This paper analyses theoretically the response of a solid for fast moving trains using models related to real situations: a load moving in a tunnel and a load moving on a surface. The mathematical model is described by Navier's elastodynamic equation of motion for the soil and Euler-Bernoulli equation for the beam with appropriate boundary conditions. Two modelling approaches are investigated: the model with half space under the beam and the model with finite thickness of supporting medium. The problem of singularities for displacements calculation is discussed in relation with boundary conditions and types of considered loads: harmonic and constant, point and distributed moving loads. The analysis in frequency-time and frequency-velocity domains is presented and discussed with regard to critical velocities.A wavelet approximation method based on application of coiflet filters is used for the derivation of displacements in physical domain. A new, modified filter is used in numerical calculations allowing to alleviate numerical difficulties related to the form of approximating sequences based on classical filters, formulated in previous publications. The effectiveness of the coiflet approach is discussed for filter coefficients with different order of accuracy. This methodology is very efficient for the analysis in the range of relatively high and low load frequencies (treated as an approximation of a constant load) which are especially important for the analysis of vibrations generated by trains moving with velocities near critical values.Results of numerical simulations are presented, demonstrating their utility for modelling and preliminary analysis of complex models.