LINTON, C.M. and MCIVER, M., 2002. Periodic structures in waveguides. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 458 (no.2028), pp.3003-3021.
We consider structures of period 2 spanning a two-dimensional waveguide of width
2N. Scattering problems, where Neumann conditions are imposed on the boundary
of the structure and either Neumann or Dirichlet conditions are applied on the guide
walls, are decomposed into N +1 independent problems. The existence of at least N
trapped modes is proved for the Neumann guide case, and for the Dirichlet case we
prove that at least N - 1 such modes exist, this number increasing to N if a certain
geometrical condition is satisfied.