Vibrating Drumhead

Differential Equations · Axiom Academy

Strike a circular drum and it doesn't ring one clean pitch — it rings a whole family of Bessel-function shapes at once, and none of them are simple multiples of each other. A guitar string's overtones are whole-number multiples of its fundamental — that's what makes a plucked string sound musical. A drumhead is a 2-D membrane fixed around a circle, and its vibration modes are — Bessel functions, not sines. Pick a mode below and watch it ring. Every mode is labeled (m,n) : m nodal diameters (straight lines through the center that never move) and n nodal circles (rings that never move). Select a mode and watch the drum breathe in that exact shape. Bigger drum, lower pitch — by how much? The frequency of any mode is — the SAME from Beat 1, now turned into a pitch. Drag the radius and tension and watch the frequency respond. A drum's overtones don't line up like a string's A string's overtones sit exactly on — whole-number multiples of the fundamental. Toggle the drum's modes on the same number line and watch how far each one misses. A circular membrane's modes are — fixed by the rim into a discrete ladder of Bessel zeros, with frequency . Because those zeros aren't integer multiples of each other, the same math that shapes drums, timpani, and cymbals also shapes speaker cones, circular antenna apertures, and the diffraction pattern behind a round aperture in optics.

This is the written version of the interactive lesson above. See the full Differential Equations course.