On the evaluation of effective density for plate- and membrane-type acoustic metamaterials without mass attached.

The effective densities of plate- and membrane-type acoustic metamaterials (AMMs) without mass attached are studied theoretically and numerically. Three models, including the analytic model (based on the plate flexural wave equation and the membrane wave equation), approximate model (under the low frequency approximation), and the finite element method (FEM) model, are first used to calculate the acoustic impedance of square and clamped plates or membranes. The effective density is then obtained using the resulting acoustic impedance and a lumped model. Pressure transmission coefficients of the AMMs are computed using the obtained densities. The effect of the loss from the plate is also taken into account. Results from different models are compared and good agreement is found, particularly between the analytic model and the FEM model. The approximate model is less accurate when the frequency of interest is above the first resonance frequency of the plate or membrane. The approximate model, however, provides simple formulae to predict the effective densities of plate- or membrane-type AMMs and is accurate for the negative density frequency region. The methods presented in this paper are useful in designing AMMs for manipulating acoustic waves.