<p>This is a continuation of Hu and Kirsch’s previous work (2024) 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 Green’s function and prove new well-posedness results for scattering of plane waves at a propagative number. In such a case, there exist guided waves to the unperturbed problem, which are also known as bound states in the continuum (BICs) in physics. In this paper, the uniqueness of the forward scattering follows from an orthogonal constraint condition enforced 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 sources and plane waves, depending on whether <i>a priori</i> information on the size and height of the defect is available.</p>