Sound waves in continuum models of periodic sonic black holes

Abstract

In this work we address the study of continuum models of periodic sonic black holes (SBHs) at duct terminations. A conventional SBH combines a decreasing power-law profile with a wall admittance in such a way that sound waves entering the SBH slow down, while their amplitude increases and their wavelength decreases as they approach the duct termination. In an ideal scenario, the wave is trapped and dissipated inside the SBH so that no reflection occurs. Instead of dealing with conventional SBHs, we are interested here in periodic SBHs. These have the potential to extend the application range of SBHs below the cut-on frequency thanks to the formation of stopbands, among other significant aspects. First, the case of an SBH cell placed between semi-infinite ducts is analyzed and it is shown how it reduces not only sound reflection but also sound transmission. The acoustic pressure inside the SBH is governed by a modified Webster equation that is solved in weak form using a basis of Gaussian functions. The dependence of the reflection, transmission and absorption coefficients for different SBH parameters and boundary conditions is analyzed in detail. Next, the case of an infinite periodic SBH lattice is considered. The nullspace method (NSM) is used to impose the periodic boundary conditions of the problem. Complex dispersion curves are computed and the separate roles played by the SBH power-law profile and wall admittance on the dispersion curves and pressure distribution inside the cells are examined. Bragg scattering turns out to be the main mechanism behind bandgap formation and wall admittance is essential to achieve the SBH effect. A parametric study follows, showing the influence of the damping, the residual radius, the SBH order and its length on the imaginary part of the complex dispersion curves. Finally, the reflection, transmission and absorption coefficients for a finite 3-cell SBH are investigated. Although most of the research on SBHs has focused on practical application designs, it is believed that a thorough understanding of the performance of continuum models of SBHs could be very useful for the former.