The problem of kinematic analysis of the propagation of axisymmetric elastoelectric waves in a hollow layered cylinder from metal and piezoceramic layers polarized in the radial direction is considered. The surfaces of the cylinder are free from tractions and undergo the action of a harmonically electrostatic potential $\pm V_{0} e^{i(kz-\omega t)}$. The numerical-analytical method is offered for solving this problem. After the separation of variables and the representation of a solution in the form of waves running along the cylinder, the initial problem of the theory of electroelasticity in partial derivatives is reduced to a non-homogeneous boundary the value problem for the system of ordinary differential equations for the radial coordinate. The problem obtained is solved by the stable numerical method of discrete orthogonalization. The numerical results are presented for a layered cylinder from metal and piezoceramic PZT layers.