  This is a continuation of the authors' previous work (A. Kirsch, Math. Meth.
Appl. Sci., 45 (2022): 5737-5773.) on well-posedness of time-harmonic
scattering by locally perturbed periodic curves of Dirichlet kind. The
scattering interface is supposed to be given by a non-self-intersecting
Lipschitz curve. We study properties of the Green's function and prove new
well-posedness results for scattering of plane waves at a propagative wave
number. In such a case there exist guided waves to the unperturbed problem,
which are also known as Bounded States in the Continuity (BICs) in physics. In
this paper uniqueness of the forward scattering follows from an orthogonal
constraint condition enforcing on the total field to the unperturbed scattering
problem. This constraint condition, which is also valid under the Neumann
boundary condition, is derived from the singular perturbation arguments and
also from the approach of approximating a plane wave by point source waves. For
the inverse problem of determining the defect, we prove several uniqueness
results using a finite or infinite number of point source and plane waves,
depending on whether a priori information on the size and height of the defect
is available.
