Bead Sheet Cymbal a square sheet of beads: stretching links + bending + dome, pinned edges, struck once

The FPUT chain made two-dimensional. Beads on a square sheet move up and down; the links between neighbours stiffen when stretched (which couples the modes), the sheet resists bending (which gives it a plate's spectrum, about 550 modes between 100 Hz and 17 kHz, denser toward the top), and it can be built as a dome. Unlike a wire, a sheet has more and more modes per octave as the pitch rises, so a hard strike can spread its energy into a dense high-frequency wash, as a cymbal or gong does.

Physical setupThe sheet seen from above. Each square is a bead; its color is its height (warm up, cool down), drawn magnified. The edges are pinned. Click to move the strike (gold); shift-click to move the listener (ring).

Each bead is tied to its four neighbours by links that pull harder as they stretch: a height difference D between neighbours stretches the link by about D²/2, so its stiffness grows with D². This is the FPUT chain's nonlinearity, now in two directions, and it plays the role of a cymbal's midplane stretching, passing energy from mode to mode.

The sheet also resists bending, so short ripples vibrate much faster than long ones, as in a metal plate. Built as a dome, it stiffens and gains the three-wave coupling of curved shells. The strike gives a patch of beads a sudden upward velocity; damping is friction plus dashpots between neighbours, strongest for the shortest ripples.

Where the energy isShare of strike A's energy in each frequency band (modes grouped by pitch, log scale) over time. A cascade shows as energy climbing toward the top.
How high it reachesShare of each strike's energy above 4 kHz (solid) and above 1 kHz (thin), for all 8 strikes. Shaded: the chaotic phase of strike A.
The spectrum it rings withEach strike's final share of energy per frequency band.
Strike AStrike B6 more strikes
How fast nearby strikes separateLocal Lyapunov exponent of strike A (per second, 50 ms windows). The dashed line is the energy in dB.
What you hearSpectrogram of the selected strike at the listener. Click a strike, or use the ← and → keys, to flip between them; the color scale is shared, so differences in level are real.