On asymptotics of a wave field near the origin of an interference head wave

Let an incident wave propagates in a medium with wave-speed c₁ and illuminates a convex interface to a medium with propagation speed c₂, c₂ > c₁ . Let at some point O an incident ray meets an interface under critical reflection angle, then such a point looks like a source of an interference head wave which propagates in the first medium along the interface with the speed c₂. The wave field in a vicinity of O is rather complicated. In the paper, we discuss its high-frequency asymptotics.