Propagation of a Localized Impulse on a One-Dimensional Lattice

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Abstract
A chain of N atoms, each of mass M, with an interatomic distance a and coupled by identical massless ideal springs is considered to be in an equilibrium state with the interconnecting springs at equilibrium length at time t=0 . With the system in this state of equilibrium an impulse is given to one of the atoms in the chain. The atomic displacements and corresponding velocities for the disturbed chain are derived and expressed in terms of the dispersion relation for the chain by using Fourier expansions and by assuming periodic boundary conditions. It is shown that the k=0 mode of the Fourier expansion involved in the solutions for the displacement and velocity carries all of the momentum given to the chain during the impulse. The propagator which describes the atomic displacements in the chain resulting from the application of a space and time dependent force is presented. Graphical displays of the results of computations of the atomic displacements at various times illustrate the propagation through the chain of the initially localized pulse.

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