Uniform asymptotics of the wave diffracted by a cone of arbitrary cross section

In this paper, inspired by V. A. Borovikov's pioneering work, we study a wave diffracted by a convex cone of arbitrary cross section. Our analysis is focused on the shadow boundary of the wave reflected by the cone surface, where geometric optics yields nonphysical weak singularity. The uniform asymptotic expansion constructed here, which is expressed in terms of the parabolic cylinder functions, is regular at the geometric shadow boundary. Outside the penumbra region, it perfectly matches the ray expansions of the reflected wave, yielding in addition a regular spherical wave, would-be scattered by the cone vertex.