Review of the theory of the compression wave generated when a high-speed train enters a tunnel

Abstract

A review is given of the analytical theory of the compression wave generated when a high-speed train enters a tunnel. The waveform can be calculated exactly at subsonic train Mach numbers for a tunnel consisting of an unflanged cylinder of semicircular cross-section, which has been the subject of extensive experimental study. An analysis of this case reveals how an approximate theory can be developed for arbitrary tunnel portals and train Mach numbers as large as 0.4. Details are given for flanged portals of both semicircular and rectangular cross-sections. Non-linear steepening in a very long tunnel is responsible for an intense, environmentally harmful Micropressure wave, which propagates as a pulse from the distant tunnel exit when the compression wave arrives, with amplitude proportional to the maximum gradient in the compression wavefront. The theory permits the design of tunnel portals that greatly increase the initial wave thickness (thereby tending to inhibit wave steepening), either by flaring the portal or by installing a tunnel entrance ‘hood’, which allows high-pressure air produced by the train to be vented away through windows in the sidewalls. A formula is given for the ‘optimum’ flared portal, which produces a pressure gradient across the wavefront that is constant and an overall minimum, so that the pressure in the wavefront increases linearly and provides the maximal protection against shock formation.