ACOUSTIC ANALYSIS OF THE PRESSURE FIELD GENERATED BY ACCELERATION OF A TRAIN IN A LONG TUNNEL

Abstract

This paper solves the acoustic wave equation for a pressure field generated by acceleration of a train in a circular tunnel of infinite length. The motion of the train is taken into account through the source term of the equation in the form of a pair of positive and negative monopoles, moving along a center axis of the tunnel, whose velocity is given by a step function of time. The solution is presented in the form of the eigenfunction-expansion and is evaluated asymptotically. The pressure field consists of the steady part due to the train moving constantly and of the transient part due to the impulsive acceleration by the step function. The former is simply the subsonic flow field around the train, while the latter is characterized by propagation of singular surfaces which are generated by the initial acceleration in the form of spherical wavefront and are reflected repeatedly at the cylindrical tunnel wall. The singularity is classified into two types, one being the type of a delta function and the other of a cotangent function. Intersections of the singular surfaces with a plane containing the axis exhibit a "diamond pattern". It is shown that this pattern disappears eventually and a distant field settles down to the one-dimensional field calculated simply by the wave equation averaged over the tunnel's cross-section.