🥁 Drum Membrane Vibration
A circular drum membrane satisfies the 2-D wave equation ∂²u/∂t² = c²∇²u with Dirichlet boundary conditions u = 0 on the rim. The solutions are product eigenmodes Jm(kmnr)·cos(mθ)·cos(ωmnt), where Jm is a Bessel function of order m and kmn is its n-th zero. Select modes with buttons (m = angular, n = radial node count) or superpose several at once to form complex nodal patterns. Click the membrane to excite it at a point. Nodal lines (yellow) show where the membrane stays still — these are the Chladni patterns of a drum.
Eigenmode (m, n)
m = angular nodes, n = radial nodes
Superposition
Parameters
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∂²u/∂t² = c²(∂²u/∂r² + (1/r)∂u/∂r + (1/r²)∂²u/∂θ²)
Eigenfrequencies:
ωmn = c·jmn/R
jmn = n-th zero of Jm
Why Drums Are Not Harmonic
A vibrating string has harmonics f, 2f, 3f, … because its zeros of the eigenfunction (sin(nπx/L)) are evenly spaced. A circular membrane's eigenfrequencies are proportional to the zeros of Bessel functions: j01=2.405, j11=3.832, j21=5.136, … These are not integer multiples of the fundamental, making drum tones percussive and somewhat inharmonic. This is why drums have a less definite pitch than strings or organ pipes. Ernst Chladni (1756–1827) demonstrated experimentally that sand sprinkled on a vibrating plate collects on nodal lines — the boundaries of destructive interference where the plate doesn't move.