Origin of negative density

We present a new physical interpretation for the effective densities. We introduce the idea of hidden force: the effective density is negative when the hidden force is larger than, and operates in antiphase to the applied force. We demonstrate this picture for some established acoustic metamaterials with elements based on membranes, springs, and masses. The hidden force for membrane-based acoustic metamaterials, for instance, is the force from the membrane tension. We also explain the analogous concepts for pure mass-and-spring systems, in which case, the hidden force can arise from masses and springs fixed inside other masses. This new picture provides a powerful tool for conceptual understanding and design of new acoustic metamaterials, and avoids common pitfalls involved in determining the effective parameters of such materials.