  We are interested in the Helmholtz equation in a junction of two periodic
half-spaces. When the overall medium is periodic in the direction of the
interface, Fliss and Joly (2019) proposed a method which consists in applying a
partial Floquet-Bloch transform along the interface, to obtain a family of
waveguide problems parameterized by the Floquet variable. In this paper, we
consider two model configurations where the medium is no longer periodic in the
direction of the interface. Inspired by the works of G\'erard-Varet and
Masmoudi (2011, 2012), and Blanc, Le Bris, and Lions (2015), we use the fact
that the overall medium has a so-called quasiperiodic structure, in the sense
that it is the restriction of a higher dimensional periodic medium.
Accordingly, the Helmholtz equation is lifted onto a higher dimensional problem
with coefficients that are periodic along the interface. This periodicity
property allows us to adapt the tools previously developed for periodic media.
However, the augmented PDE is elliptically degenerate (in the sense of the
principal part of its differential operator) and thus more delicate to analyse.
