Search

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.

Beyond classical Bragg diffraction, we report on new sub-Bragg phenomena to achieve enhanced sound wave tunability in the low-frequency range in a multilayered acoustic metamaterial system. Remarkably, we reveal and study the formation of genuinely sub-wavelength Bragg-like band-splitting induced bandgaps, generalizing the band-folding induced bandgaps in the literature. Additionally, we propose a methodology to widen sub-wavelength local resonance bandgaps by simultaneously hosting two local resonances within the same bandgap. These sub-Bragg bandgaps are realized in an axisymmetric vibroacoustic phononic metamaterial consisting of repetitive multilayered unit cells, each composed of two layers of membrane-cavity resonators. The resulting coupled sound-structure interaction system is analytically solved in closed form. Furthermore, the studied multilayered vibroacoustic metamaterial exhibits sub-wavelength acoustical transparency, akin to electromagnetically induced transparency, and an acoustic analogue of low-frequency plasma oscillations, resulting in a plasma bandgap where wave propagation is entirely prohibited below the plasma frequency. These sub-wavelength phenomena, achieved predictively and well below the ubiquitous first-order Bragg diffraction range, provide broadband attenuation and superior wave manipulation capabilities for low-frequency sound. This highlights the potential of utilizing sound-structure interaction across diverse physical applications. Moreover, our findings can be extrapolated to wave propagation in broader classes of physical systems where sub-wavelength phenomena are crucial, including phononic and photonic crystals of more general configurations.

This study investigates a vibroacoustic phononic metamaterial system composed of repeated monolayered membrane-air cavity unit-cells to assess its efficacy in controlling sound waves. Assuming low-frequency axisymmetric modes, the coupled membrane-cavity vibroacoustic system for a representative unit-cell is solved entirely analytically. Unlike previous research that relied on an infinite series of eigenfunctions, our analysis offers a single-term exact solution for the membrane's displacement field, fully accounting for coupling with the acoustic cavities. Utilizing the transfer matrix method and the Bloch-Floquet theorem, we offer a comprehensive analytical characterization of the band structure, including closed-form analytical expressions for determining the bounding frequencies of the bandgaps and the dispersion branches. Interaction between Bragg and local resonance bandgaps is examined by adjusting Bragg bandgap positions, with detailed mathematical descriptions provided for their overlapping and transition. Additionally, a "plasma bandgap" analogous to metallic plasma oscillations is identified, with a derived analytical expression for its frequency. First two passbands remain robust against cavity depth variations, limiting wave manipulation capabilities. Analysis of the finite phononic system involves constructing the global transfer matrix to study natural frequencies and scattering coefficients. Interaction between Bragg and local resonance bandgaps in finite systems results in ultra-narrow passbands, creating transparency windows analogous to electromagnetically induced transparency by quantum interference. This theoretical framework enables precise characterization and engineering of bandgaps in the monolayered vibroacoustic phononic metamaterial, highlighting its potential for controlling low-frequency sound wave propagation across multiple frequencies.