Linear Dynamics of Oscillating Lattices – A Spatial Discretization of the Wave Equation∗

This work examines solutions of a spatial discretization of the classical wave equation as a model for the linearized dynamics of an infinite system of particles indexed by and having equilibrium positions x ∈ hZn, with each particle linked to its 2n nearest neighbours by harmonic springs. We numerically observe the existence of a spherical monotone wavefront corresponding to the continuum light cone, the propagation of regions of maximal displacement along the principal diagonals of the lattice, and prove the emergence of second-nearestneighbour synchronized oscillations in long time. We prove a leadingorder t−n/2 decay law for the amplitude of oscillation of each particle. We also show that the group and phase velocities of the system are bounded above by the corresponding continuum wave speed, and determine the long-time behaviour of the local and global kinetic and potential energies. ∗Originally submitted on March 26, 2004, for MA469 Project at the University of Warwick, supervised by Prof. Gero Friesecke.