  We consider the propagation of surface water waves in a straight planar
channel perturbed at the bottom by several thin curved tunnels and wells. We
propose a method to construct non reflecting underwater topographies of this
type at an arbitrary prescribed wave number. To proceed, we compute asymptotic
expansions of the diffraction solutions with respect to the small parameter of
the geometry taking into account the existence of boundary layer phenomena. We
establish error estimates to validate the expansions using advances techniques
of weighted spaces with detached asymptotics. In the process, we show the
absence of trapped surface waves for perturbations small enough. This analysis
furnishes asymptotic formulas for the scattering matrix and we use them to
determine underwater topographies which are non-reflecting. Theoretical and
numerical examples are given.
