  In the present study, we revisit the theme of wave propagation in locally
resonant granular crystal systems, also referred to as Mass-in-Mass systems. We
use 3 distinct approaches to identify relevant traveling waves. The first
consists of a direct solution of the traveling wave problem. The second one
consists of the solution of the Fourier tranformed variant of the problem. or,
more precisely, of its convolution reformulation (upon an inverse Fourier
transform) of the problem in real space. Finally, our third approach will
restrict considerations to a finite domain, utilizing the notion of Fourier
series for important technical reasons, namely the avoidance of resonances,
that will be discussed in detail. All three approaches can be utilized in
either the displacement or the strain formulation. Typical resulting
computations in finite domains result in the solitary waves bearing symmetric
non-vanishing tails at both ends of the computational domain. Importantly,
however, a countably infinite set of resonance conditions is identified for
which solutions with genuinely monotonic decaying tails arise.
