The article deals with elastic homogeneous isotropic half-space filled with the Cosserat medium. At the initial instant of time and at infinity, there are no perturbations. At the boundary of the half-space, normal pressures are given. All the components of the stress-strain state are supposed to be limited. A cylindrical coordinate system is used with an axis directed inward into the half-space. With allowance for axial symmetry, the resolving system of equations includes three hyperbolic equations with respect to the scalar potential and the non-zero components of the vector potential and the rotation vector. The components of displacement vectors, rotation angle, stress tensors and stress moments are related to the potentials by known relationships. The solution of the problem is sought in the form of generalized convolutions of a given pressure with the corresponding surface influence functions. To construct the latter, Hankel transformations  along the radius and Laplace transformations  are applied in time. We use the expansion in power series for a small parameter characterizing the connection between the shear and rotation waves. The images of the first two coefficients of these series are found. The corresponding originals are determined by the connection between the plane and axisymmetric problems. Examples of calculations of the regular components of the influence of a granular composite from an aluminum shot in an epoxy matrix are given.