A plucked string or an organ pipe has a clear pitch; a struck drumhead just gives a 'thud'. Why? Are a drum's overtones the harmonic series (1,2,3,4×) like a string's — or something else?
▶ Launch the interactive simulationModel an ideal circular membrane under tension: the 2-D wave equation ∇²u=(1/c²)∂²u/∂t² on a disk clamped at the rim. Separating u=R(r)cos(mθ)e^{iωt} gives the radial ODE R''+(1/r)R'+(k²−m²/r²)R=0 with R(a)=0. Recover the eigenvalues k·a WITHOUT ever evaluating a Bessel function: RK4-integrate the ODE outward from a tiny r₀ (regular series start) and bisect the trial wavenumber k until the rim residual R(a;k) crosses zero for the n-th time. Do this for the first modes (0,1),(1,1),(2,1),(0,2),(3,1),(1,2), check the answer is invariant across four grid discretizations, and run the SAME shooting code on a 1-D string (u''+k²u=0) as a harmonic control.
the Bessel-function zeros α_mn (j_{0,1}=2.4048, j_{1,1}=3.8317, j_{2,1}=5.1356, j_{0,2}=5.5201, …) — recovered to ~1e-11 from the membrane eigenvalue problem, with no Bessel call in the solver (the zeros are loaded only to score). The overtone ratios f_mn/f_01=α_mn/α_01 come out 1 : 1.593 : 2.136 : 2.295 : 2.653 : 2.917 — irrational and NON-integer. The drum's RMS distance from the nearest harmonic (integer) is ≈0.32, versus exactly 0 for the 1-D string control, whose overtones are the perfect series 1,2,3,4. The first overtone is 1.593× the fundamental (a dissonant near-minor-sixth), not the 2× octave a string gives — which is precisely why a drumhead has no definite pitch. Perturbation: change the radius a and wave speed c=√(T/σ); the dimensionless α_mn stay fixed while every physical frequency scales as c/a (a bigger or looser drum sounds lower but keeps the same inharmonic timbre). Rayleigh's 1877 circular-membrane analysis and the physics behind Kac's 1966 'Can one hear the shape of a drum?'