  We consider the time-harmonic scattering wave problem in a 2D waveguide at
wavenumber $k$ such that one mode is propagating in the far field. In a first
step, for a given $k$, playing with one scattering branch of finite length, we
demonstrate how to construct geometries with zero transmission. The main
novelty in this result is that the symmetry of the geometry is not needed: the
proof relies on the unitary structure of the scattering matrix. Then, in a
second step, from a waveguide with zero transmission, we show how to build
geometries supporting trapped modes associated with eigenvalues embedded in the
continuous spectrum. For this second construction, using the augmented
scattering matrix and its unitarity, we play both with the geometry and the
wavenumber. Finally, the mathematical analysis is supplemented by numerical
illustrations of the results.
