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We theoretically, numerically, and experimentally analyze the Density-Near-Zero (DNZ) regime of a one-dimensional acoustic metamaterial. This acoustic metamaterial is composed of thin elastic plates periodically clamped in an air-filled waveguide, and the effective dynamic zero mass density is obtained from the strong dispersion around the bandgaps associated with the resonances of the plates. We emphasize the importance of the impedance mismatch between the acoustic metamaterial and the surrounding waveguide at the frequency of the zero effective density in addition to the consequences of the inherent losses. As a result, the frequency of the zero phase propagation, i.e., the acoustic propagation with zero phase delay, is not exactly the frequency of the zero density and lies in the frequency bandgap where the effective density is negative. Considering these limitations, the zero phase propagation is still experimentally observed and a subwavelength acoustic dipole is numerically designed, thus demonstrating the possible realistic implementations of DNZ acoustic metamaterials.