Analytical Study of a Monolayered Vibroacoustic Metamaterial

Abstract: 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.