Dynamic mass density may be defined as
Negative mass density (NMD) metamaterials can be useful in attenuating low frequency sound, which has very high penetrating power through solid walls because of the mass density law that states that the sound transmission amplitude is inversely proportional to the product of (areal mass density) times (sound frequency). By moving out of phase with the incident wave, NMD metamaterials can exponentially attenuate the low frequency sound within its effective frequency range, thus breaking the mass density law. In the ultimate limit of such materials it would be desirable to have a thin and light-weight membrane that can operate effectively in the 100 - 1,000 Hz range. However, stopping low frequency sound with a thin membrane is against simple intuition, as total reflection requires the formation of a node at the reflecting surface, and a membrane with weak elastic restoring force is an unlikely candidate to be a low frequency sound reflector.
We show that precisely because of the weak elastic moduli of the membrane, there can be various low-frequency oscillation patterns even within a small and finite sample with fixed boundaries as defined by a rigid grid. Such vibrational eigenmodes can be tuned by placing a small mass at the center of the membrane sample, and near-total reflection is achieved at a frequency in-between two eigenmodes where the in-plane average displacement (normal to the membrane) is zero, leading to exponentially small far-field transmission .
The fact that the effective mass density can be negative at finite frequencies does not answer the question about whether the dynamic mass density always has to equal its static counterpart, i.e., the volume average value, in the limit of frequency approaching zero. Mathematically, this is equivalent to asking the question about homogenization of the elastic wave operator/equation consisting essentially the inertia part and the Laplacian part. Homogenization is usually done by first taking the limit of frequency approaching zero, so that the homogenization is just on the Laplace operator only, and the inertia part with the mass density does not participate. In that case the effective mass density is generally taken to be that of the volume averaged value. We have shown that for a composite with a fluid matrix [3,4], one can obtain a different limiting expression (compared to the static volume average value) for the effective mass by letting the dimensionless ratio of (viscous boundary layer thickness)/(typical feature size in the composite) first, equivalent to the inviscid case. This difference in the static and dynamic mass densities is shown to explain some puzzling experimental results [3,4].
 Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan and Ping Sheng, “Locally Resonant Sonic Materials,” Science 289, 1734-1736 (2000).
 Z. Yang, J. Mei, M. Yang, N. H. Chan and Ping Sheng, “Membrane-Type Acoustic Metamaterial with Negative Dynamic Mass,” Phys. Rev. Lett. 101, 204301 (2008).
 J. Mei, Z. Liu, W. Wen and Ping Sheng, “Effective Mass Density of Fluid-Solid Composites,” Phys. Rev. Lett. 96, 024301-024304 (2006).
 J. Mei, Z. Liu, W. Wen and Ping Sheng, “Effective Dynamic Mass Density of Composites,” Phys. Rev. B76, 134205 (2007).