Hayterwave tools

Drum Modes

Every way an ideal round drum skin can vibrate, drawn as a heatmap of displacement, with each mode's pitch as a ratio to the lowest one.

Ideal membrane maths, not a recording. This is a uniform, perfectly flexible, undamped circular membrane clamped at its rim. Real drums have air loading, stiffness, a shell and tuning, so their partials sit elsewhere. Nothing here was measured from any instrument.
Mode (m, n)
(0, 1)
Frequency ratio to fundamental
1.000

0%

All modes

Pick one. Arrow keys and Tab move between them.

Table

Mode (m, n)Bessel zeroRatioNodal diametersNodal circles

How it is computed

The displacement of mode (m, n) is u(r, a) = J_m(j_mn r) cos(m a) for radius r from 0 to 1. The rim is fixed, so j_mn must be a zero of the Bessel function J_m. The frequency is proportional to j_mn, so the ratio is j_mn divided by j_01 = 2.4048.

J_m(x) is evaluated here from its power series, the sum over k of (-1)^k (x/2)^(2k+m) / (k! (k+m)!), summed until the terms fall under 1e-17. That is accurate to about 1e-11 for x up to 15, which covers every mode shown. Zeros are found by stepping along x in 0.05 increments to catch each sign change, then bisecting to 1e-13. No library and no table is used.

Open this page with ?selftest=1 to check the zeros against published values (2.4048, 3.8317, 5.1356, 5.5201) and the ratios (1.000, 1.593, 2.136, 2.295).