ИСТИНА

The problem of diffraction of a scalar wave by a curved surface is revisited. This problem was studied by V. A. Fock more than 50 years ago. He analyzed the problem of diffraction by a spherical surface using the parabolic equation method. He showed that solution can be represented as a contour integral containing the Airy's functions. Also he obtained an expression for the creeping waves in the shadow. Later, the results of V. A. Fock have been generalized for a much wider range of surfaces. Some recent progress concerns the problems of diffraction by highly prolate bodies. Such surfaces are characterized by two curvatures, such that the longitudinal curvature is much smaller than the transversal one. In this case it is not easy to say whether propagation is ray-like or not. The authors of the current study suppose that answer on this question can be found with the help of the boundary integral equation derived for the parabolic equation of the diffraction theory. This equation is known, and it is of Volterra type. The authors analyze this equation for two particular problems. They are 2D problem of diffraction by an arbitrary smooth curve and 3D problem of diffraction by a circular cone in the case of an axially-symmetric incidence. Both problems are reduced to one-dimensional Volterra-type integral equations. For the case of diffraction by a parabolic curve the equation is solved analytically and solution is represented as the Fock's integral. For the circular cone the equation is studied using the iteration method.