When the Springs Stop Being Simple - Nonlinearity and Chaos
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.