Band Gaps and Dispersion — Reading the Map
At the end of Post 3, we arrived at a precise physical picture of how a material’s effective parameters can go negative. Near a resonant frequency, the internal dynamics of a locally resonant inclusion reverse the macroscopic response: effective density goes negative above the resonant frequency of the internal mass, effective bulk modulus goes negative above the Helmholtz resonant frequency. When either parameter is negative, k² is negative, k becomes imaginary, and waves decay exponentially rather than propagate. We called the resulting frequency window a band gap.