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We have developed an effective two-stage time-integration technique for elastodynamics and acoustic wave propagation problems solved with explicit and implicit time-integration methods and different space-discretization methods. For the first time, we have quantified the range of spurious oscillations for different space-discretization methods and have effectively filtered the spurious oscillations at the filtering stage. We have developed new finite elements with reduced dispersion for explicit time-integration methods as well as an analytical procedure for the selection of the size of time increments for the stage of basic computations and the filtering stage for the new finite elements with reduced numerical dispersion. The solution of 1D, 2D and 3D benchmark wave propagation problems showed that the new technique yields accurate and non-oscillatory results without interaction between computer code and user and reduces the computation time by a factor of 10-1000 and more compared with the standard finite element approaches. A surprisingly good agreement between experiments by group of Dr. Foley from AFRL/RWMF, Eglin and the simulations with the new numerical technique has been obtained for wave propagation in the components of the Hopkinson Pressure Bar.