Summary: 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 Dirichiet 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.