# Vibrating Drum

> A circular drumhead with a fixed rim, vibrating in three normal modes at once: Bessel functions in motion.

When you strike a drum, its skin vibrates as a combination of normal modes: shapes that keep their form and only change in amplitude as they swing up and down. For a circular membrane with a fixed rim, each mode is a Bessel function in the radius times a cosine in the angle: $J_m(k r)\cos(m\theta)$. The rim stays still only when $k$ is a zero of $J_m$, and each mode then vibrates at the frequency $k$.

This document adds three modes: the fundamental ($m = 0$, $k = 2.4048$), where the whole skin moves up and down together, and the first modes with one and two nodal lines across the drum ($m = 1$, $k = 3.8317$ and $m = 2$, $k = 5.1356$). The colors show the displacement at each instant: magenta is up, blue is down, and white is level with the rim. A white point is crossing its rest height, not standing still; only the rim never moves. The amplitudes $a_1$, $a_2$, $a_3$ are editable: set two of them to $0$ to watch one mode alone.

The frequencies are in the ratios $1 : 1.593 : 2.136$, which are not whole numbers, so the shape never repeats. That is why a drum has a less definite pitch than a guitar string, whose overtones are exact multiples of its fundamental.

After a Desmos graph by MYoung, which shows a membrane stretched between two moving rings.

- [Open the interactive version](https://graph-paper.io/showcase/en/vibrating-drum)
- [Learn more on Wikipedia](https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane)