The passing through the elastic wave barrier velocity is investigated for the case of a uniform load motion along a string on an elastic foundation. The string and foundation parameters vary 'smoothly' in such a manner that a transition from sub- into supercritical load motion occurs as the load passes a region of inhomogeneity. The transient solution for the string vibrations is derived and studied as a function of a characteristic length of the region of inhomogeneity. It turns out that during the passing the string displacements remain limited. It is shown that the smaller the characteristic length the smaller is the maximum amplitude of the string vibrations.

The steady-state response of an axially compressed Euler-Bernoulli beam on an elastic half-space due to a uniformly moving lateral load has been investigated. It is assumed that the beam has a finite width and that the half-space and beam deflections are equal along the center line of the beam. To analyze the problem, firstly the equivalent lateral stiffness of the half-space is derived as a function of the phase velocity of waves in the beam. Then using the expressions for the equivalent stiffness, a dispersion relation is obtained for the lateral waves in the beam. Analyzing this equation, it is shown that lateral waves can propagate in the beam only when the axial force in the beam is larger than a 'cut-off compressional force'. The critical (resonance) velocities of a uniformly moving constant and harmonically varying load are determined as functions of the axial compressional force in the beam. It is shown that the critical velocity of the harmonically varying load is always smaller than that of the constant load. A comparison is made between the critical velocity of a vertical and lateral constant load showing that the lateral constant load is smaller.

Generation of waves due to a constant load moving uniformly along an infinite string resting on an inhomogeneous elastic foundation is studied. Two types of inhomogeneity are considered: (1) an abrupt change of the foundation stiffness: (2) a smooth change of the foundation stiffness. It is shown that transition radiation arises as the load passes the region of inhomogeneity. Expressions for the spectral density of the radiation energy forwards (in the direction of the load motion) and backwards are found and analysed as a function of the load velocity, the ratio of the foundation stiffness and a characteristic length of the inhomogeneity. To visualise the process of radiation the transient vibrations of the string are determined for an abrupt change of the stiffness.

The stability problem of two oscillators moving uniformly along an Euler-Bernoulli beam on a viscoelastic foundation has been studied. It is assumed that the masses and the beam are in continuous contact and that the velocity of the oscillators exceeds the minimum phase velocity of waves in the supported beam. Stability regions are found. It is shown that a range of velocities exists for which unstable vibrations of the two oscillators will occur for all elastic-inertial properties.

The critical (resonance) velocities of a harmonically varying point load moving uniformly along an elastic layer are determined as a function of the load frequency. It is shown that resonance occurs when the velocity of the load is equal to the group velocity of the waves generated by the load. The critical depths of the layer are determined as function of the load velocity in the case the load frequency is proportional to the load velocity. This is of importance for high-speed trains where the loading frequency of the train wheel excitations is mainly determined by the ratio between the train velocity and the distance between the sleepers (ties). It is shown that the critical depths are decreasing with increasing train velocity. It is concluded that the higher the train velocity, the more important are the properties of the ballast and the border between the ballast and the substrate.

An alternative way of passing through the characteristic wave velocity in a waveguide is investigated by a uniformly moving object along an elastic system with a change in parameters such that a transition from subcritical into supercritical object motion is taking place. To demonstrate this way of passing, a uniformly moving constant load along an infinite elastically supported string with a sudden change in density is studied. The results of this investigation are compared with the more usual way of passing by a fast accelerating load along a homogeneous infinite string on the same foundation. For both cases transient solutions for the string vibrations are derived showing the wave processes, which are a combination of the main features of Mach radiation and transitional radiation in the alternative case and Mach radiation and radiation due to a non-uniform load motion in the usual case. It is shown that the alternative case provides a slightly smaller amplification in displacements than the usual case. The alternative way can be more practical since it is known that the faster the critical velocity is passed the smaller the dynamical amplification will be. Evidently a sudden change in the elastic parameters is much more simply organised than a large acceleration of the train.

The uniform motion of a mass along an axially compressed Euler-Bernoulli beam on a viscoelastic foundation is investigated. It is assumed that the mass is subjected to a constant vertical load and that the beam and mass are in continuous contact. The velocity of the mass after which the vibrations of the system are unstable is found. The instability implies that the amplitude of the mass vibrations is growing exponentially and that the problem does not have a steady state solution. It is shown that the instability starts at lower velocities as the compresional force increases. The instability occurs even for over-critical viscosities of the foundation when there is no dynamical amplification of the steady state vibrations due to resonance.

Three-dimensional vibrations of a Euler-Bernoulli beam on an elastic half-space are investigated. In the model the beam has a finite width and the half-space and beam deflections are equal along the centre line of the beam. It is shown that the vertical and longitudinal beam vibrations are uncoupled from the lateral ones. The dispersion relations for the lateral and vertical-longitudinal waves in the beam are derived and the respective dispersion curves are plotted. These curves can cross each other due to the different equivalent stiffnesses of the half-space in vertical and lateral directions and different vertical and lateral bending stiffnesses of the beam. The existence of a crossing point implies that if the vertical-longitudinal and lateral beam vibrations are coupled for some reason (half space inhomogeneity, beam asymmetry, etc.), the energy of the vertical vibrations of the beam can be resonantly transferred into the energy of lateral vibrations. This transfer will take place if the frequency of vibrations is close to the frequency determined by the crossing point. The dependency of the frequency of the crossing point on axial compressional stresses in the beam is studied. It is shown that this frequency decreases as the stresses increase.

The equivalent vertical stiffness of an elastic half-space interacting with a beam with finite width is derived. Both vertical and shear stresses are taken into account at the interface of the beam and the half-space. A comparison is made with the case of the equivalent stiffness of a smooth contact. It is shown that the differences are small if the phase velocities of the waves in the beam are smaller than the Rayleigh wave velocity in the half-space. For phase velocities larger than R-wave velocity the differences can be substantial. The analysis shows that the shear stresses at the interface have to be taken into account, if a load moving along the beam can generate waves with phase velocities larger than the Rayleigh wave velocity.

The steady-state displacements of an Euler-Bernoulli beam on an elastic half-space due to a uniformly moving constant load are determined using the concept of the equivalent stiffness of the half-space. The displacements are calculated for four different cases of beam and half-space parameters. The displacements in each case are derived for five relevant load velocities. The lowest is small with respect to the shear (S)-wave velocity in the half-space, the subsequent velocities are near the lowest critical velocity, between the critical velocity and the Rayleigh (R)-wave velocity, between the R-wave and S-wave velocity and larger than S-wave velocity. The beam displacement under the load is also determined for each case for all load velocities. Near the critical velocity the effect of an external viscous damping along the beam on the displacement under the load is studied.

Expressions are derived for the equivalent stiffness of an elastic half-space interacting with a beam with finite width. It is shown that the equivalent stiffness depends on the frequency and the wave number of the waves in the beam. The dependency is severe when the phase velocity of the waves in the beam is close to the velocity of the Rayleigh waves in the half-space. Expressions for the equivalent stiffness for long waves relative to the beam width are derived from general results and are approximated in terms of elementary functions. Critical velocities of a constant load moving at constant speed along a Euler-Bernoulli beam are determined. It is shown that there are two critical velocities, the first one equal to the velocity of the Rayleigh waves in the half-space and the second one mostly slightly smaller.

The wave radiation in an elastically supported infinite string due to a sudden change in the velocity of a constant moving vertical load is studied. It is shown that radiation results from the transition of the eigenfield moving with the load. In this process both the vertical load and the horizontal, force maintaining the motion after the velocity change, are performing work. The energy variation law has been verified, showing that the change of the energy in the eigenfields of the load and the radiated energy is equal to the work done by the moving vertical load and the horizontal force maintaining the motion after the velocity change. To visualize the process of radiation after the velocity jump the transient vibrations of the string have been determined. It shows that the eigenfield of the load before the velocity change is transferred to a new eigenfield by going through a so-called formation zone in which waves are radiating out in both directions.