本篇論文主要探討聲波在超穎材料介質 (metamaterial media) 中的波傳特性。我們首先回顧超穎材料發展史及介紹超穎材料各項熱門研究主題，接下來介紹在本論文研究中會使用到的各項理論模擬計算方法以及算式的推演，包括：平面波展開法、Dirichlet to Neumann 法、多重散射法、聲波電路圖以及有限時域差分法 (使用Z transformation)。此篇論文的研究主題包括：探討不同晶格排列的光子晶體平板對點波源聚焦點像的影響、聲子晶體平板負折射、聲子晶體波導耦合條件限制，以及管中聲波能量的傳遞特性及分佈之探討。最後總結此篇論文的主要結論以及探討未來的展望。

In this thesis we discuss the properties of acoustic wave in metamaterial media. We first review the history of metamaterials and introduce activity topics of metamaterial research. The numerical methods are introduced and relevant formulas are derived in chapter 2. These numerical methods include: Plane wave expansion method, Dirichlet to Neumann method, Multiple scattering method, Effective circuit diagram of acoustic system, and Finite difference time domain (FDTD) method based on Z transforms. Four topics are included in this thesis: the stability of subwavelength imaging of a photonic crystal slab, the off-plane shear waves focusing of a phononic crystal slab, phononic crystal waveguide of array cylinders, and acoustic field propagation in a dispersive medium. Finally, we summarized the main conclusions and discuss the future work.

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