  We consider an explicitly solvable model (formulated in the Riemannian
geometry terms) for a stationary wave process in a specific thin domain with
the Dirichlet boundary conditions on the boundary of the domain. The transition
from the solutions of the scattering problem to the solutions of a problem on
the limiting quantum graph is studied. We calculate the Lagrangian gluing
conditions at vertices for the problem on the limiting graph. If the frequency
of the incident wave is above the bottom of the absolutely continuous spectrum,
the gluing conditions are formulated in terms of the scattering data of a
problem in a neighborhood of each vertex. Near the bottom of the absolutely
continuous spectrum the wave propagation is generically suppressed, and the
gluing condition is degenerate (any solution of the limiting problem is zero at
each vertex).
