本文以彈簧與質量塊組成的週期系統為基礎，提出單共振模型與雙共振模型，並將模型化作等效質量或等效彈簧常數，發現在共振頻附近時會出現等效負質量或等效負彈簧常數，並以數值方法探討負等效材料性質的範圍與色散能隙的對應情形，發現兩者會對應不上，並提出數學上與物理上的解釋，再以有限元素軟體ANSYS分別探討兩個模型在不同頻率下的波傳行為，然後計算晶格內的做功情形以及解釋波衰減的原因；而雙共振模型在適當的材料參數下可出現一條負群速度色散曲線，該條色散曲線經ANSYS驗證後發現有相速度與群速度相反的狀況，即是該條色散曲線具有雙負現象。
為了達成雙負聲學超常材料，將兩模型以串聯結構或並聯結構結合，並調整材料參數，使兩模型的負等效材料性質的範圍重疊，再以有限元素軟體ANSYS分別探討各結構的波傳行為，發現不管在串聯或並聯下，只有兩個單共振模型組合無法有雙負現象，其他的組合皆能達成雙負現象，有些組合甚至有多條色散曲線具有雙負現象，而雙負現象不只會出現在兩個子結構的負等效材料性質的重疊區間內，也會出現在兩個子結構的能隙重疊區間內。

This research bases on period system which consist of spring and mass. We propose single resonance model and dual resonance model. An effective mass or effective spring constant is developed to represent these models. Negative mass or negative spring constant is found in the vicinity of the resonance frequency. We compare range of negative property with band gap by numerical methods, and find ranges of the two were inconsistent. We propose an explanation of the mathematics and physics. Finite element software ANSYS is employed to investigate wave propagation of the two models at different frequencies. Work done of unit cell is calculated to explain wave attenuation. Dual resonance model could have one negative group velocity dispersion curve in appropriate material parameters. We verify the dispersion curve by ANSYS and find phase velocity opposite to group velocity. In other words, the dispersion curve has double negative phenomenon.

For procuring double negative acoustic metamaterial, we combine the two models in series or parallel and adjust material constant to overlap range of negative property. We use ANSYS to investigate wave propagation on each structure and find that no matter in series or in parallel, only two single-resonance combination does not have a double negative phenomenon, other combinations have double negative phenomenon, and even some combinations have multiple dispersion which possessing double negative phenomenon. Double negative phenomenon not only appear in overlap of negative property of two model but also appear in overlap of band gap of two model.

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