Eigendrum: a client-side FEM solver that plays the modes of any drum shape

Eigendrum is a browser tool that computes and sonifies the vibration modes of a clamped membrane of arbitrary shape. It treats the physics as the Helmholtz eigenvalue problem (−∇²u = λu inside the region, u = 0 on the boundary), which has no closed form for most shapes, so it meshes the region into triangles, assembles finite-element stiffness and mass matrices, and extracts the smallest eigenvalues of Kφ = λMφ, with audible frequency scaling as √λ. Shapes can be traced by hand or written as equations—polar r(t) or parametric x(t)/y(t)—reaching curves like nephroids, superellipses, and many-lobed flowers that freehand drawing can’t produce accurately.

The author leans hard on verifiability. The solver is regression-tested on every change against the few shapes with exact spectra—Bessel-function zeros for the circle, π²(m²/a² + n²/b²) for the rectangle—matching to better than a tenth of a percent, and because conforming FEM minimizes energy over a restricted space, its results are provably slight overestimates rather than random error. The interaction models real acoustics too: strike location determines which modes are excited (via projection of the mallet onto each mode, so nodal lines stay silent), mallet width and contact time are tunable, and Rayleigh damping makes high overtones decay first. It even ships the Gordon–Webb–Wolpert isospectral drums as a playable answer to Kac’s 1966 question—two visibly different shapes that sound identical.

The engineering story is the notable part for a technical audience: no build step and no backend—meshing, solving, and audio all run locally in the browser. The drawn shape lives in the URL fragment after the ’#’, which browsers never transmit, and analytics is configured not to capture it, so shared links stay client-side by construction. The full solver and its verification tests are open source on GitHub, with the hosted site monetized through Google/Vercel analytics and AdSense.