Conical Wave Propagation and Diffraction in 2D Hexagonally Packed Granular Lattices

TL;DR

This paper investigates how conical waves propagate and diffract in 2D hexagonally packed granular lattices, revealing linear and nonlinear mechanisms, and transitions between different wavefront behaviors under varying precompression conditions.

Contribution

It provides a comprehensive analysis of conical wave propagation in granular lattices, including dispersion relations, diffraction properties, and the transition between linear and nonlinear regimes.

Findings

Conical diffraction occurs near the linear regime under strong precompression.

Weak precompression leads to non-oscillatory expanding circular wave fronts.

The transition between linear and nonlinear wave propagation regimes is characterized.

Abstract

Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wavepacket, as well as via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a non-oscillatory nature, resulting from the complex interplay between the discreteness, nonlinearity and geometry of the packing. The transition between these two types of propagation is explored.