Scattering of an edge wave by the domain of critical change of the wedge's angle

The paper studies the scattering of a wave running along the edge of a wedge with Robin (i.e. impedance type) and Neumann boundary conditions, respectively, on the faces and localized near it, by the region of decreasing its angle to a new constant value. The wedge's angle, when changing smoothly, passes through the so-called critical value in such a way that for a new reduced angle of the wedge, the possibility arises of propagating of two wedge waves, localized near the edge, and, in the wider part of the wedge, the edge wave, reflected by the region of the change in the angle, propagates.
The described scattering process in the open waveguide under study is additionally accompanied by the appearance of other components in the scattered field: a spherical wave and surface waves, localized near the edge of the wedge with the Robin condition on it, including surface waves running from the edge, as well as cylindrical waves from the edge.
The paper uses and develops a method for studying this type of problems in cone-shaped domains with boundary conditions of the third kind, based on integral representations of the solution of the Watson–Bessel and Sommerfeld types, on incomplete separation of variables.