DISPERSION CHARACTERISTICS OF SOUND-WAVES IN A TUNNEL WITH AN ARRAY OF HELMHOLTZ RESONATORS

This paper examines dispersion characteristics of sound waves propagating in a tunnel with an array of Helmholtz resonators connected axially. Assuming plane waves over the tunnel's cross section except a thin boundary layer, weakly dissipative effects due to the wall friction and the thermoviscous diffusivity of sound are taken into account. Sound propagation in such a spatially periodic structure may be termed “acoustic Bloch waves.” The dispersion relation derived exhibits peculiar characteristics marked by emergence of “stopping bands” in the frequency domain. The stopping bands inhibit selectively propagation of sound waves even if no dissipative effects are taken into account, and enhance the damping pronouncedly even in a dissipative case. The stopping bands result from the resonance with the resonators as side branches and also from the Bragg reflection by their periodic arrangements. In the “passing bands” outside of the stopping bands, the sound waves exhibit dispersion, though subjected intrinsically not only to the weak damping due to the dissipative effects but also to the weak dispersion due to the wall friction. Taking a plausible example, the dispersion relation and the Bloch wave functions for the pressure are displayed. Finally the validity of the continuum approximation for distribution of the Helmholtz resonators is discussed in terms of the dispersion relations.© 1995, Acoustical Society of America. All rights reserved.