how it works

A drumhead clamped at its rim can only vibrate in certain shapes, at certain frequencies. Those shapes and frequencies are the solutions of

−∇²u = λu inside the shape, u = 0 on the edge

Each solution u is a mode, a standing wave, and each λ gives a frequency proportional to √λ. This is an eigenvalue problem, and for almost every shape it has no formula. So Eigendrum solves it numerically: it covers your shape with a mesh of triangles, builds the finite element stiffness and mass matrices, and finds the smallest eigenvalues of Kφ = λMφ .

why you can trust the numbers

A few shapes have spectra that can be written down exactly, and the solver is tested against them on every change. A circle's frequencies are the zeros of Bessel functions; a rectangle's are π²(m²/a² + n²/b²) . The solver reproduces both to better than a tenth of a percent, and because a conforming finite element method minimises energy over a restricted space, its answers are guaranteed slight over estimates, never under.…