The Dirichlet Laplacian −Δ on a bounded planar domain Ω has a discrete spectrum 0<λ1<λ2≤λ3≤… with eigenfunctions ϕk vanishing on ∂Ω: −Δϕk=λkϕkin Ω,ϕk=0on ∂Ω.
Physically these are the pure tones of a drumhead clamped along its rim. On the unit disk the eigenproblem separates in polar coordinates, and the eigenfunctions are exactly ϕn,m(r,θ)=Jn(λn,mr)cos(nθ),
where Jn is the Bessel function of the first kind of order n and λn,m is its m-th positive zero. The corresponding eigenvalue is λn,m2 and the time evolution from the wave equation utt=c2Δu is cos(ωn,mt) with ωn,m∝λn,m, so higher modes vibrate faster — that is the physics of drum overtones. The animation cycles through the first 16 modes ordered by λ; red is positive deflection, blue negative, and the cream curves are the nodal lines {ϕn,m=0} (Courant's bound: at most k nodal domains for the k-th eigenfunction). Mark Kac asked in 1966 *can one hear the shape of a drum?* — the answer turned out to be no in general (Gordon–Webb–Wolpert, 1992), but the spectrum still encodes a great deal: λ1 scales as 1/area (Faber–Krahn), Weyl's law recovers area and perimeter from the counting function. Inspired by Gabriel Peyré's spectral-geometry visualizations.