Solitary Waves in 1:N Dimer Chains

In the present work we report the discovery of new families of solitary waves in a 1:N (N>1) granular dimers (a heavy bead followed and preceded by N light beads) wherein the Hertzian interaction law governs the interaction between spherical beads. We consider the dimer chain with zero precompression. The dynamics of such a dimer chain is governed by two system parameters, the stiffness ratio and the mass ratio between the light and the heavy beads. In particular we study in detail the solitary waves in 1:2 dimer chains [11]. The solitary waves in a 1:2 dimer are contrastingly different from that in a homogeneous chain and 1:1 dimer chain. Solitary waves realized in homogeneous and 1:1 dimer chains possess symmetric velocity waveforms. In contrast, in a 1:2 dimer chain we realize solitary waves that have symmetric velocity waveforms on the heavy beads, whereas that on the light beads is non-symmetric. The existence of families of solitary waves in these systems is attributed to the dynamical phenomenon to ‘anti-resonance’. This leads to the complete elimination of radiating waves in the trail of the propagating pulse. Anti-resonances are associated with certain symmetries of the velocity waveforms of the beads of the dimer. We conjecture that a countable infinity of family of solitary waves can be realized in 1:2 dimer chains. Interestingly, solitary waves in a general 1:N (N>2) dimer chain are far more difficult to realize. For the case of 1:2 dimers, we can vary the two parameters to satisfy conditions such that the oscillatory tails in the trail of the primary pulse of the two light beads decay to zero. In contrast, for a 1:N (N>2) dimer chain, we have the same two parameters but need to satisfy the decaying conditions on N light beads simultaneously. This leads to a mathematically ill-posed problem and as such rigorously no solitary waves can be realized in general.