Reflection and transmission of a compression wave at a tunnel portal

Analyses are made of the interaction of the nonlinearly steepened, compression wavefront generated by a high-speed train in a tunnel with the tunnel portal ahead of the train. The 'micro-pressure' pulse emitted from the portal can rattle structures in nearby buildings, and the expansion wave reflected back towards the train can cause discomfort to passengers. It is concluded that the usual simplified approximation of one-dimensional propagation within the tunnel provides an adequate representation of interactions of the wave with the portal, and also with 'windows' in the tunnel wall near the portal. It is shown how a discrete distribution of windows can be used to produce a reflected expansion wave that varies linearly across the wavefront, and how the thickness of that wavefront can be made many times larger than the thickness of the incident compression wave profile. A detailed analysis of the wave radiated from the portal reveals that cumulative nonlinear effects of propagation over long distances make little or no contribution to the free-space radiation of the micro-pressure wave.