Hierarchical Band Gaps in Complex Periodic Systems

Complex periodic structures inherit spectral properties from the constituent parts of their unit cells, chiefly their spectral band gaps. Exploiting this intuitive principle, which is made precise in this work, means spectral features of periodic systems with very large unit cells can be predicted without numerical simulation. We study a class of difference equations with periodic coefficients and show that they inherit spectral gaps from their constituent elements. This result shows that if a frequency falls in a band gap for every constituent element then it must be in a band gap for the combined complex periodic structure. This theory and its instantaneous utility is demonstrated in a series of vibro-acoustic and mechanical examples.