Vibrating Drums and Bessel Functions

A Bessel function is simply the shape a round drumhead takes as you go out from its center to its rim, a wave that slowly fades, the way a sine is the shape of a vibrating string. They're named after Friedrich Bessel, an astronomer who studied them while working on planetary orbits, long before anyone used them on drums. They're good for anything round that vibrates or spreads: drums, speaker cones, heat in a pipe. All we're really saying is that a drumhead shakes in a handful of basic patterns, each with its own pitch, and the sound is a mix of them. The mode formula $u_{mn} = J_m\!\left(\frac{j_{mn}r}{R}\right)\cos(m\theta)\cos\left(\frac{c\,j_{mn}}{R}\,t\right)$ looks frightening, but it's three simple pieces multiplied together: $J_m(\cdot)$ (the shape going out from the center to the rim), $\cos(m\theta)$ (how the shape changes as you go around the drum) and $\cos(\ldots t)$ (the up-and-down shaking in time). With $j_{mn}$ (where the shape hits zero, chosen so the rim stays still), it's just the string's $\sin\frac{n\pi x}{L}\cos\frac{n\pi ct}{L}$ in round clothing, so bear with it.

Separation. Think back to the vibrating string: $\sin\frac{n\pi x}{L}$ was forced on us because the ends couldn't move, and $\frac{n\pi}{L}$ were the places where sine hits zero. A drum works the same way, with the rim playing the role of the ends. Write $u = R(r)\Theta(\theta)T(t)$. Going once around the drum brings you back to the same spot, so periodicity forces $\Theta = \cos m\theta$ or $\sin m\theta$ with integer $m$, just like on a circular ring. The radial equation

is Bessel's equation. It looks like the Euler equation from the disk, except for the extra $\lambda r^2$, and that one term is what makes its solutions oscillate. The solutions bounded at the center are $J_m(\sqrt\lambda\,r)$, and the fixed rim requires $\sqrt\lambda R$ to be a zero of $J_m$. So Bessel zeros do exactly the job that $n\pi$ did for the string.

Frequencies. Mode $(m, n)$ vibrates at $\omega_{mn} = \frac{c\,j_{mn}}{R}$. Relative to the fundamental:

Notice these aren't whole numbers, unlike a string's $1, 2, 3, \dots$. Your ear hears whole-number ratios as one clear note, so drum overtones clash a little and the pitch gets muddy. Timpani makers shape the kettle and air load to pull the main modes closer to harmonic ratios.

Nodal lines. Mode $(m, n)$ has $m$ nodal diameters and $n - 1$ nodal circles inside the rim. The diameters come from $\cos m\theta$ hitting zero as you go around, and the circles come from $J_m$ hitting zero on the way out, at the earlier Bessel zeros. It's the drum version of a string's nodes, the points of a standing wave that never move.

A common mistake. It's tempting to think the Bessel zeros are evenly spaced like the zeros of sine, because $J_0$ looks like a fading cosine. They're close, and the gaps settle toward $\pi$ further out, but not at first and never starting from 0. That small offset is exactly what keeps the frequency ratios from landing on whole numbers.

Drum and cymbal design, loudspeaker cones, microphone diaphragms, and the vibration of circular plates in engines and turbines are Bessel-mode problems.