In the present work we study the dynamics of spatially periodic traveling waves in granular 1:1 (each bead is followed and preceded by a bead of different mass and/or stiffness) dimer chain with no pre-compression. The dynamics of a 1:1 dimer chain is governed by a single parameter, the mass ratio of the two beads forming each dimer pair of the chain. In particular, we demonstrate numerically the formation of special families of traveling waves with spatially periodic waveforms that are realized in semi-infinite dimer chains with the application of an arbitrary impulse. These traveling waves were first observed in the form of oscillatory tails in the trail of the propagating primary pulse. The energy radiated by the propagating primary pulse manifests in the form of traveling waves of varying spatial periodicity depending on the mass ratio. These traveling waves depend only on the mass ratio and are rescalable with respect to any arbitrary applied energy. The dynamics of these families of traveling waves is systematically studied by considering finite dimer chains (termed the ‘reduced systems’) subject to periodic boundary conditions. We demonstrate that these waves may exhibit interesting bifurcations or loss of stability as the system parameter varies. In turn, these bifurcations and stability exchanges in infinite dimer chains are correlated to previous studies of pulse attenuation in finite dimer chains through efficient energy radiation from the propagating pulse to the far field, mainly in the form of traveling waves. Based on these results a new formulation of attenuation and propagation zones (stop and pass bands) in semi-infinite granular dimer chains is proposed.
Existence and spectral instability of bounded spatially periodic traveling waves for scalar viscous balance laws
