聲子晶體是由兩種或兩種以上不同的彈性材料或流體進行週期性排列所形成的結構。此聲子晶體結構存在聲子能隙現象，在任何角度或頻率入射下，可隔絕聲波或彈性波在聲子晶體上的傳遞，許多應用元件因此被設計出，如濾波器或隔音裝置。而破壞完美聲子晶體的週期性結構，會在能隙裡面產生缺陷模態，利用此缺陷模態可用來做為波導或共振腔的設計。接著進一步研究發現在其傳導區擁有異常之頻散現象，發現聲子晶體具有負折射特性。而近年來聲學梯度材料在操縱聲波或彈性波這方面扮演了很重要的角色，但聲學梯度材料有非均質的特性且實際上不容易實現，所以可利用聲子晶體來等效聲學梯度材料。在不同的頻率入射到不同填充率或不同材料的聲子晶體下所得到的折射率會不同，藉由此原理設計梯度折射率聲子晶體去操縱聲波的行進方向，進而設計許多聲波元件，如透鏡、波導、吸收器等等，使得該方面的研究更成為最近學術界的熱門話題。
本文以平面波展開法求得二維聲子晶體的色散曲線，並利用色散曲線求得分別在不同的無因次化頻率及不同填充率下的聲子晶體折射率，進而利用梯度聲子晶體來設計聲學黑洞，再利用有限元素軟體模擬其聲波折射情形。此黑洞結構分為內層和外層兩部分，當一平面聲波入射進外層結構時，會被折射進入內層結構並且加以吸收。並且討論其適當的工作頻寬。此結構可達到噪音控制、聲波吸收的效果，並更加擴展了聲子晶體在工程上的應用面。

Phononic crystals (sonic crystals) are periodic elastic composite materials. Phononic crystals can typically exhibit complete band gaps in which acoustic/elastic waves are forbidden in any direction. Many applied devices have been designed by using band gap effects, such as elastic/acoustic filters and noise/vibration isolations. By creating crystal defects in a perfect phononic crystal, the elastic/acoustic waves can be confined in localized modes. There exist the defect modes in the absoulate band gap. During the past years, there has a great deal of attention in dispersion characteristics. Negative refraction behavior has been investigated in phononic crystals. Recently, a graded index medium plays an important role in controlling elastic/acoustic wave propagation, but this will lead to difficult in the practical implementation. A possible solution needs to rely on the phononic crystals to replace the graded index medium. The effective refractive index of a phononic crystal can be tuned by carefully adjusting the filling fraction, the lattice constant, or choosing appropriate material parameter. The characteristic has been employed to develop many applications for acoustic devices, such as lenses, waveguides, and absorbers.
The plane wave expansion method are used to obtained the dispersion relations and the effective refractive index of the phononic crystal. An acoustic black hole is designed by using graded index phononic crystals. An acoustic black hole consists of the outer shell and the inner core. The finite element method (FEM) is employed to simulate the acoustic wave propagation in the acoustic black hole. Numerical simulations show that the acoustic waves can be bent toward the central area in the outer shell and absorbed by the inner core. Thus, the operating frequency is broadband. Such a mechanism of an omnidirectional acoustic absorber could have more potential to design various applications, such as sound absorption and noise control.

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