Vibrating Drum Membrane
The normal modes of a circular drum membrane are solutions to the 2D wave equation in polar coordinates. Each mode is characterized by Bessel functions, producing beautiful nodal patterns of circles and radial lines where the surface remains still.
Mode Properties
How it works
When a circular drum membrane vibrates, its motion is governed by the 2D wave equation. In polar coordinates, separation of variables yields solutions involving Bessel functions of the first kind, Jm(x). The integer m determines the angular structure (how many nodal diameters), while n determines the radial structure (how many nodal circles, not counting the boundary).
The boundary condition — the membrane is fixed at its rim — requires Jm(jm,n) = 0, where jm,n is the n-th zero of Jm. These zeros determine the allowed vibration frequencies: ωm,n = jm,n c/a, where c is the wave speed and a is the drum radius.
Unlike a vibrating string, the overtone frequencies of a drum are not integer multiples of the fundamental. This is why drums produce less definite pitch than strings. The ratios jm,n/j0,1 give 1.000, 1.593, 2.136, 2.296, 2.653, … — a non-harmonic series.
The nodal lines — where the displacement is always zero — form circles (from radial nodes of Jm) and straight diameters (from the cos(mθ) term). These are the drum’s analog of Chladni figures. If you sprinkle sand on a vibrating drum, it collects along these nodal lines, making the pattern visible.