Acoustic analysis of the pressure field in a tunnel generated by entry of a train

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This paper determines the pressure field in a tunnel, generated by the entry of a travelling train. A theoretical model of the problem is first given within the framework of linear acoustic theory. The train's motion is taken into account through a source term in the wave equation, in the form of a pair of acoustic monopoles of the same magnitude but of opposite sign. An axially symmetric pressure field in a tunnel of circular cross–section and of semi–infinite length is obtained in closed form. Radiation of pressure waves into free space outside the tunnel is discarded by imposing an undisturbed boundary condition at the tunnel entrance. It is shown by asymptotic evaluation of the solutions that one–dimensional and non–dispersive wave propagation survives after the initial transients decay. This provides an estimate of the maximum pressure observed far ahead of the train. It is found from the transient solutions that two types of singularity, due to a delta function and to a cotangent function, emerge in the pressure field. The singular surfaces form a ‘diamond pattern’ in the field by repetition of reflections, at the cylindrical surface of the tunnel wall, of the spherical wavefront generated on entry into the tunnel. The magnitudes of the singularities are evaluated for the asymptotic behaviour of the pattern as time elapses. In order to check the validity of the theoretical model and analytical results, numerical computations are carried out by solving Euler equations directly, taking account of the radiation into free space. Under a situation corresponding to the theoretical model, the numerical results are found to agree well with the analytical results. As long as the blockage ratio is as small as 0.01 or 0.1, the linear acoustic theory is sufficient to describe the pressure field, even for a train Mach number 0.44. But when scattering of pressure waves by the wall edge at the tunnel entrance as the train approaches it is taken into account, the diamond pattern disappears. Instead, the transient field appears to be almost one–dimensional, which may be described by the model derived by averaging the source over the tunnel's cross–section.