Keywords: non-linear waves, ballast and slab tracks, railway tunnels, boundary layer.

The evolution of the profile of the compression wave produced by the entrance of a train in a tunnel is smoother in ballast tracks as compared with concrete slab tracks. Early authors suggested that the pores of the ballast could play an important role but, to our knowledge, no compelling explanation has yet been put forward to describe that mechanism. Some considered that unusually high shear stresses would appear over the rough ballast surface in the region of the front wave, due to the flow unsteadiness; others suggested that a Helmholtz resonance could take place inside the pores of the ballast.

This paper proposes that in the region of the front wave, a special kind of boundary layer exists, that produces a conically diverging flow that ultimately affects the pressure distribution. The pressure is almost uniform over the cross section but not the fluid velocity. Wall effects (shear, heat or mass transfer) only reach the flow close to the wall, leaving a central core of flow potential and adiabatic. That central streamtube is affected by the wall indirectly, in virtue of a "displacement thickness" of the boundary layer.

The characteristics of that boundary layer depend on (1) a quasi-steady state quasi-isentropic compression of the air inside the pores of the ballast; (2) on a greater heat transfer rate between the air compressed inside the pores and the solid boundary, because of the much larger contact surface; (3) the Archimedes action of the pressure gradient of the front wave on the stones of the ballast; and (4) on a greater head loss, although not directly at the wave front region but as a cumulative effect over the length travelled by the pressure wave behind the front wave.

The problem is addressed in the stationary frame of reference of the flow, that of the front wave. In that frame of reference the fluid and the walls move at the speed of sound but the steady-state analysis is much simpler.

A numerical example is presented of a front wave, in the stationary frame of reference, inside a tunnel with no wall effects and a tunnel with boundary displacement. The first case is the limit of a smooth tunnel without skin friction, the second case represents a ballast tunnel. The results show that the pressure gradients are be smoothed in the ballast tunnel and enhanced in the smooth tunnel.

As a means to better attenuate the gradient of the wave front it is suggested that all the surfaces of tunnels be cast with artificial roughness elements similar to the superficial irregularities of the ballast. This method could turn slab tunnels equivalent to ballast tunnels with respect to the attenuation of pressure waves.

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