Effective medium theory of thin-plate acoustic metamaterials

Effective dynamic properties of acoustic metamaterials made of multilayered flexible thin-plates with periodically attached mass-spring resonators are studied. By using the transfer matrix method, the thin-plate acoustic metamaterial under the plane wave incidence is characterized by a homogeneous effective medium with anisotropic mass density. An approximate analytic expression of effective mass density is derived for a single-layer metamaterial in the normally incident case, and it is shown that the effective mass density can follow either Lorentz or Drude medium models. For the obliquely incident case, it is found that effective mass density is dependent on the lateral wave number of incident waves. Such spatial dispersion comes from the coincidence effect between the incident acoustic wave and flexural wave in the thin plate, and it occurs at much lower frequencies than that for a uniform plate without resonators. Based on the observed spatial dispersion, an acoustic device made of thin-plate metamaterials is designed for frequency-controlled acoustic directive radiation in the low-frequency regime.