Hear the Shape of a Drum Spectral geometry — how a membrane's shape sets its vibration modes and its sound

Mark Kac (1966) asked: “Can one hear the shape of a drum?” Pick a shape, see its eigenmodes and nodal lines (the Chladni figures), and hear why drums sound un-pitched.

− displacement + displacement

Dark curves are nodal lines — the membrane stays still there.

Drum shape

Circle: mode (m,n) = J_m(k r)·cos(mθ), frequency ∝ the n-th zero of J_m.

Pick a mode (m, n)

Rows = m (angular), columns = n (radial). Numbers are the frequency ratio to the fundamental.

Selected mode

Mode (0, 1) — frequency ratio

1.00×

= 220.0 Hz demo pitch

Cymatics particles

The bars are the lowest modes' frequencies. They are not integer multiples — the drum spectrum is inharmonic, so a struck drum has no clear pitch.

The competition-level math read more

The eigenvalue problem

A membrane's height u(x,y,t) obeys the 2-D wave equation u_tt = c² ∇² u, with u = 0 on the clamped boundary. Separating u = φ(x,y) cos(ω t) turns it into the Helmholtz eigenvalue problem:

-∇² φ = λ φ, φ|_boundary = 0, ω = c √λ

Each eigenfunction φ is a mode; its eigenvalue λ fixes the frequency. The set of all λ is the drum's spectrum — literally what Kac meant by “hearing” it.

Rectangle a × b

Separation gives φ_mn = sin(mπx/a) sin(nπy/b), so λ_mn = π²[(m/a)² + (n/b)²] and therefore

f_mn ∝ √( (m/a)² + (n/b)² )

The square-root-of-a-sum-of-squares is why the ratios are irrational and change as you drag the aspect slider. Some ratios can even collide (degenerate modes) at special aspect ratios.

Circle radius R

In polar coordinates the radial factor solves Bessel's equation, giving φ_mn = J_m(k_mn r) cos(mθ). The clamp J_m(k R) = 0 forces k_mn R = j_mn, the n-th positive zero of J_m:

f_mn ∝ j_mn

The fundamental is j_0,1 ≈ 2.4048. The next modes sit at ratios 1.594, 2.136, 2.296, 2.653, ... — nowhere near 2, 3, 4. That is why a drum is inharmonic, unlike a string whose modes are at exact integers 1, 2, 3, ...

Answer to Kac's question: NO (in general)

In 1992 Gordon, Webb & Wolpert built two differently-shaped polygonal drums with the identical spectrum — isospectral but not congruent. So you cannot always recover the shape from the sound. You can hear some things (area and perimeter, via Weyl's law and heat-kernel asymptotics), just not the full shape.