GoKawiilhow 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 overestimates, never under. The measured error is in “the numbers”.
where you strike it matters
Striking a spot drives each mode in proportion to how much that mode moves there. Hit a line where a mode stands still and you cannot excite it at all. That was not programmed in; it falls out of projecting the mallet onto the modes.
So a strike is never one mode: it is every mode at once, in a mixture set by where your mallet landed. The rules along the mode list are that mixture, and the modes marked with a square were the ones your mallet could not reach. Pressing a row instead plays that single mode alone - something no mallet can do, and the only way to hear what one frequency of a shape actually sounds like.
drums from equations
... continue reading