Struck FPUT Chain f(Δ) = Δ + βΔ³ between neighbours, β = 1, walls at both ends

A row of masses joined by springs that stiffen as they stretch, hit near one end and left to ring down: the many-mode version of the coupled-oscillator toy. A gentle strike keeps its energy in the few modes it started in. A strong one spreads it over all of them, chaotically, until damping takes the energy back below the threshold, and the chain then rings on with whatever spectrum the chaos left behind. Strikes A and B differ by a tiny change in strength.

Physical setupMasses on a line between two walls, each joined to the next by a spring and a dashpot. The masses move along the line; here each mass's displacement is drawn sideways so the waves are visible.

The springs push back with f(Δ) = Δ + βΔ³ when stretched by Δ, so each one stiffens as it stretches: its stiffness is 1 + 3βΔ² (line thickness). That is what couples the modes, exactly as stretching the midplane couples the modes of a plate. The strike gives the marked masses a sudden velocity; the sound is the velocity of the mass at the pickup ring.

Where the energy isShare of strike A's energy in each normal mode over time (log scale, lowest mode at the bottom). A cascade shows as energy climbing to the higher modes.
How many modes share itEffective fraction of modes holding the energy (1 for equal shares; about 0.7 for random equipartition), for 8 strikes. Shaded: the chaotic phase of strike A.
The spectrum it rings withEach strike's final share of energy per mode. Where the strikes differ, the fingerprint is chaos's doing.
Strike AStrike B6 more strikes
How fast nearby strikes separateLocal Lyapunov exponent of strike A (per second). The dashed line is the energy in dB.
What you hearSpectrogram of the selected strike at the pickup. Click a strike, or use the ← and → keys, to flip between them; the color scale is shared, so differences in level are real. The selected strike is also drawn bold in the two plots above.