Propagation of nonlinear acoustic waves in a tunnel with an array of Helmholtz resonators

It is proposed that an array of Helmholtz resonators connected to a tunnel in its axial direction will suppress the propagation of sound generated by a travelling train and especially the emergence of shock waves in the far field. Under the approximation that the resonators may be regarded as continuously distributed, quasi-one-dimensional formulation is given for nonlinear acoustic waves by taking account of not only the resonators but also the wall friction due to the presence of a boundary layer and the diffusivity of sound. For a far-field propagation, the spatial evolution equation coupled with the equation for the response of the resonator is then derived. The linear dispersion relation suggests that the resonators, if appropriately designed, enhance the dissipation and give rise to the dispersion as well. By solving initial-value problems for the evolution equation, the array of resonators is proved to be very effective in suppressing shock waves in the far field. The resonators themselves fail to counteract shock waves once formed, but rather prevent their emergency by rendering acoustic waves dispersive. By this dispersion, it becomes possible, in a special case, for an acoustic soliton to be propagated in place of a shock wave.