed the first good ones in 1909). We use the usual approximation: the shapes of a free membrane, cos(nπx)·cos(mπy) ± cos(mπx)·cos(nπy) on the square and Jn(j′n,k r)·cos(nθ) on the circle, with the frequency of a stiff plate, proportional to n² + m² (to j′² on the circle). The patterns look like the real ones; the exact frequencies of a real free plate are different.
Strings are harmonic, drums are not
A string's frequencies are 1, 2, 3, 4… times the lowest one. A round drum's are the zeros of Bessel functions, divided by the first: 1, 1.59, 2.14, 2.30, 2.65… The ones without nodal diameters go 1, 2.295, 3.598. Whole-number ratios sound like one note; others sound like a thud.
Can one hear the shape of a drum?
Hermann Weyl showed in 1911 that the frequencies tell you the area of the drum. Later it was shown they also tell the perimeter. Mark Kac asked in 1966 whether they tell everything. Gordon, Webb and Wolpert answered in 1992: no, two different drums can sound exactly the same. You can check it in the "Your drum" tab.
A bit of history
In 1680 Robert Hooke ran a violin bow along a glass plate covered with flour and saw patterns. In 1787 Ernst Chladni published hundreds of them, made with sand on metal plates. In 1808 he showed them to Napoleon in Paris, who paid for a prize: a medal of one kilogram of gold for the mathematical theory of vibrating plates.
Only one person tried seriously: Sophie Germain, who taught herself mathematics. After two attempts she won in 1816, the first woman to win a prize of the Paris Academy of Sciences. Her equation had a flaw, fixed by Gustav Kirchhoff in 1850.
Challenges
- Find the + and the − version of mode (0, 2) on the square. One is a diamond, the other a cross.
- Put the shaker exactly at the center of the square. Which modes disappear from the resonance curve? Why?
- On the circle, shake at the center. Only rings appear. Why?
- Make the rectangle twice as wide as it is tall. Two different modes now share a frequency. Find them.
- Pluck the string at 1/3. Which harmonics are missing? Then touch at 1/2: what is left?
- Listen to the drum and the string with the same lowest note. Which one could play a melody?
- Draw a drum with a narrow neck between two rooms. Where do the lowest modes live?
- In the ripple tank, detune the source in the closed box. Does the standing wave still grow?
About
Science in Motion is created and maintained by Eduardo Correia, sponsored by Apliant.
Shortcuts
Plate: space sweep · ←→ frequency ±1% · ↑↓ next/previous resonance · R scatter the sand · V view. Drum: space strike the center. String: space pluck. Ripple tank: space pause · R calm the water · V view. Everywhere: M sound on/off · H hide/show the menu.