Natural Modes Explorer

Amplitude = |φ_n(x)| — the eigenfunction value at the strike point. Striking a node of mode n gives zero excitation of that mode.

Decay τ_n = τ₁ / ratio_n² — internal (thermoelastic) damping scales as f², so overtones die away much faster than the fundamental. Mode 5 at 13.3f₁ decays ~177× faster than mode 1.

Strike excites modes ∝ |φₙ(x)| — hitting a node silences that mode.

Supports suppress modes ∝ α·|φₙ(p)| — a support at an antinode damps that mode; at a node it has no effect. α = 0 removes all suppression; α = 1 is full contact.

Final amp = |φₙ(strike)| · (1−α|φₙ(p₁)|) · (1−α|φₙ(p₂)|)

Default p₁≈0.224, p₂≈0.776 are the nodes of mode 1, so the fundamental survives while modes with antinodes there are attenuated — the same principle used to position marimba and xylophone frame supports.

Amplitude = |J_m(β_mn r/R)| — the total energy deposited in mode (m,n) is J_m²(β·r/R)·(cos²(mθ)+sin²(mθ)) = J_m²(β·r/R). The angle-independent Pythagorean identity means dragging azimuthally rotates the pattern without changing the recipe. Moving r to a nodal ring of J_m zeros that mode entirely.

Decay τ_n = τ₁ / ratio_n — radiation-dominated damping scales as f (not f²), so overtones decay more gradually relative to the fundamental than in a bar. The inharmonic spectrum combined with this milder roll-off gives drums their characteristic pitch ambiguity.

Beating — for m≥1 the two degenerate partners (cos and sin eigenfunctions) have slightly different frequencies on any real drum. A random split δf ~ 0.5–2.5 Hz is applied in synthesis, producing amplitude beating whose rate depends on drum imperfection.