Non-linear acoustic analysis of the pressure rise of the compression wave generated by a train entering a tunnel

Abstract Hara's formula is used to predict the pressure rise of a compression wave generated by a high-speed train entering a tunnel. The acoustic theory is used to predict its waveform. Hara's formula was derived from the shock theory. Meanwhile, the acoustic theory produces the same non-dimensional formula as Hara's formula up to order O M 4 , where M is the train Mach number, when the non-linear acoustic effect in the vortex sound theory is considered. However, the relationship between the shock and acoustic theories remains unclear because the closed formula produced by the vortex sound theory apparently differs from that of Hara's formula. This study shows that Hara's formula can be deductively obtained from the non-linear acoustic theory. A general equation used to predict the compression waveform with a relative error of O p / ρ 0 c 0 2 , where p / ρ 0 c 0 2 is the non-dimensional acoustic pressure to the atmospheric density ρ 0 and the sound speed c 0 , is shown herein using the non-linear acoustic theory.