A method of fundamental solutions (MFS) is employed for the numerical solution of the initial boundary value problem for the elastodynamic equation in annular planar domains. Using the Laguerre transformation in the time, the non-stationary problem is semi discretized to a sequence of stationary Dirichlet problems with an inhomogeneous equation, for which the sequence of fundamental solutions is known. The solutions of stationary problems are found by the MFS, when the unknown functions are approximated by a linear combination of narrowing of the elements from the fundamental sequence, and the source points are placed uniformly on articial boundaries, located at xed distances from the boundaries of the domain. The unknown coecients in the MFS-approximations are found using the collocation method, taking into account the Dirichlet conditions on the boundaries of the domain. As a result, we obtain a sequence of recurrent SLAEs with the same matrix and recurrent right-side parts, that depend on the solutions from previous iterations. In general, for the numerical solution of a problem with an inhomogeneous equation by the method of fundamental solutions, it is necessary to nd a partial solution of the inhomogeneous equation, for example, by the method of radial basis functions, however, according to our approach, this is not necessary. A step-by-step algorithm for the numerical solution of the given problem is described and the algorithm for the distribution of collocation points and source points is shown. The results of numerical experiments for dierent domain congurations are presented, which conrm the applicability and eectiveness of the proposed approach.