本研究分為兩部分，一部份在探討類週期結構的一維聲子晶體的能帶結構及波傳行為，一部分在探討一維梯度厚度排列之層狀結構內波傳現象，構成層狀材料分別為正、負質量密度，負質量密度可藉由聲學超常材料所達成。
第一部分將對類週期結構的一維聲子晶體的能帶結構作探討，先以轉移矩陣來模擬彈性波垂直入射週期性結構，討論材料參數間的差異以及晶格厚度間的比例對能帶結構的影響，可以發現選用材料波速差異大或是改變其填充比例都可以增加其聲子能隙的寬度，接著探討類週期結構聲子晶體，瞭解厚度調變對波傳行為及能帶結構的影響。
第二部分在探討一維層狀正負質量密度介質以梯度厚度變化的方式堆疊內的波傳行為，在長波極限之假設下，可將其等效為一異向性材料，而此異向性材料具有特殊之波傳準直性，進而探討不同厚度調變對波傳行為的影響，分別觀察出結構邊界最大強度及沿結構傳遞方向上的壓力場強度分佈，發現在具厚度調變的層狀結構內波傳具有方向性，並觀察其雙向穿透率之落差，此波傳方向的選擇性行為可作為聲學二極體應用，並研究質量密度參數對聲學二極體之雙向穿透率的影響。

This research was divided into two parts. In the first part, we investigated the band structure of one-dimensional quasi-periodic phononic crystal using transfer matrix method. The elastic wave propagation inside both the periodic and quasi-periodic structure was analyzed. The quasi-periodic structures with various thickness modulations were studied and the band structures were compared with the periodic one in order to understand the modulation effect. Meanwhile, the effects of material parameters and the filling ratios of the periodic thickness on band structure were examined. It can be found that both the constituted materials and the filling ratio would change the band gaps.
In the second part, we investigated one-dimensional graded thickness layer structure constituting with positive and negative mass density medium. Under the assumption of long-wavelength limit, the layer structure can be modeled as an anisotropic material which have collimation phenomenon. Thus, we investigated the collimation phenomenon of the anisotropic structure with periodic arrangement and various thickness modulations. The propagation behavior was analyzed by examining the maximum intensity on the structure boundary and the pressure field distribution along the propagation direction. From finite element simulation, we found that the transmissions were direction dependent for the layer structure with graded thickness. The results showed that when the anisotropic structure with various thickness modulations, this structure can act as an acoustic diode. Meanwhile, we also discussed the mass density effect on the transmission contrast ratio of the diode.

[1] E. Yblonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. Vol. 58, no. 20, pp. 2059-2062, 1987.
[2] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. Vol. 71, no. 13, pp. 2022-2025, 1993.
[3] M. Ruzzene and A. Baz, “Attenuation and localization of wave propagation in periodic rods using shape memory inserts,” Smart Mater. Struct., Vol. 9, pp. 805-816, 2000.
[4] M. L. Wu, L. Y. Wu, W. P. Yang and L. W. Chen, “Elastic wave band gaps of one-dimensional phononic crystals with functionally graded materials,” Smart Mater. Struct., Vol. 18, pp. 115013, 2009.
[5] X. L. Su, Y. W. Gao, Y. H. Zhou, “The influence of material properties on the elastic band structures of one-dimensional functionally graded phononic crystals,” Journal of Applied Physics, Vol. 112, pp. 123503, 2012.
[6] A. L. Chen, Y. S. Wang, “Study on band gaps of elastic waves propagating in one-dimensional disordered phononic crystals,” Physica B., Vol. 392, pp. 369-378, 2007.
[7] Z. H. Wang, C. Guo, and W. Jiang, “Omnidirectional reflection extension in a one-dimensional superconducting-dielectric binary graded photonic crystal with graded geometric layers thickness,” Progress In Electromagnetics Research Letters, Vol. 42, pp. 13-22, 2013
[8] Y. Z. Xie, H. F. Qi, M. Zhao, H. Fang, J. Gao and X. G. Zhang, “Thickness-modulated one-dimensional periodic phononic crystal,” Advanced Materials Research, Vol. 750-752, pp. 1207-1210, 2013.
[9] J. B. Pendry, A. J. Pendry and W. J. Stewart, “Extremely Low Frequency Plasmons in Metallic Mesostructures”, Phys. Rev. Lett. 76, 4773, 1996.
[10] J. B. Pendry and A. J. Holden, “Low frequency plasmas in thin wire structures”, J. Appl. Phys. 10, pp. 4785-4809, 1998.
[11] J. B. Pendry, A. J. Holden and D. J. Robbins, “Magnetism from Conductors and Enhanced Nonlinear Phenomena”, IEEE 47, pp. 2075-2084, 1999.
[12] D. R. Smith, D. C. Vier and W. Padilla, “Loop wire medium for investigating plasmons at microwave frequencies”, Appl. Phys. Lett. 75, pp. 1425-1427, 1999.
[13] D. R. Smith, R. A. Shelby and S. Schultz, “Experimental Verification of a Negative Index of Refraction”, Science 292, pp. 77-79, 2001.
[14] D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors”, Phys. Rev. Lett. 90, 077405, 2003.
[15] Z. Liu and P. Sheng, “Locally Resonant Sonic Materials”, Science 289, pp. 8-10, 2000.
[16] C. Sun, N. Fang and X. Zhang, “Ultrasonic metamaterials with negative modulus”, Nature Materials 5, pp. 452-454, 2007.
[17] L. Jensen and C. T. Chan, “Double-negative acoustic metamaterial”, Phys. Rev. E 70, 055602, 2004.
[18] P. Sheng, “Introduction to wave Scattering, Localization and Microscopic Phenomena, 2nd ed”, Springer, Berlin, 2006.
[19] M. Ambati, N. Fang, C. Sun, and X. Zhang, “Surface resonant states and superlensing in acoustic metamaterials”, PHYSICAL REVIEW B 75, 195447, 2007.
[20] T. Y. Chiang, L. Y. Wu, C. N. Tsai, L. W. Chen, “A multilayered acoustic hyperlens with acoustic metamaterials”, Appl Phys A, 2011.
[21] 黃婉婷。螺旋型聲學雙曲透鏡與超常材料波傳現象之探討。碩士論文。國立成功大學製造資訊與系統研究所。2013。