The wave equation with symmetric velocity on the hybrid manifold obtained by gluing a ray to a three-dimensional sphere

In the paper, the Cauchy problem for the wave equation with variable (symmetric) velocity on the hybrid manifold obtained by gluing a ray to a three-dimensional sphere is considered. It is assumed that the initial conditions are localized on the ray and the velocity on the sphere depends only on the geodesic distance to the gluing point. The asymptotic series of the solution of the problem as parameter characterizing the initial perturbation tends to zero is given. Since the sphere is compact, then the wave propagating over the sphere is reflected at the pole opposite to the gluing point and returns to the ray. Thus, the question of the distribution of wave energy at every moment of time is also interested and discussed in this work.