Asymptotics of waves diffracted by a cone and diffraction series on a sphere

Diffraction of a plane harmonic scalar wave by a cone with ideal boundary condition is studied. A flat cone or a circular cone is chosen as a scatterer. It is known that the diffarcted field contains different components: a spherical wave, geometrically reflected wave, multiply diffracted cylindrical waves (for a flat cone), creepind waves (for a circular cone). The main task of the paper is to find a uniform asymptotics of all wave components. This problem is solved by using an integral representation proposed in the works by V. M. Babich and V. P. Smyshlyaev. This representaition uses a Green's function of the problem on a unit sphere with a cut. This Green's function can be presented in the form of diffraction series. It is shown that different terms of the series correspond to different wave components of the conical diffraction problem. A simple formula connecting the leading terms of the diffraction series for the spherical Green's function with the leading terms of different wave components of the conical problem is derived. Some important particular cases are studied.