Majdi O. Gzal ^1* Joshua R. Tempelman ² Kathryn H. Matlack ¹ Alexander F. Vakakis ¹

• ¹ Department of Mechanical Science and Engineering, The Grainger College of Engineering, University of Illinois at Urbana–Champaign, Urbana, United States
• ² Los Alamos National Laboratory (DOE), Los Alamos, New Mexico, United States

We present a systematic and rigorous analytical approach, based on the transfer matrix methodology, to study the existence, evolution, and robustness of subwavelength topological interface states in practical multilayered vibroacoustic phononic lattices. These lattices, composed of membrane-air cavity unit cells, exhibit complex band structures with various bandgaps, including Bragg, band-splitting induced, local resonance, and plasma bandgaps. Focusing on the challenging low-frequency range and assuming axisymmetric modes, we show that topological interface states are confined to Bragg-like band-splitting induced bandgaps. Unlike the Su-Schrieffer-Heeger model, the vibroacoustic lattice exhibits diverse topological phase transitions across infinite bands, enabling broadband, multi-frequency vibroacoustics in the subwavelength regime. We establish three criteria for the existence of these states: the Zak phase, surface impedance, and a new reflection coefficient concept, all derived from transfer matrix components. Notably, we provide an explicit expression for the exact location of topological interface states within the band structure, offering insight for their predictive implementation. We confirm the robustness of these states against structural variations and identify delocalization as bandgaps narrow. Our work provides a complete and exact analytical characterization of topological interface states, demonstrating the effectiveness of the transfer matrix method. Beyond its analytical depth, our approach provides a useful framework and design tool for topological phononic lattices, advancing applications such as efficient sound filters, waveguides, noise control, and acoustic sensors in the subwavelength regime. Its versatility extends beyond the vibroacoustic systems, encompassing a broader range of phononic and photonic crystals with repetitive inversion-symmetric unit cells.