Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices

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.

Nonlinear coherent structures in granular crystals

“…We also introduce the notation c = ω (k 0 ). The approximation (32) represents a standing DB if c = 0 (i.e., when the dispersion curve has vanishing slope). For a granular chain, this occurs at the edge of the spectrum (i.e., at k 0 = π).…”

“…10[b]). Unlike many 2D systems (such as photonic systems [3,4]), one can compute the dispersion relation explicitly (using a 2D discrete Fourier transform) instead of relying on numerical computations [32], see Fig. 11(b).…”

“…( 1) and take into account the relative positions of the particles in contact. For example, the force along the horizontal axis that results from contact between particles at positions {m, n} and {m, n − 1} is (such as photonic systems [3,4]), one can compute the dispersion relation explicitly (using a 2D discrete Fourier transform) instead of relying on numerical computations [32], see Fig. 11(b).…”

“…Such diffraction results from the excitation of linear modes near a Dirac point. [Panels (b,c) are reprinted with permission from[32]. Copyrighted (2016) by the American Physical Society.…”

Wave propagation and pattern formation in two-dimensional hexagonally-packed granular crystals under various configurations

“…5, with the middle region playing the role of the purple region in Figure 21, which was shown to often trap wave energy (see Figs. [22][23][24]. Such a configuration could prove useful for acoustic insulation or impact mitigation.…”