Discrete wave dynamics in vibrating membranes: A cellular automaton approach

Abstract

We introduce a two-dimensional cellular automaton (CA) framework in which wave-like membrane dynamics emerge from purely local, discrete interactions. The model evolves second-order local rules with explicit velocity, acceleration, and optional damping, combined with an asynchronous update scheme.Quantitative comparison with the analytical Bessel-function solution of a circular membrane shows that, within a well-defined parameter regime, the automaton accurately reproduces the oscillatory dynamics of the central node, with systematic improvement under grid refinement across multiple convergence metrics. This establishes a quasi-continuum correspondence between the discrete model and its continuous analogue.Beyond this regime, numerical experiments reveal resolution-dependent frequency scaling, symmetry breaking induced by asymmetric initial conditions and boundary geometries, and spatially heterogeneous stiffness patterns emerging from local Hookean interactions.A bifurcation diagram obtained by varying the damping coefficient γ shows an abrupt transition in steady-state amplitude at a critical value γc, accompanied by a shift from single-frequency to broadband spectral behaviour. This transition is absent from the linear damped wave equation and arises from the discrete, rule-based update structure.These results highlight both the correspondence between discrete and continuous descriptions and its breakdown, positioning cellular automata as a minimal and extensible framework for studying emergent wave phenomena in nonlinear, heterogeneous, and bio-inspired active membranes.