  In this work, we study the wave propagation in a recently proposed acoustic
structure, the locally resonant granular crystal. This structure is composed of
a one-dimensional granular crystal of hollow spherical particles in contact,
containing linear resonators. The relevant model is presented and examined
through a combination of analytical approximations (based on ODE and nonlinear
map analysis) and of numerical results. The generic dynamics of the system
involves a degradation of the well-known traveling pulse of the standard
Hertzian chain of elastic beads. Nevertheless, the present system is richer, in
that as the primary pulse decays, secondary ones emerge and eventually
interfere with it creating modulated wavetrains. Remarkably, upon suitable
choices of parameters, this interference "distills" a weakly nonlocal solitary
wave (a "nanopteron"). This motivates the consideration of such nonlinear
structures through a separate Fourier space technique, whose results suggest
the existence of such entities not only with a single-side tail, but also with
periodic tails on both ends. These tails are found to oscillate with the
intrinsic oscillation frequency of the out-of-phase motion between the outer
hollow bead and its internal linear attachment.
