Exact Solutions for Dynamics of Finite, Semi-Infinite, and Infinite Chains with General Boundary and Initial Conditions

Abstract

The exact analytic solutions in closed form for the dynamics of finite, semi-infinite, and infinite linear chains of identical masses and ideal springs are presented. In addition to the harmonic interaction between nearest neighbors, each particle is harmonically bound to its equilibrium position and is subjected to frictional and other external time-dependent forces. The motion of all the particles is expressed in terms of the given initial conditions and applied forces. In contrast with previous studies on the finite chains, very general (i.e., not necessarily ``periodic'') boundary conditions are used and the resulting solutions contain terms representing all the higher order multiple reflections from the two ends. The properties of the solutions are studied for all possible values of physical constants including the limiting values for uncoupled oscillators. By specializing some results of the present work, those of previous derivations on simpler systems by other authors are recovered. Some useful applications of the results are suggested.