A circular membrane of radius $R$, fixed at its boundary, obeys the two-dimensional wave equation \[ \frac{\partial^2 z}{\partial t^2} \;=\; c^2 \nabla^2 z \;=\; c^2 \!\left(\frac{\partial^2 z}{\partial r^2} + \frac{1}{r}\frac{\partial z}{\partial r} + \frac{1}{r^2}\frac{\partial^2 z}{\partial \theta^2}\right) \] where $z(r,\theta,t)$ is the transverse displacement and $c$ is the wave speed in the membrane.
Separation of variables $z = \mathcal{R}(r)\,\Theta(\theta)\,T(t)$ splits this into three ordinary equations. The angular part gives $\Theta(\theta) = \cos(m\theta)$ with $m = 0, 1, 2, \ldots$, and the temporal part gives $T(t) = \cos(\omega\, t)$. The radial part becomes Bessel's equation \[ r^2 \mathcal{R}'' + r\,\mathcal{R}' + \left(\frac{\omega^2}{c^2}\, r^2 - m^2\right)\mathcal{R} = 0 \] whose physically admissible solutions are the Bessel functions of the first kind, $\mathcal{R}(r) = J_m\!\!\left(\frac{\omega}{c}\, r\right)$.
The boundary condition $z(R, \theta, t) = 0$ requires $J_m\!\left(\frac{\omega R}{c}\right) = 0$. Denoting the $k$-th positive zero of $J_m$ by $j_{m,k}$, the eigenfrequencies are \[ \omega_{m,k} \;=\; \frac{j_{m,k}\; c}{R} \] and the corresponding mode shapes are \[ z_{m,k}(r,\theta,t) \;=\; J_m\!\left(\frac{j_{m,k}\, r}{R}\right) \cos(m\theta)\;\cos(\omega_{m,k}\, t). \] The integer $m$ counts the number of nodal diameters (angular nodes) and $k$ counts the number of nodal circles (radial nodes, excluding the fixed boundary). Drag to orbit; use the arrow keys or the +/− buttons to zoom.