Extensive efforts are being exerted to develop various types of acoustic metamaterials to effectively control the flow of acoustical energy through these materials. However, all these efforts are focused on passive metamaterials with fixed material properties. In this paper, the emphasis is placed on the development of a class of one-dimensional acoustic metamaterials with tunable effective densities in an attempt to enable the adaptation to varying external environment. More importantly, the active metamaterials can be tailored to have increasing or decreasing variation of the material properties along and across the material volume. With such unique capabilities, physically realizable acoustic cloaks can be achieved and objects treated with these active metamaterials can become acoustically invisible. The theoretical analysis of this class of active acoustic metamaterials is presented and the theoretical predictions are determined for an array of fluid cavities separated by piezoelectric boundaries. These boundaries control the stiffness of the individual cavity and in turn its dynamical density. Various control strategies are considered to achieve different spectral and spatial control of the density of this class of acoustic metamaterials. A natural extension of this work is to include active control capabilities to tailor the bulk modulus distribution of the metamaterial in order to build practical configurations of acoustic cloaks.