BrainFlux

Drums and billiards

Click the drum to strike it there (you hear it too); click a peak of the spectrum to see that mode

Spectrum of the motion

Click a peak: the mode behind it appears below

Mode

Shape

Draw your own shape

What a click on the picture does:

The modes are recomputed when you lift the pen.

Display and grid

The membrane lives on the grid points: more points follow the shape and short waves better, but every step and the eigensolver take longer.

Experiments

Start here

A drum. Strike it and it rings in its modes, each a fixed pattern swinging at its own frequency; the picture is the membrane (red up, cyan down), slowed down a few hundred times. The spectrum beside it is built from the motion as it goes on, and grows a peak at every mode.

Click the drum to strike it somewhere else (you hear it too). Click a peak of the spectrum and the mode behind it appears below, with its nodal lines. Choose another shape on the right.

Look for the peaks sharpening as the run goes on, for where you strike deciding which peaks ring, and for the lowest mode moving all together while the higher ones are cut up by nodal lines. The tab above the picture reads the same shape as a quantum billiard: drag on it to launch a wave packet.

1. A struck drum: its motion is a sum of modes

A round drum, 29 cm across, struck a little off centre and slowed down about 170 times. The ripple spreads from the mallet, reflects off the rim and settles into a slosh that never quite repeats: a sum of modes $\phi_k(x,y)$, each swinging at its own frequency $f_k$. The strip above the spectrum records the membrane under the mallet; the spectrum is its Fourier transform, and grows a peak at every $f_k$ in the motion. The green ticks are the frequencies a direct eigensolver finds for this drum: each peak sits on one. Click the lowest peak, near 266 Hz: the fundamental, the whole membrane up and down together. Click the next, near 424 Hz: one half up while the other is down, a nodal line across the middle.

2. Where you strike it changes the sound

The same drum struck dead centre. It rings only in its modes with $m=0$, the ones that are round (every other mode has a nodal line through the centre, so a mallet there cannot move it): few peaks, a hollow sound. Now click near the edge: many more modes, a brighter, more rattling sound. Click on a nodal line of a mode (see the Mode panel) and that mode drops out of the spectrum.

3. The square: modes $(m,n)$ and degeneracy

A square drum of side $a$, with its mode 2 shown. Its modes are $\phi=\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{a}$ with $\lambda=\frac{\pi^2}{a^2}(m^2+n^2)$, and the list below the spectrum shows each mode's $(m,n)$. Modes 2 and 3, $(1,2)$ and $(2,1)$, have the same frequency (they are degenerate): any combination of them is a mode too, so the curved nodal line of mode 2 is just the solver's choice. Click that peak in the spectrum: the motion makes its own combination, set by where the drum was struck. Some numbers are sums of two squares in two ways, $5^2+5^2=1^2+7^2$: then three modes share a frequency (on the grid they split slightly: the grid's error depends on direction).

4. The circle: zeros of Bessel functions

The round drum with its fundamental shown. A circle of radius $R$ has the modes $\phi=J_m(j_{m,k}r/R)\cos m\theta$ (and $\sin m\theta$), with $\lambda=(j_{m,k}/R)^2$, where $j_{m,k}$ is the $k$-th zero of the Bessel function $J_m$; $m$ counts the nodal diameters, $k$ the nodal circles (the rim included). The list compares the grid's eigenvalues with these. The grid cannot draw a circle: its edge is a staircase, which costs a fraction of a percent here, less on a finer grid. The overtones are not whole multiples of the fundamental: $j_{1,1}/j_{0,1}=1.59$, $j_{2,1}/j_{0,1}=2.14$. That is why a drum, unlike a string, has no clear pitch.

5. Nodal lines

The square with mode 21 shown as sand. Chladni scattered sand on vibrating plates; it collects where the plate stays still, on the nodal lines. Click modes in the list (or peaks) and look at the patterns. In a square they form grids and, for degenerate pairs, curves. Try a stadium or a drawn shape: the lines wander and never cross, a sign that nothing is separable there. (Chladni's plates obey a different, fourth-order equation; the membrane's lines are the same idea.)

6. Longer runs, sharper peaks

The round drum struck off centre, from the start. The bar at the top of the spectrum is how wide a peak is for the length of the run so far; it shrinks as $1/T$. A run of length $T$ cannot tell apart frequencies closer than about $1/T$ (in hertz; $2\pi/T$ in angular frequency): here the window makes a peak $\Delta f=1.9/T$ wide, 73 Hz after 26 ms. Two modes closer than that make a single peak until the run is long enough. Press Fast and watch the peaks narrow onto the green ticks. For the billiard, with $E=\hbar\omega$, the same is $\Delta E=1.9\times2\pi\hbar/T$: the time-energy uncertainty relation $\Delta E\,T\sim h$, which is a property of Fourier transforms, true of drums too.

7. The same method on a quantum billiard

A stadium read as a quantum billiard (a particle in a box of that shape), started in a random wave function and run fast. Its stationary states are the drum's modes, with energies $E_k=\hbar^2\lambda_k/2m$ in place of frequencies ($\hbar=m=1$ here), and the recorded signal is now the overlap of the state with its start, $\langle\psi(0)|\psi(t)\rangle$, which drops at once and then flickers: the overlap never comes back. Watch the spectrum: blurred bumps at first, then separate peaks, each sitting on a green tick. The peaks are the stadium's energy levels, read off from the motion alone, the way a spectroscopist reads a molecule's levels from the light it emits.

8. Build a mode from the motion

The same random state in the stadium, with a peak already clicked. The Mode panel shows $$\phi_E\propto\int\psi(t)\,e^{iEt}\,w(t)\,dt,$$ accumulating as the run goes on: every mode except those at $E$ turns against $e^{iEt}$ and averages away, and the window $w(t)$ makes them die off as $1/(T\Delta E)^3$. The note says how much of what was built is the exact eigenmode of the direct solver; it climbs to 99.9% and beyond. A peak made of two close modes gives a mixture until the run is long enough. Click any other peak to build that one instead.

9. Weyl's law: you can hear the area

An eighth of the Sinai billiard, with its lowest 200 modes counted. Count the modes below $\lambda$. Weyl (1911) proved that $$N(\lambda)\approx\frac{A}{4\pi}\lambda-\frac{P}{4\pi}\sqrt\lambda+K$$ for any drum of area $A$ and perimeter $P$ ($K$ comes from the corners and the curvature). So the spectrum knows the area, and the perimeter too. Press "Take away the area term": what is left of the staircase, plotted against $\sqrt\lambda$, falls along a straight line of slope $-P/4\pi$, wobbling about it. The panel fits the staircase and reports the area and the perimeter it "hears". (The grid lowers short waves' eigenvalues a little, by $\lambda^2h^2/16$ on average over directions; the curves and the fit allow for that.)

10. Can one hear the shape of a drum?

Kac asked in 1966. No: Gordon, Webb and Wolpert (1992) found two different drums with the same eigenvalues. Each is seven right isosceles triangles, glued in two different ways. The first drum's eigenvalues are pinned (red ticks at the top of the spectrum); the second's are green, and they match. On this grid they agree to $10^{-14}$, because the grid lines and diagonals run along every edge of the triangles. Under "More" in the drum panel, press "Play the pinned one", then "Play its spectrum": the same sound. Strike them at a point, though, and they differ: a strike excites the modes by different amounts in the two drums.

11. A regular drum: spacings like random numbers

A rectangle whose sides have a ratio far from a simple fraction, with 200 modes counted. Its eigenvalues $\pi^2(m^2/a^2+n^2/b^2)$ come from two independent motions, so they fall like independent random numbers: the spacings between neighbours (in units of the mean spacing) follow $P(s)=e^{-s}$, and small gaps are the most common.

12. A chaotic drum: level repulsion

A quarter of Bunimovich's stadium (a rectangle with round ends): a classical ball in it is chaotic. Now the spacings avoid zero: neighbouring levels repel, and $P(s)$ is close to Wigner's $\frac{\pi s}{2}e^{-\pi s^2/4}$, the law of random symmetric matrices (Bohigas, Giannoni and Schmit, 1984, with the Sinai billiard; try "Sinai, ⅛"). Why a quarter: a symmetric shape's spectrum is a mixture of independent families of modes (even or odd about each axis), and modes of different families do not repel. Count the whole stadium: about 14% of its spacings fall below ¼, twice as many as for the quarter. (A rectangle with a simple ratio of sides, or the circle, has exactly repeated eigenvalues: spacings of zero.)

13. A packet in a square comes back; in a stadium it does not

A wave packet launched across a square billiard, with a classical particle (the yellow dot) launched the same way. The square's energies $\frac{\pi^2}{2}(m^2+n^2)$ are all whole multiples of $\pi^2/2$, so after $T_\text{rev}=4/\pi=1.27$ every phase is back where it started and the packet reassembles. Press Fast and watch $|\langle\psi(0)|\psi(t)\rangle|$: it falls to nearly nothing and returns to 0.87 at $t=1.28$ (on this grid short waves have slightly lower energies, which delays and blurs the revival). At $T_\text{rev}/2$ it reforms on the other side, reflected through the centre: every mode has turned by $(-1)^{m+n}$. The classical particle's path keeps to a few directions. Now choose Stadium: the particle's path soon covers the whole stadium, the packet spreads into a speckle of random-looking waves, and nothing comes back.

14. Scars

A packet launched along the stadium's long axis, an orbit that is unstable: nearby paths fly off. Yet the signal comes back each time the packet returns, every $T=2L/v\approx0.065$, weaker and weaker; so the spectrum's peaks gather in bumps about $2\pi\hbar/T\approx100$ apart. Click the tallest peaks, or the exact modes under them: many of those modes are concentrated along the axis, a "scar" of the unstable orbit (Heller, 1984), where a chaotic mode should spread evenly. Measured: the dozen modes this packet excites most keep about a third of their intensity within 0.04 of the axis, a band that is a fifth of the stadium; a typical mode of the same symmetry keeps a quarter there.

15. Draw your own drum

The round drum with the pen ready: paint on the picture to add membrane (Erase, in the "Draw your own shape" panel, adds wall; Clear starts from nothing). The exact modes are recomputed when you lift the pen. Try a dumbbell, two equal rooms joined by a narrow corridor: the low modes come in close pairs, one even and one odd, and a packet takes a long time to leak from one room to the other. Or type a shape: Typed takes any condition on $x$, $y$, $r$ and $\theta$, and every new letter in it gets a slider.

How it works

One eigenvalue problem, two kinds of physics. Inside a shape, with zero on its edge, $$-\nabla^2\phi_k=\lambda_k\phi_k .$$ A drum (a membrane, $u_{tt}=c^2\nabla^2u$) vibrates in these modes at the frequencies $\omega_k=c\sqrt{\lambda_k}$; a quantum billiard (a particle in a box with that shape, $i\hbar\psi_t=-\frac{\hbar^2}{2m}\nabla^2\psi$) has them as its stationary states, with energies $E_k=\hbar^2\lambda_k/2m$. Here $\hbar=m=1$ and the box is 1 wide; for the sound, the box is 30 cm wide and $c$ is a slider.

The demo finds the $\lambda_k$ the way a spectroscopist would (Feit, Fleck and Steiger, 1982): set the system going, record a signal, and Fourier-transform it. Every mode in the motion turns at its own frequency, so the transform has a peak at each one. For the billiard the signal is the autocorrelation $\langle\psi(0)|\psi(t)\rangle=\sum_k|c_k|^2e^{-iE_kt}$, the overlap of the state with its start; for the drum it is the displacement under the mallet. The green ticks are the same eigenvalues found directly (shift-invert Lanczos on the grid, an eigensolver for the lowest eigenvalues of a large sparse matrix), to check the peaks.

The shape lives on a $128\times128$ grid (h = 1/127), with the 5-point Laplacian. A grid point is free if it lies more than $h/2$ inside the shape: straight edges along grid lines or diagonals are exact, curved edges are staircases. The drum uses the leapfrog scheme, the billiard Visscher's staggered scheme for $\mathrm{Re}\,\psi$ and $\mathrm{Im}\,\psi$; both conserve an energy exactly (the "drift" in the readouts is how much it has changed, a check on the arithmetic) and turn each mode at a known rate, which is undone when the peaks are read off, so the peaks match the direct solver to a small fraction of their width. The direct solver runs in a background thread with an envelope Cholesky factor. The sound adds up the modes, each as loud as the strike makes it, each ringing for the same number of cycles.

New in 2026 for BrainFlux. Source: github.com/gallicchio/brainflux, MIT licence.