Traveling Waves and their Tails in Locally Resonant Granular Systems

TL;DR

This paper investigates traveling wave solutions in locally resonant granular systems using three different analytical approaches, revealing conditions for both symmetric tails and monotonic decay in solitary waves.

Contribution

It introduces three distinct methods to identify traveling waves in Mass-in-Mass systems and analyzes resonance conditions for decay properties of the waves.

Findings

Traveling waves with symmetric non-vanishing tails are found.

Resonance conditions lead to genuinely monotonic decaying tails.

Three approaches provide complementary insights into wave behavior.

Abstract

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.