General Method for Evaluation of Stop-Bands of Periodic Structures with Symmetric Unit Cells
The mirror symmetries of a periodic unit cell are exploited to decompose the standing-wave eigenproblem at the high-symmetry vertices of the Brillouin zone into four independent sub-problems on a quarter-cell, each governed by Neumann (sound-hard) or Dirichlet (sound-soft) boundary conditions. Sorting and pairing the resulting eigenfrequencies by index along each segment of the irreducible Brillouin zone boundary yields an explicit formula for the stop-band intervals without computing the full dispersion diagram. The decomposition is exact, following directly from the representation theory of the little group at each high-symmetry point. It applies to any unit cell whose material distribution is invariant under the mirrors normal to the cell faces. The method is validated on two configurations: a phononic crystal of lead cylinders in an epoxy matrix, analyzed using the plane-wave expansion, and a lattice of coupled C-shaped Helmholtz resonators, analyzed using finite-element analysis. For both systems, the reconstructed stop-band boundaries agree with the full Floquet dispersion calculation to within 1% for the lowest bands, requiring eigenvalue solutions at only three discrete wavevectors. Avoided crossings within a Brillouin zone segment can cause bands to exhibit non-monotone behavior, rendering the pairing rule approximate; the spectral conditions for this are identified. Flat bands common to both boundary-condition types are identified as bound states in the continuum.