Spatial distributions of railway-generated ground vibrations at and around sleeper passage frequencies

In this paper, spatial distributions of ground vibrations generated by railway trains travelling at conventional speeds on straight tracks are investigated theoretically. The main attention is being paid to calculations of spatial distributions of ground vibrations generated by a single axle load at and around sleeper passage frequencies that are defined by train speeds and sleeper periodicity. It is demonstrated that, at sleeper passage frequencies, generated ground vibrations represent plane waves propagating symmetrically away from the track in the normal directions to it. At frequency components that are slightly above or below sleeper passage frequencies, generated ground vibrations still remain plane waves, but they are now propagating at certain angles in respect of the normal directions to the track.


Introduction
Over the last decades, high-speed railways underwent rapid development throughout the world. As many other means of transportation, high-speed railways encounter a number of environmental problems. In particular, ground vibrations, mostly Rayleigh surface waves, generated by high-speed trains is one of the major environmental problems that must be mitigated to allow high-speed trains to be used in densely populated areas (see e.g. the recent book [1]).
The intensity of railway-generated ground vibrations generally becomes larger at higher train speeds. The increase in amplitudes of generated ground vibrations can be especially large when train speeds approach the velocity of Rayleigh surface wave in the supporting ground. As was theoretically predicted by the present author [2][3][4][5], if a train speed v exceeds the Rayleigh wave velocity cR in the supporting soil, a ground vibration boom occurs. This phenomenon is similar to a sonic boom from supersonic aircraft, and it is associated with a very large increase in generated ground vibrations, as compared to the case of conventional trains. The existence of ground vibration boom at trans-Rayleigh train speeds has been later confirmed experimentally [6,7].
The increased attention to the problems of ground vibrations from high-speed trains is reflected in a growing number of theoretical and experimental investigations in this area (see e.g. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). In addition to ground vibration problems typical for trains approaching or exceeding Rayleigh wave velocity in the ground, there are also many problems associated with trains travelling at lower (conventional) train speeds. In particular, for the important case of generation of ground vibrations at the so-called sleeper passage frequencies, very little or no research into the structure and directions of generated waves has been carried out so far.
The aim of the present paper is to investigate spatial distributions of ground vibrations generated at and around sleeper passage frequencies by high-speed trains travelling at conventional speeds, i.e. at speeds not exceeding Rayleigh wave velocity in the supporting ground. It will be demonstrated that spatial distributions of generated Rayleigh waves in this case can change significantly, depending on train speeds, frequency components, and distance between adjacent sleepers.

Theoretical background
The main mechanisms of railway-generated ground vibrations are the wheel-axle pressure onto the track, the effects of joints in unwelded rails, the dynamically induced forces of carriage-and wheel-axle vibrations excited due to random roughness of wheels and rails, as well as variations in sleeper-ground contact from sleeper to sleeper. In this paper we limit our consideration with only the first mentioned generation mechanism, which is always present, even for ideally flat rails and perfectly round wheels -namely, the quasi-static pressure of wheel axles acting onto the track. This universal generation mechanism is responsible for the above-mentioned railway-generated ground vibration boom, and it is also responsible for ground vibrations generated at sleeper passage frequencies considered in this paper. It should be noted that at other frequencies the unevenness of real wheels and rails usually plays a dominant role in generation of ground vibrations (see e.g. [14,15,[18][19][20]).
The quasi-static pressure generation mechanism results from load forces applied to the railway track from each wheel axle. These load forces cause downward deflections of the track that are moving together with the loads, thus producing a wave-like motion along the track at speed of the train, which results in a distribution of each axle load over all the rail sleepers that are within the track deflection distance. Each sleeper, in turn, acts as a vertical force applied to the ground during the time necessary for a deflection curve to pass through the sleeper. These vertical forces can be approximately considered as point forces applied to the ground's surface, assuming that shortest wavelengths of generated ground vibrations are still essentially larger than sleeper dimensions. To determine the track deflection curve and thus the time dependence of the forces applied from each sleeper to the ground the system of track and ground is modelled as an Euler-Bernoulli beam resting on Winkler foundation.
According to the earlier developed general theory [2][3][4][5][9][10][11], in order to calculate ground vibrations generated by a train due to the quasi-static pressure mechanism, one needs to take into account the superposition of waves generated by each elementary source of ground vibrations (sleeper) activated by wheel axles of all carriages, with the time and space differences between sources (sleepers) being taken into account. In doing so we take into account only the contribution of generated Rayleigh surface waves, because they make the main contribution to ground vibrations causing environmental problems.
Being interested in fundamental features of railway-generated Rayleigh waves, in this paper, for the sake of simplicity, we will consider ground vibrations generated by a single axle load only. In this case, we can use the following expression for the vertical vibration velocity vz of Rayleigh waves generated on the ground surface (z = 0) at the point of observation having the coordinates x, y by a single axle load moving along the straight track (located on the x-axis) at constant speed v (see e.g. [9][10][11]): Here ρn is the distance from each sleeper characterised by a number n to the point of observation, ω is a circular frequency, v is the train speed, d is a sleeper periodicity, cR is the Rayleigh wave velocity, and γ = 0.001 -0.1 is a non-dimensional loss factor describing the attenuation of Rayleigh waves in soil. The function P(ω) in Equation (1) represents the Fourier spectrum of a force acting from each sleeper to the ground. The expression for P(ω) has the form (see [5,10,11] for detail): where T is the axle load, cmin is the minimal phase velocity of track flexural waves propagating in a track/ground system (this velocity is related to cR via the parameters of Winkler foundation expressed in terms of the ground elastic parameters, and it is usually larger than cR by 10-20 %), β is the parameter dependent on the elastic properties of track and ground [10] and measured in m -1 , and η is a non-dimensional track damping parameter. For relatively low train speeds, i.e., for v < cR, the 'dynamic' solution (2) for the force spectrum P(ω) goes over to the quasi-static one [10,11]. When train speeds increase and approach or exceed the minimal track wave velocity cmin, the spectra P(ω) become broader and larger in amplitudes, and a second peak appears at higher frequencies.
The analysis of the Equation (1) shows that maximum radiation of ground vibrations takes place if the train speed v is larger than Rayleigh wave velocity cR [2][3][4][5][9][10][11] (such trains are often called 'trans-Rayleigh trains'). Under this condition, a ground vibration boom takes place, i.e., ground vibrations are generated as quasi-plane Rayleigh surface waves symmetrically propagating at angles Θ = cos -1 (cR/v) in respect to the track, and with amplitudes much larger than in the case of conventional trains. This phenomenon is similar to the well-known phenomenon of sonic boom from supersonic aircraft.
It should be noted that for trans-Rayleigh trains these symmetrically propagating Rayleigh surface waves are generated equally well on tracks with and without railway sleepers, whereas for conventional (sub-Rayleigh) trains the presence of sleepers for the possibility of generation of Rayleigh waves is paramount. Without them no propagating waves are generated by a force moving at constant sub-Rayleigh speeds in the framework of the quasi-static pressure generation mechanism. Only quasi-static localised displacements are present in this case that are moving along with the force. However, if the same force is moving along a railway track supported by discrete sleepers, Rayleigh waves are generated even at sub-Rayleigh load speeds. For a single moving load, the efficient generation takes place mainly at and around the so-called sleeper passage frequencies fsp (m) where v is the load speed, d is the sleeper periodicity, and m = 1, 2, 3... . In this paper, we will be interested in spatial distributions of ground vibrations (Rayleigh surface waves) generated by a single moving load at and around sleeper passage frequencies.

Numerical calculations and discussion
In what follows, we describe the results of the calculations of spatial distributions of railwaygenerated ground vibration fields vz(x,y , ω) (i.e. wave snapshots shown in arbitrary linear units) over a certain surface area with a railway track in the middle of it (located along the xaxis) for some interesting situations corresponding to frequency components at and around sleeper-passage frequencies. Calculations have been carried out according to Equations (1)-(4) over the surface area of 40 × 40 m to obtain spatial distributions of ground vibrations (Rayleigh waves) generated at different frequency components f by a single axle load travelling along a straight railway track at speeds v. The summation in Equation (1) has been taken over 120 sleepers in all cases for each point of observation, and the magnitude of the axle load has been considered to be the same in all cases. The results will be presented as greyscale contour plots. It will be assumed that the velocity of Rayleigh waves in the ground is cR = 80 m/s for all cases. Other relevant parameters are as follows: d = 0.7 m, β = 1.28 m -1 , which is typical for British railway tracks, and γ = 0.001.
It is instructive to start with the distinctive case of ground vibrations generated by a single axle load travelling at a trans-Rayleigh speed v > cR, which is associated with generation of ground vibration boom by high-speed trains. Because the amplitudes of generated ground vibrations are the largest in this case, the obtained results can be used as a useful reference for comparison with ground vibrations generated at lower (conventional) train speeds. As it can be seen from Fig. 1, the spatial distribution of ground vibrations (Rayleigh waves) shows a typical picture of plane waves generated symmetrically in respect of the track at Mach angles Θ = cos -1 (cR/v). It should be noted that Mach angles do not depend on frequency in the case under consideration. Therefore, for the same trans-Rayleigh speed, ground vibrations will be generated at the same Mach angles for any frequency components.
The amplitudes of generated ground vibrations (in arbitrary linear units) can be estimated from the surface plot corresponding to Fig. 1 (not shown here for shortness). The vertical scale in this surface plot gives the range of vertical components of ground vibration velocities between -1.551 and 1.819, which can be compared with the amplitudes of ground vibrations generated at sub-Rayleigh speeds that will be calculated below.  As it can be seen from Fig. 2, the spatial distribution of ground vibrations generated by a single load at the first sleeper passage frequency represents plane waves propagating symmetrically away from the track in the normal directions to it. This is due to the fact that at sleeper passage frequencies all sleepers are radiating in phase. The amplitudes of generated ground vibrations here (in arbitrary linear units) can be also estimated from the surface plot corresponding to Fig. 2 (not shown here). The vertical scale in the surface plot in this case gives the range of vertical components of vibration velocities between -0.031 and 0.009, which demonstrates that the amplitudes of generated Rayleigh waves in this case are roughly by 85 times smaller than in the case of generation of ground vibration boom by a single load illustrated in Fig. 1.
Distribution of ground vibrations generated by the same single axle load at the frequency component f = 21 Hz, which is slightly above the first sleeper passage frequency for this case, is shown in Fig. 3. One can see that the wave fronts in this case change symmetrically their directions of propagation from the normal directions to the track, as in the case of the first sleeper passage frequency (see Fig. 2), to the directions that are slightly deflected from the normal directions. The reason for this deflection is the phase differences between vibrating sleepers at this frequency. The amplitudes of generated ground vibrations in this case are roughly the same as in Fig. 2. For even higher frequency component f = 23 Hz, the ground vibrations generated by the same single axle load undergo further evolution, as shown in Fig. 4. It can be seen that the directions of propagation of the wave fronts in this case deflect symmetrically even further from the normal directions to the track, and the amplitudes of generated ground vibrations in this case are roughly the same as in Figs. 2 and 3.
It should be noted that spatial distributions of ground vibrations in Figs. 3 and 4 look similar to that shown in Fig.1 for the case of ground vibration boom. One should remember though that the associated physical mechanisms are absolutely different. Rayleigh wave radiation by a load travelling at sub-Rayleigh speeds illustrated in Figs. 3 and 4 is entirely due to the presence of sleepers. Therefore, radiation angles of generated waves depend here on frequency and on sleeper periodicity, as it follows from the comparison of Figs. 3 and 4.  Figure 6 shows the spatial distribution of ground vibrations generated by the same axle load at the lower frequency component f = 18 Hz. It can be seen that ground vibrations in this case represent slightly distorted plane waves that propagate at larger angles in respect of the normal directions to the track.
For even lower frequency component, f = 17 Hz, some changes in the spatial distribution of generated ground vibrations become obvious, as can be seen from Fig. 7. Namely, the generated Rayleigh waves cease to be plane and become cylindrical, which indicates the end of constructive interference between waves generated by different sleepers for frequency components that are far away enough from the sleeper passage frequencies. The amplitudes of generated ground vibrations are further reduced in this case, as it can be expected from consideration of the frequency spectra of generated ground vibrations at the sides of the frequency peaks associated with sleeper passage frequencies.
Thus, resuming the above, it can be concluded that ground vibrations at and around the first sleeper passage frequency are generated as plane waves that are steering around the normal directions to the track either in the direction of the load motion -for higher frequencies, or in the opposite direction -for lower frequencies. Such a behaviour is very similar to the behaviour of electromagnetic 'travelling-wave antennas' used in microwave technology.   So far we discussed ground vibrations generated at and around the first sleeper passage frequency fsp (1) [Hz]. It can be shown that for the second sleeper passage frequency fsp (2) = 2(v/d), which is equal to 40 Hz in the case under consideration, the situation is very similar. This is illustrated by Fig. 8 showing the spatial distribution of generated ground vibrations by the same moving load at the frequency component f = 40 Hz.  A similar behaviour can be observed also for the third sleeper passage frequency and for higher order sleeper passage frequencies. However, because of the limited widths of railwaygenerated ground vibration spectra (usually in the range of 0 -100 Hz), only a small number of lowest order sleeper passage frequencies can be important.

Conclusions
It has been demonstrated in this paper that, at sleeper passage frequencies, railway-generated ground vibrations represent plane waves propagating symmetrically away from the track in the normal directions to it. At frequencies slightly above or slightly below sleeper passage frequencies, generated ground vibrations still remain plane waves, but their radiation takes place at certain angles in respect of the normal directions to the track. These radiation angles depend on train speed, on frequency and on sleeper periodicity. For frequencies slightly above sleeper passage frequencies, the above-mentioned radiation angles are counted towards the direction of the train (axle load) motion, whereas for frequencies slightly below sleeper passage frequencies, the radiation angles are counted toward the opposite direction.
The above-mentioned properties of railway-generated ground vibrations at and around sleeper passage frequencies could be used for their more efficient observation and monitoring.