Domino chain - 2D toppling wave

Model

Each domino (height h, thickness t, depth h/2, density 700 kg/m³) is a rigid block that pivots without slipping about its front bottom edge. Its angle θ follows Iθ'' = m g (h/2 sinθ - t/2 cosθ) + τ_contact with I = m(h² + t²)/3 about the pivot. The upper front corner of a falling domino meets the back face of the next one; the contact is a stiff spring-damper (force kd - c v_n, damping ratio ζ sets how inelastic the hit is) integrated with a 20 µs time step. A lone domino has a tipping threshold θ_c = arctan(t/h): the centre of mass must pass over the pivot edge. The wave speed is measured from the times at which successive dominoes pass 20°, a quantity that depends on spacing: wider gaps give faster, lower-energy waves until the corner can no longer reach the neighbour (gap about the height) and the wave dies.

Honest simplifications: no sliding or friction at the contacts, no pivot slip, dominoes can only fall forward, the ground stops a domino at 90 degrees without bounce, and the contact stiffness is a numerical choice (overlaps are shown in the readouts). The wave speeds are model values, not a fit to a lab measurement.

• Standard preset, then raise the gap from 0.25 to 0.9 height and compare the measured wave speed.
• Choose 'Wide gap stalls' and see the chain stop part-way.
• Raise the damping ratio to turn the hits into soft, sticky ones and watch the dissipated share climb.