Dynamics of a system of remote punches on an elastic half-space

The dynamic interaction (contact) between an elastic half-space and several smooth punches is studied. It is assumed that the dimensions of the contact regions $\Omega_i$ are much smaller than the distances between them and the scale of time of the process considered is comparable with the time required for an elastic wave to travel from one region to another. An asymptotic approach to the solution of the problem is proposed and the first two terms of the asymptotic representation of the displacement in the contact region and its neighborhood are constructed.