On the validity of periodic boundary conditions for modelling finite plate-type acoustic metamaterials

Plate-type acoustic metamaterials (PAMs) are thin structures that exhibit antiresonances with high sound transmission loss (STL) values, making PAMs a promising new technology for controlling tonal noise in the challenging low-frequency regime. A PAM consists of rigid masses periodically attached to a thin baseplate. The periodicity of PAM can be exploited in simulations, allowing to model only a single unit cell using periodic boundary conditions. This approach essentially represents the PAM as an infinite structure, but real PAM implementations will always be finite and influenced by boundary conditions. In this paper, extensive numerical simulations of different PAM configurations have been performed to study the performance of finite PAM compared to infinite PAM. The results indicate that as the number of unit cells in a finite PAM increase, the STL converges toward that of an infinite PAM. The impact of the finite PAM edge boundary conditions becomes negligible at some point. Based on the numerical results, a simple criterion is proposed to determine a priori how many unit cells are required in a finite PAM design to consider it quasi-infinite. This criterion aids in justifying unit cell models with periodic boundary conditions for efficient design optimizations in practical PAM applications.