Detailed study of the Malyuzhinets-Popov diffraction problem

The problem under consideration is the 2D high-frequency diffraction of a plane wave incident along a planar boundary that turns into a smooth convex contour in such a way that the curvature undergoes a jump. Asymptotic analysis based on the classical parabolic-equation method is developed, allowing formulas for the wavefield in the illuminated area, shadow, and the penumbra. The penumbral field is described by means of previously unseen in diffraction theory special functions showing a certain similarity to the Fock's integrals.