Acoustic solitary waves in a tunnel with an array of Helmholtz resonators

摘要

It is demonstrated theoretically that an acoustic solitary wave can be propagated in a tunnel with a periodic array of Helmholtz resonators, if the dissipative effects are made negligibly small. As wave propagation in such a periodic system is known as the Bloch waves, the array can give rise to the dispersion necessary to formation of the solitary wave. Explicit profiles of the solitary waves are obtained by solving the steady-wave solutions to the nonlinear wave equations derived previously. It is found that the solitary wave is compressive and its propagation speed is slower than the usual sound speed a0, i.e., subsonic, but faster than a0(1 – /2) in the linear long-wave limit, being a small parameter representing the ratio of the cavity's volume to the tunnel's volume per axial spacing between the neighboring resonators. As the propagation speed approaches the upper bound, the height of the solitary wave increases to approach the limiting height, while as the speed approaches the lower bound, the solitary wave tends to be the soliton solution of the Korteweg–de Vries equation. It is also found that while no solutions exist for the speed below the lower bound, the shock wave may be propagated for the speed above the upper bound, i.e., supersonic, but accompanying nonlinearly oscillatory wavetrain downstream.