One of main issues in the application of the finite element method and other numerical methods to wave propagation and structural dynamics problems is the appearance of high-frequency spurious oscillations in a numerical solution. A new numerical approach based on a new solution strategy and new time-integration methods is developed for elastodynamics. The finite element method is used for the space discretization. The new approach filters spurious oscillations and retains actual oscillations. 1-D and 2-D impact problems are used for the calibration of numerical dissipation for the new approach as well as for the validation of the new numerical technique. It is shown that a simple modification of the mass matrix of the semi-discrete finite element equations leads to the essential reduction of the space-discretization error. For the first time a reliable, fast, accurate and non-oscillatory solution of acoustic and elastic wave propagation problems is possible. In contrast to existing approaches, the new technique does not require any guesswork for the selection of numerical dissipation or artificial viscosity and retains the accuracy of the basic solution at low modes. New fundamental results have been obtained due to the new approach: e.g., in contrast to textbooks on finite elements, for long-term integration, the size of time increments for explicit methods should be much smaller than the stability limit (rather than close to it).