单层振动声学超材料的解析研究
Analytical Study of a Monolayered Vibroacoustic Metamaterial
AI总结：
本研究针对单层薄膜-空气腔振动声学声子超材料，提出薄膜位移场的单项精确解析解，结合传递矩阵法与Bloch-Floquet定理表征带结构，分析多种带隙的相互作用及有限系统的超窄通带效应，为低频声波调控提供理论支撑。
摘要：
本研究探究了由重复的单层薄膜-空气腔单元胞组成的振动声学声子超材料系统，以评估其在调控声波方面的效能。在低频轴对称模式的假设下，我们对代表性单元胞的薄膜-空腔耦合振动声学系统进行了完全解析求解。与此前依赖无穷级数本征函数的研究不同，我们的分析给出了薄膜位移场的单项精确解，充分考虑了其与声学空腔的耦合作用。结合传递矩阵法和Bloch-Floquet定理，我们对带结构进行了全面的解析表征，包括用于确定带隙边界频率和色散分支的闭式解析表达式。通过调整Bragg带隙的位置，我们研究了Bragg带隙与局域共振带隙之间的相互作用，并对二者的重叠和转变过程给出了详细的数学描述。此外，我们还识别出一种类似于金属等离子体振荡的“等离子体带隙”，并推导了其频率的解析表达式。前两个通带不受空腔深度变化的影响，这限制了波的调控能力。针对有限声子系统的分析通过构建全局传递矩阵来研究固有频率和散射系数。有限系统中Bragg带隙与局域共振带隙的相互作用会产生超窄通带，形成类似于量子干涉引发的电磁诱导透明的透明窗口。该理论框架能够对单层振动声学声子超材料的带隙进行精确表征和设计，凸显了其在多频段调控低频声波传播方面的潜力。
英文摘要：
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.
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