A wavelet based approach is proposed in this paper for analysis and optimization of the dynamical response of a multilayered medium subject to a moving load with respect to the material properties and thickness of supporting half-space. The investigated model consists of a load moving along a beam resting on a surface of a multilayered medium with infinite thickness and layers with different physical properties. The theoretical model is described by the Euler-Bernoulli equation for the beam and the Navier's elastodynamic equation of motion for a viscoelastic half-space. The moving load is modelled by a finite series of distributed harmonic loads. A special method based on a wavelet expansion of functions in the transform domain is adopted for calculation of displacements in the physical domain. The interaction between the beam and the multilayered medium is analyzed in order to obtain the vibration response at the surface and the critical velocities associated. The choice of the specific values of the design parameters for each layer, which minimize the vibration response of the multilayered medium, can be seen as a structural optimization problem. A first approach for using optimization techniques to explore the potential of the wavelet model is presented and briefly discussed. Results from the analysis of the vibration response are presented to illustrate the dynamic characterization obtained by using this method. Numerical examples reflecting the results of numerical optimizations with respect to a multilayered medium parameters are also presented.