  We study the propagation of time-harmonic acoustic or transverse magnetic
(TM) polarized electromagnetic waves in a periodic waveguide lying in the
semi-strip $(0,\infty)\times(0,L)$. It is shown that there exists a Riesz basis
of the space of solutions to the time-harmonic wave equation such that the
translation operator shifting a function by one periodicity length to the left
is represented by an infinite Jordan matrix which contains at most a finite
number of Jordan blocks of size $> 1$. Moreover, the Dirichlet-, Neumann- and
mixed traces of this Riesz basis on the left boundary also form a Riesz basis.
Both the cases of frequencies in a band gap and frequencies in the spectrum and
a variety of boundary conditions on the top and bottom are considered.
