Mode Selection
∇²u = (1/c²)ü (wave eq.)
u_{mn}(r,θ,t) = J_m(k_{mn}r)cos(mθ)cos(ω_{mn}t)
J_m(k_{mn}R) = 0 (Dirichlet BC)
Mode (m,n) = (0,1)
Zero j_{m,n} ≈ 2.405
Frequency ω_{mn} ≈ 2.405 (c/R)
Nodal circles: 0
Nodal diameters: 0
Bessel zeros J_m(j_{m,n})=0. Higher modes produce more complex nodal patterns — stripes and rings like Chladni figures.