Negative Density

At the end of Post 2, we derived the wave equation and discovered something unsettling sitting inside it. If the effective density of a material goes negative, the character of the equation changes. Instead of oscillatory solutions that propagate energy through space, you get exponential ones: a wave entering such a region decays rather than travels. We called this evanescent decay, named the resulting frequency band a band gap, and noted it was central to everything interesting in acoustic metamaterials.