<p>We construct an approximate numerical-analytic solution of a dynamic problem for an infinite elastic layer containing a cylindrical rigid inclusion with conditions of rigid fastening imposed on its cylindrical surface. One face of the layer is subjected to the action of an axisymmetric normal compressive load, whereas the other face is coupled with the absolutely rigid foundation. To construct the fields of displacement and stresses in the layer, we apply the Laplace and Weber integral transformations to the axisymmetric equations of motion, which leads to an inhomogeneous one-dimensional vector boundary-value problem for the unknown transforms of displacements. This problem is solved by using the matrix differential calculus. The obtained integral equation on a finite interval is solved by the method of orthogonal polynomials with determination of the character of singularities of the solution at the ends of the interval. The normal stresses formed on the cylindrical surface of the inclusion and on the bottom face of the elastic layer are investigated. The solution is analyzed for the case of steady-state oscillations</p>