  The present paper deals with the wave propagation in a particular two
dimensional structure, obtained from a localized perturbation of a reference
periodic medium. This reference medium is a ladder like domain, namely a thin
periodic structure (the thickness being characterized by a small parameter
$\epsilon > 0$) whose limit (as $\epsilon$ tends to 0) is a periodic graph. The
localized perturbation consists in changing the geometry of the reference
medium by modifying the thickness of one rung of the ladder. Considering the
scalar Helmholtz equation with Neumann boundary conditions in this domain, we
wonder whether such a geometrical perturbation is able to produce localized
eigenmodes. To address this question, we use a standard approach of asymptotic
analysis that consists of three main steps. We first find the formal limit of
the eigenvalue problem as the $\epsilon$ tends to 0. In the present case, it
corresponds to an eigenvalue problem for a second order differential operator
defined along the periodic graph. Then, we proceed to an explicit calculation
of the spectrum of the limit operator. Finally, we prove that the spectrum of
the initial operator is close to the spectrum of the limit operator. In
particular, we prove the existence of localized modes provided that the
geometrical perturbation consists in diminishing the width of one rung of the
periodic thin structure. Moreover, in that case, it is possible to create as
many eigenvalues as one wants, provided that $\epsilon$ is small enough.
Numerical experiments illustrate the theoretical results.
