On the spatial sampling and beat effects in discrete wave profiles of lumped acoustic metamaterials

Acoustic metamaterials are sub-wavelength locally resonant structures known for their band gap behavior and unique response. To capture their working mechanisms, the analysis typically discretizes the continuum model into lumped cells at the interface with the resonators with a cell size chosen appropriately to satisfy homogenization limits. This paper investigates steady-state wave profiles computed from the numerically obtained displacement field of the adjacent discrete cells. It is shown that predicted wave properties often deviate from those obtained via dispersion analysis of the unit cell. For a metamaterial comprised of a finite series of locally resonant cells, the resolution of the discretized waves on both sides of the band gap depends heavily on the shape of the dispersion branches, excitation frequencies, spacing, and properties of the cell constituents. A few examples are used to show the effect of these parameters on the spatial sampling of the propagating wave at both acoustic and optic modes, and the consequences of inadequate resolution on the harmonic response such as apparent modulation of longer wavelengths and beat-like effects in the resultant profiles. These effects are explained in light of defined parameters such as the number of cells per wavelength and the equivalent spatial Nyquist rate.