This paper studies propagation of non-stationary axially symmetrical kinematic or electromagnetic disturbances applied on the surface of a ball. To this end, linear equations of motion of an elastic ball together with Maxwell equations are used as well as linearized generalized Ohm law and Lorentz force equation. The required functions are expanded in series in terms of Legendre and Gegenbauer polynomials. Laplace integral time transformation and expansion of coefficients of series into power series in small parameter linking mechanical and electromagnetic properties of the medium enabled finding recurrent sequence of boundary value problems with respect to components of mechanical and electromagnetic fields. The solution of each problem is represented in the form of generalized convolution of functions corresponding to previous members of the recurrent sequence with Green functions.