Recipe

Find the modes of a circular drumhead

Use when: you have $u_{tt}=c^2\nabla^2u$ on a disk of radius $a$ with $u(a,\theta,t)=0$

Goal: the natural frequency $\omega_{m,n}$ of any mode, and (if needed) that mode's own coefficient from the initial data

Taught in The vibrating circular drumhead

Quick recipe

Identify the mode's two integer labels, $m$ and $n$
Look up the matching Bessel zero $j_{m,n}$
Compute $\omega_{m,n}=cj_{m,n}/a$
If you need a mode's own coefficient, apply BOTH orthogonality relations

1
$m$ counts the angular dependence: how many full diameters split the mode. $n$ counts the radial dependence: which zero of $J_m$ is being used.

Example

The lowest, purely radially symmetric mode has $m=0$ , $n=1$ .

Why: a circular membrane's modes are indexed by two integers. The domain has two genuinely independent directions: angular and radial.
2
Find the $n$ -th positive zero of $J_m$ . Use the earlier section's table, or a numerical method, for that particular $m$ .

Example

$j_{0,1}\approx2.405$ . A later mode like $(m,n)=(1,2)$ uses $j_{1,2}\approx7.016$ .

Why: the clamped rim $R(a)=0$ forces $k_{m,n}a$ to land exactly on a zero of $J_m$ . So $j_{m,n}$ is what the boundary condition actually selects.
3
Divide the Bessel zero by the radius $a$ to get $k_{m,n}$ . Then multiply by the wave speed $c$ to get the natural frequency.

Example

For $a=1,c=1,(m,n)=(0,1)$ : $\omega_{0,1}=1\times2.405/1\approx2.405$ .

Why: this matches the 1-D and rectangular cases. An eigenvalue only feeds the spatial equation. The second-order time equation turns it into a frequency, through this same $c\sqrt\lambda$ pattern.
4
Check whether the initial shape is already a sum of pure modes. Or is it a genuinely arbitrary function of $r$ and $\theta$ ?

Example

$f(r,\theta)=3J_0(j_{0,1}r)+2J_1(j_{1,2}r)\sin\theta$ is already a sum of two pure modes. So $A_{0,1}=3$ and $B_{1,2}=2$ directly. Every other coefficient is zero.
- the initial shape is already a sum of pure modes — read each mode's coefficient directly off its own scale factor. Every other coefficient is zero.
- the initial shape is a general function — extract the angular dependence first, with the ordinary Fourier integral. Then extract the radial dependence, with the Bessel-zero orthogonality integral from earlier in this chapter.
Why: one orthogonality relation only ever isolates one of the two independent directions. A genuinely 2-D domain needs both.

Common mistakes

Reporting $j_{m,n}$ or $k_{m,n}$ itself as the frequency, instead of $\omega_{m,n}=cj_{m,n}/a$ .
Using only the angular Fourier integral and treating the result as the whole coefficient. That leaves the radial dependence unresolved.
Expecting the drumhead's overtones to be integer multiples of the fundamental, the way a string's are. Bessel zeros aren't evenly spaced.
Mixing up $m$ (angular) and $n$ (radial) when looking up the Bessel zero table.