Each atom here is a real point mass coupled to its four nearest neighbours by identical springs of stiffness K. The simulation integrates Newton's second law for every atom, mü = KΣ(u_neighbour − u_self) − γu̇, every frame — nothing here is a canned sine wave. A discrete chain like this has a maximum vibration frequency, the cutoff ω_max = 2√(K/m): drive it below cutoff and a travelling wave propagates cleanly through the lattice; drive it above cutoff and the disturbance instead decays exponentially with distance (an evanescent wave) because no real propagating mode exists at that frequency in a discrete medium.

• Governing equation: mü_i,j = K(u_i+1,j+u_i-1,j+u_i,j+1+u_i,j-1−4u_i,j) − γu̇_i,j
• Cutoff frequency ω_max = 2√(K/m) — no plane wave propagates above it in a discrete lattice.
• Dispersion relation: ω(k) = 2√(K/m)·|sin(ka/2)|, non-linear unlike a continuous string.
• Group velocity v_g = a√(K/m)·cos(ka/2) — goes to zero at the zone edge (k = π/a).
• Boundaries are lightly damped (absorbing) to suppress standing-wave reflections at the edges.