Shocks in non-loaded bead chains with impurities
We numerically investigate the problem of the propagation of a shock in an horizontal non-loaded granular chain with a bead interaction force exponent varying from unity to large values. When $α$ is close to unity we observed a cross-over between a nonlinearity-dominated regime and a solitonic one, the latest being the final steady state of the propagating wave. In the case of large values of $α$ the deformation field given by the numerical simulations is completely different from the one obtained by analytical calculation. In the following we studied the interaction of these shock waves with a mass impurity placed in the bead chain. Two different physical pictures emerge whether we consider a light or a heavy impurity mass. The scatter of the shock wave with a light impurity yields damped oscillations of the impurity which then behave as a solitary wave source. Differently an heavy impurity is just shifted by the shock and the transmitted wave loses its solitonic character being fragmented into waves of decreasing amplitudes.