Fig. 10: Schematic representation of the COMSOL model used to simulate the reflection experiment in the impedance tube for the samples shown in Fig. 2d–f.

The simulations were carried out in the frequency domain using the Pressure Acoustics module of COMSOL Multiphysics, which solves the Helmholtz equation for the acoustic pressure field. Neumann boundary conditions were applied to the lateral walls to emulate the rigid boundaries of the impedance tube, while a Perfectly Matched Layer (PML) was implemented at the left end of the model to absorb outgoing waves. A harmonic plane wave was introduced in the air domain using a “background pressure field” excitation. In the perforations corresponding to the first, second, and third hierarchical levels (HL1, HL2, HL3), the Thermoviscous Acoustics (TvA) module was used to capture viscous and thermal losses within narrow channels. This figure shows one-eight of the modeled geometry for HL3. Additional labels in the figure include: IP = Integration Plane, where the reflected pressure is calculated; PA = domain where the Poroacoustics module is applied.