Coefficient of restitution for one-dimensional harmonic solids

Using a numerical algorithm based on the time evolution of normal modes, we calculate the coefficient of restitution eta for various one-dimensional harmonic solids colliding with a hard wall. We find that, for a homogeneous chain, eta=1 in the thermodynamic limit. However, for a chain in which weaker springs are introduced in the colliding front half, eta remains significantly less than one even in the thermodynamic limit, and the "lost" energy goes mostly into low frequency normal modes. An understanding of these results is given in terms of how the energy is redistributed among the normal modes as the chain collides with the wall. We then contrast these results with those for collisions of one-dimensional harmonic solids with a soft wall. Using perturbation theory, we find that eta=1 for all harmonic chains in the extremely soft wall limit, but that inelasticity grows with increasing chain size in contrast to hard wall collisions.