Effects of higher-order wave modes on periodic array of circular Helmholtz resonators

The Helmholtz resonator is a reactive-type acoustic silencer characterized by its narrow band of sound attenuation at the frequency of resonance. As a branch element, the acoustic performance can be significantly improved by setting periodically several Helmholtz resonators along with a pipe system. This arrangement provides the Bragg bandgap where the wave cannot propagate in a specific frequency region. However, the assumption of plane wave propagation may not describe correctly the reactive behavior at duct-branch and branch-cavity interfaces. For the present work, the effects of nonplanar wave modes are numerically investigated in a periodic arrangement of circular Helmholtz resonators coupled at piping systems. The acoustic impedance of a single resonator is calculated by applying the bidimensional analytical solution in order to take into account the nonplanar wave behavior at the discontinuity regio between the main duct and the branch region and its cavity. The transfer matrix approach is implemented in order to compute the periodic condition efficiently. From the transfer matrix of the resonator-duct system, the identification of bandgaps is performed by calculating the Bloch wavenumber. Moreover, by applying the Floquet-Bloch theorem, the transmission loss and the sound pressure level can be computed directly from the transfer matrix. The results are compared with other numerical methods.