Wave scattering in the joint of a straight and a periodic waveguide

The asymptotic expansions of the reflection and transmission coefficients (the scattering matrix) for waves propagating in a planar compound acoustic waveguide are investigated. Semi-infinite straight walls smoothly turn into gently sloped periodic one. In the periodic outlet the waves become Floquet waves, and gaps can appear in the spectrum, these being stopping zones impeding wave propagation in the corresponding frequency range. Differences in the diffraction patterns for the spectral parameter at some distance from an open or closed spectral gap or near it or inside it are examined. The scattering characteristics vary exclusively due to variation of the shape of one outlet. The leading terms of the asymptotic expansions of the elements of the scattering matrices are calculated. Sizes of these matrices depend on the position of the spectral parameter. Effects of moving the spectral parameter from one side of the gap to the other are discussed.