Nonlocal interaction engineering of 2D roton-like dispersion relations in acoustic and mechanical metamaterials

The interior of the synthetic unit cells and their interactions determine the wave properties of metamaterials composed of periodic lattices of these cells. While local interactions with the nearest neighbors are well appreciated, nonlocal beyond-nearest-neighbor interactions are often considered as a nuisance. Here, by introducing a versatile effectively two-dimensional metamaterial platform for airborne sound and elastic waves, we exploit nonlocal effects as a powerful design tool. Within a simplified discrete model, we analytically show that the lowest band can be engineered by Fourier synthesis, where the N-th order Fourier coefficient is determined by the N-th nearest-neighbor interaction strength. Roton-like dispersion relations are an example. The results of the discrete model agree well with a refined model and with numerical calculations. In addition, we engineer the passage of waves from a local metamaterial into a nonlocal metamaterial by carefully tailoring the transition region between the two.