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Vibrating Drumhead Modes

Strike a circular drum and it does not ring on a single clean pitch the way a string does, and the reason is geometry. Solving the wave equation on a disk, the displacement separates into an angular part $\cos(m\theta)$ and a radial part that turns out to be a Bessel function $J_m(kr)$, the natural standing wave of a round membrane. Clamping the rim forces $J_m(ka)=0$, so the only allowed wavenumbers are $k_{mn} = j_{m,n}/a$ where $j_{m,n}$ is the $n$-th zero of $J_m$, and the frequencies scale with those zeros. The scene paints the mode as it breathes in and out, red rising and blue falling, with the still lines drawn on top: $m$ nodal diameters from the angular factor and $n-1$ nodal circles from the interior zeros of the Bessel function, the same patterns sand grains settle into on a Chladni plate. The bottom panel is that radial profile, and its zeros are exactly the nodal circles you see in the disk, with the last zero pinned to the rim. The catch, and the reason a drum sounds more like a thud than a note, is in the frequencies: the Bessel zeros are not in any simple integer ratio, so the overtones are inharmonic, scattered rather than stacked like a string's $1, 2, 3, \ldots$. Step the angular and radial mode numbers and watch the pattern and its pitch rearrange.

Figure 1. Vibrating drumhead modes. Top: the (m,n) mode displacement on the disk (red up, blue down), oscillating in time, with m nodal diameters and n-1 nodal circles drawn dark. Bottom: the radial profile J_m(k r), whose zeros are the nodal circles, clamped to zero at the rim. Method: Bessel functions and their zeros from series and root-finding; frequencies scale as j_{m,n}. Source: Arfken, Weber, Harris, Mathematical Methods for Physicists, 7th ed.
m = 1
n = 2

WHAT TO TRY

  • Step the angular number $m$: each increment adds a nodal diameter, a still line straight across the drum.
  • Step the radial number $n$: each increment adds a nodal circle, a still ring, matching a new zero in the radial profile below.
  • Read the frequency ratio: the Bessel zeros are not integer multiples, so the overtones are inharmonic, the reason a drum has no clear pitch.
  • Watch the rim: every mode is clamped to zero there, the boundary condition that selects the allowed frequencies.