Band Gaps

Everything in the last four posts has rested on a single assumption, stated quietly but used constantly: the oscillations are small. Small enough that restoring forces are proportional to displacement. Small enough that the wave equation stays linear, that superposition holds, that the dispersion relation describes behaviour across all amplitudes equally. The band gaps we derived in Post 4 are real, they work in experiments, and they are beautiful. But they belong to a world where amplitude doesn’t matter.

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.