Speaker Cone Breakup Simulator

1 — The piston illusion

Below the first modal frequency every point on the cone moves in lockstep. The air load sees one area, one velocity, one phase. Push the button — the whole disk goes. This holds while wavelength in the cone material is long compared to radius and while bending stiffness keeps the shape rigid. Once the cone can fit a half bending wavelength across its radius, different rings start moving opposite to each other. Cancellation on-axis, lobes off-axis. Your flat piston response is gone, replaced by modal peaks and dips that shift with every material choice.

2 — Bending waves, not sound waves

Sound in air travels at constant speed. Bending waves in a plate don't — they are dispersive. Their speed grows with frequency. That matters because it sets when breakup hits.

Bending stiffness for a thin isotropic plate: $$B = \frac{E t^3}{12(1-\nu^2)}$$ where $E$ is Young's modulus, $t$ thickness, $\nu$ Poisson's ratio.

Bending wave speed: $$c_b = \left(\frac{B}{\rho h}\right)^{1/4}\sqrt{\omega}$$

Stiffer ($E$ up) raises $c_b$, pushes nodal circles outward. Heavier ($\rho h$ up) lowers it. Ratio $B^2$ controls everything — which is why carbon fiber at $E=120$ GPa shreds the same diameter at 4× higher frequency than polypropylene. The explorer collapses this into a single material scale $\sqrt{E/\rho}/D$ for speed.

3 — Modal breakup: Bessel, not magic

A clamped circular plate does not ring like a guitar string. Its mode shapes are Bessel functions in radius, cosine in azimuth:

$$w_{mn}(r,\theta) \propto J_m(\alpha_{mn} r/R)\cos(m\theta)$$

$m=0$ modes are pure radial: one central dome, then a nodal circle appears at first root of $J_0$ (3.83), then two circles at 7.01, etc. Those are Modes 2 and 3 in the selector. $m=1,2$ modes add diametral nodal lines — Mode 4 splits left-right, Mode 5 mixes both with clover-leaf patterns. Breakup frequency for the (m,n) mode roughly follows:

$$f_{mn} \propto \frac{t}{R^2}\sqrt{\frac{E}{\rho(1-\nu^2)}} \cdot \alpha_{mn}^2$$

Shrink diameter $D=2R$ — $f$ climbs as $1/D^2$ for constant thickness, closer to 1/D here because we treat effective t/R as thin. In the UI: double diameter, watch $f_b$ halve. Switch paper → Aluminum → $f_b$ jumps 2.5×.

4 — Playbook: reading the explorer

• Auto mode: The solver picks the dominant mode from $f/f_b$. Below $0.6 f_b$ = piston. 0.6–1.05 = breakup onset (Mode 1 swelling). Beyond that, modal mix grows. The frequency response plot is synthetic but honest in shape: flat piston, a lift before breakup (edge anti-phase narrows directivity, on-axis rises), then jagged modal region.
• Phase coherence meter: Estimate of $\left|\sum e^{j\phi_i}\right|/N$. 100% = all points in-phase. Drops past $f_b$ as opposing rings cancel. Once coherence drops below 40%, equalization cannot fix it — you need a stiffer cone or a smaller radiating area.
• Material trick: Not just modulus. Real loss factor $\eta$ damps nodal peaks. Paper — low $E$ but high damping = gentle breakup. Aluminum — high $E$ but very low damping = vicious 8–12 dB ringing. That is why this explorer colors displacement but skips damping visualization.
• Design rule: Keep intended crossover / passband below $0.7 f_{breakup}$. If you must cross in breakup, measure off-axis and add notch filters, or switch to composite sandwich that pushes $f_b$ up while adding constrained-layer damping.

5 — Honesty: what this is not

Not FEA. No cone curvature (curvilinear profiles stiffen radial modes), no surround or spider edge termination, no voice-coil mass, no dust cap, no anisotropy (woven carbon bias stiffness differs 2× across axes), no geometric non-uniformity, no coupled membrane-bending waves, no acoustic loading. Thickness $t$ is collapsed into material scale. Real cones taper. The Bessel approximation assumes uniform clamp — real edge is compliant. Use this to build intuition for $f_{breakup} \propto \sqrt{E/\rho}/D$ and to recognize mode shapes in Klippel scans or laser vibrometry. For decisions, measure or run proper shell FEA + BEM.