随机失谐和准周期声子晶体中弹性波传播特性研究
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摘要
自从Kushwaha提出声子晶体的概念以来,许多科学家开始致力于研究这种具有声带隙特征的人工周期弹性/声晶格结构。声子晶体的本质特征是它的声子带隙(或称声波禁带),即处于声波带隙频率范围内的振动或波被禁止在声子晶体中传播。声子晶体的这一特性在声滤波器、声波导、振动隔离控制,噪音抑制和设计新的传感器等方面有着潜在的工程应用。但是在实际中由于随机参杂的杂质或制作误差往往会使周期结构出现一定程度上的失谐直至完全无序。失谐的存在会引起声子晶体中弹性波传播的局部化现象,这也为我们利用失谐实现对弹性波传播的控制提供思路。而周期性的丧失使问题的数学处理变得困难,某些表征参数也不再有效。本文研究了以下失谐声子晶体中弹性波传播的若干问题:
(1)引入局部化因子的概念,利用传递矩阵法计算分析了弹性波在一维谐调和随机失谐声子晶体中的传播和局部化特征,利用局部化因子表征了弹性波以任意角度斜入射时声子晶体的频带结构。结果表明:局部化因子是一个表征一维谐调和随机失谐声子晶体频带结构的有效参量,可以很好地描述其中的弹性波传播和局部化特性。
(2)将基于平面波展开的传递矩阵法推广用于计算局部化因子,研究了二维谐调和在某个方向随机失谐的声子晶体的频带结构和局部化特征。利用局部化因子表征了二维声子晶体沿布里渊区域的频带结构。结果表明:局部化因子可以很好地描述二维谐调和在一个方向随机失谐另一个方向周期排列的二维声子晶体的频带结构和局部化现象。
上述两问题的研究表明,局部化因子是描述一维失谐系统波动局部化行为的有效表征参量,其计算方法简便,其值在通带区间随着系统失谐度的增加而增加,在带隙区间随着系统失谐度的增加而减小,该局部化行为在高频区间尤为显著。
(3)利用超元胞法结合平面波展开法计算了二维随机失谐声子晶体的频带结构,其中随机失谐参量设为散射体的半径或位置。并将计算结果与利用有限元方法得到的响应结果做了比较,两者吻合得很好。研究表明:利用频散曲线表示的频带结构,其通带边缘出现了许多类似于缺陷态的平直带,这些平直带表示弹性波在其对应的频率下出现了局部化现象。
(4)准周期是另一类处于周期与无序之间的状态,本文将准周期声子晶体看做是谐调声子晶体以准周期特征失谐而形成的。分别利用传递矩阵法和基于平面波展开的传递矩阵法研究了一维准周期声子晶体(Fibonacci序列)和一个方向上具有Fibonacci排列、另一个方向谐调的二维声子晶体中的弹性波传播特征。并用局部化因子表示上述两种准周期系统的频带结构。计算结果表明:准周期声子晶体中弹性波传播也会表现出局部化现象,利用局部化因子可以很好地描述其中的弹性波传播行为;准周期结构与它对应的周期平均结构和随机失谐结构相比较,带隙数目相对较多但是带隙宽度小;通过向准周期结构中引入平移对称性可以得到真正的通带;通过引入镜面对称可以得到更多的频带;对准周期声子晶体来说在较低的频率区间也会出现带隙。最后利用超元胞结合平面波展开法初步探索了具有8重对称性的二维准周期声子晶体中弹性波的传播特征。
另外,本论文还对局部化因子、频散曲线、响应频谱等表征参量的特点进行了对比分析。
Since Kushwaha proposed the concept of "phononic crystal"(PNC),an artificial periodic elastic/acoustic structure that exhibits so-called "phononic band gaps",a great deal of attention has been focused on this kind of artificial lattice structures.The essential property of the PNC is its band gaps(or stopbands) which have numerous potential engineering applications such as acoustic filters,control of vibration isolation, noise suppression and design of new transducers.However in practical cases,disorder in certain degree or completely disorder always exists due to randomly distributed material defaults or manufacture errors during production process.Disorder in the PNCs will lead to the wave localization phenomenon which can be used to control the wave propagation.Because of the lack of periodicity the mathematic process becomes more difficult and some parameters used to describe the band structures fail.In this thesis,we study the following problems for elastic waves propagationg in disordered phononic crystals:
(1) The concept of the localization factor calculated by the transfer matrix method is introduced to describe the band structures and localization behaviors of elastic waves propagating obliquely in 1D perfect and randomly disordered PNCs.The results show that the localization factor is an effective parameter in characterizing the band structures and localization phenomenon of 1D perfect and randomly disordered PNCs.
(2) The plane-wave-expansion based transfer matrix method(PWE-based TMM) is generalized to calculate the localization factor which is introduced to describe the band structures and localization behaviors of 2D perfect PNCs and 2D PNCs randomly disordered in one direction and perfect in the other direction.The band structures in the first Brillouin Zone are discribed by the localization factors.The results show that the localization factor is an effective parameter in characterizing the band structures of 2D perfect PNCs and 2D PNCs randomly disordered in one direction and perfect in the other direction.
The studies of the above two problems show that the localization factor which can be calculated simplely is an effective parameter in characterizing the wave localization phenomenon of the 1D disordered systems.As the disorder of the system increases,the value of the localization factor increases in the pass bands and decreases in the band gaps.The localization behavior is more pronounced at higher frequencies.
(3) Combined with the plane wave expansion method,the supercell technique is used to calculate the dispersion curves of 2D randomly disordered PNCs in which the radius or locations of the scatterers are disordered.The band structures expressed by the dispersion curves have a good aggrernent with those by the frequency response calculated by the finite element method.And the results show that there appear many flat bands as the defected states in the band structures of the randomly disordered PNCs.This implies that the localization phenomenon exists at the corresponding frequencies of the flat bands.
(4) A quasi periodic system is another intermediate case between the perfectly ordered and completely disordered systems.Treating this system as a disordered one deviating quasi-periodically from a periodic one we study the properties of the wave propagating in the 1D(Fibonacci sequency) quasi periodic phononic crystals (QPNCs) and the 2D phononic crystal with Fibonacci sequency in one direction and translational symmetry in the other direction using the transfer matrix and PWE-based TMM,respectively.The band structures of the above two systems are expressed by the localization factors and response.The results show that the localization phenomenon appears when the wave propagates in the QPNCs,and that the localization factor is an effective parameter in characterizing the band structures of the QPNCs.The band gaps of the QPNCs are more and narrower than those of its periodic average structure and randomly disordered systems.Pure passbands appear if we introduce translational symmetry to the QPNCs.More structures in the frequency bands are obtained if the mirror symmetry is further introduced.For the QPNCs band gaps may appear in the low frequencies.At last, the supereell technique combined with the PWE method is used to conduct a primary research on the band structures of the 2D QPNC with 8-fold symmetry.
In addition,the parameters of the localization factor,the dispersion curve and the frequency response in charactering the properties of wave propagation and localization phenomenon are discussed and compared in this thesis.
(1)引入局部化因子的概念,利用传递矩阵法计算分析了弹性波在一维谐调和随机失谐声子晶体中的传播和局部化特征,利用局部化因子表征了弹性波以任意角度斜入射时声子晶体的频带结构。结果表明:局部化因子是一个表征一维谐调和随机失谐声子晶体频带结构的有效参量,可以很好地描述其中的弹性波传播和局部化特性。
(2)将基于平面波展开的传递矩阵法推广用于计算局部化因子,研究了二维谐调和在某个方向随机失谐的声子晶体的频带结构和局部化特征。利用局部化因子表征了二维声子晶体沿布里渊区域的频带结构。结果表明:局部化因子可以很好地描述二维谐调和在一个方向随机失谐另一个方向周期排列的二维声子晶体的频带结构和局部化现象。
上述两问题的研究表明,局部化因子是描述一维失谐系统波动局部化行为的有效表征参量,其计算方法简便,其值在通带区间随着系统失谐度的增加而增加,在带隙区间随着系统失谐度的增加而减小,该局部化行为在高频区间尤为显著。
(3)利用超元胞法结合平面波展开法计算了二维随机失谐声子晶体的频带结构,其中随机失谐参量设为散射体的半径或位置。并将计算结果与利用有限元方法得到的响应结果做了比较,两者吻合得很好。研究表明:利用频散曲线表示的频带结构,其通带边缘出现了许多类似于缺陷态的平直带,这些平直带表示弹性波在其对应的频率下出现了局部化现象。
(4)准周期是另一类处于周期与无序之间的状态,本文将准周期声子晶体看做是谐调声子晶体以准周期特征失谐而形成的。分别利用传递矩阵法和基于平面波展开的传递矩阵法研究了一维准周期声子晶体(Fibonacci序列)和一个方向上具有Fibonacci排列、另一个方向谐调的二维声子晶体中的弹性波传播特征。并用局部化因子表示上述两种准周期系统的频带结构。计算结果表明:准周期声子晶体中弹性波传播也会表现出局部化现象,利用局部化因子可以很好地描述其中的弹性波传播行为;准周期结构与它对应的周期平均结构和随机失谐结构相比较,带隙数目相对较多但是带隙宽度小;通过向准周期结构中引入平移对称性可以得到真正的通带;通过引入镜面对称可以得到更多的频带;对准周期声子晶体来说在较低的频率区间也会出现带隙。最后利用超元胞结合平面波展开法初步探索了具有8重对称性的二维准周期声子晶体中弹性波的传播特征。
另外,本论文还对局部化因子、频散曲线、响应频谱等表征参量的特点进行了对比分析。
Since Kushwaha proposed the concept of "phononic crystal"(PNC),an artificial periodic elastic/acoustic structure that exhibits so-called "phononic band gaps",a great deal of attention has been focused on this kind of artificial lattice structures.The essential property of the PNC is its band gaps(or stopbands) which have numerous potential engineering applications such as acoustic filters,control of vibration isolation, noise suppression and design of new transducers.However in practical cases,disorder in certain degree or completely disorder always exists due to randomly distributed material defaults or manufacture errors during production process.Disorder in the PNCs will lead to the wave localization phenomenon which can be used to control the wave propagation.Because of the lack of periodicity the mathematic process becomes more difficult and some parameters used to describe the band structures fail.In this thesis,we study the following problems for elastic waves propagationg in disordered phononic crystals:
(1) The concept of the localization factor calculated by the transfer matrix method is introduced to describe the band structures and localization behaviors of elastic waves propagating obliquely in 1D perfect and randomly disordered PNCs.The results show that the localization factor is an effective parameter in characterizing the band structures and localization phenomenon of 1D perfect and randomly disordered PNCs.
(2) The plane-wave-expansion based transfer matrix method(PWE-based TMM) is generalized to calculate the localization factor which is introduced to describe the band structures and localization behaviors of 2D perfect PNCs and 2D PNCs randomly disordered in one direction and perfect in the other direction.The band structures in the first Brillouin Zone are discribed by the localization factors.The results show that the localization factor is an effective parameter in characterizing the band structures of 2D perfect PNCs and 2D PNCs randomly disordered in one direction and perfect in the other direction.
The studies of the above two problems show that the localization factor which can be calculated simplely is an effective parameter in characterizing the wave localization phenomenon of the 1D disordered systems.As the disorder of the system increases,the value of the localization factor increases in the pass bands and decreases in the band gaps.The localization behavior is more pronounced at higher frequencies.
(3) Combined with the plane wave expansion method,the supercell technique is used to calculate the dispersion curves of 2D randomly disordered PNCs in which the radius or locations of the scatterers are disordered.The band structures expressed by the dispersion curves have a good aggrernent with those by the frequency response calculated by the finite element method.And the results show that there appear many flat bands as the defected states in the band structures of the randomly disordered PNCs.This implies that the localization phenomenon exists at the corresponding frequencies of the flat bands.
(4) A quasi periodic system is another intermediate case between the perfectly ordered and completely disordered systems.Treating this system as a disordered one deviating quasi-periodically from a periodic one we study the properties of the wave propagating in the 1D(Fibonacci sequency) quasi periodic phononic crystals (QPNCs) and the 2D phononic crystal with Fibonacci sequency in one direction and translational symmetry in the other direction using the transfer matrix and PWE-based TMM,respectively.The band structures of the above two systems are expressed by the localization factors and response.The results show that the localization phenomenon appears when the wave propagates in the QPNCs,and that the localization factor is an effective parameter in characterizing the band structures of the QPNCs.The band gaps of the QPNCs are more and narrower than those of its periodic average structure and randomly disordered systems.Pure passbands appear if we introduce translational symmetry to the QPNCs.More structures in the frequency bands are obtained if the mirror symmetry is further introduced.For the QPNCs band gaps may appear in the low frequencies.At last, the supereell technique combined with the PWE method is used to conduct a primary research on the band structures of the 2D QPNC with 8-fold symmetry.
In addition,the parameters of the localization factor,the dispersion curve and the frequency response in charactering the properties of wave propagation and localization phenomenon are discussed and compared in this thesis.
引文
[1]冯端.金国钧.凝聚态物理学(上卷)[M].高等教育出版社:北京.2003.
[2]Esaki L,Tsu.Superlattice and negative differential conductivity in semiconductors[M].IBM J Res Develop,1970.
[3]Yablonovitch E.Inhibited spontaneous emission in solid state physics and electronics[J].Physical Review Letters,1987,58:2059-2062.
[4]John S.Strong localization of photons in certain disordered dielectric superlattices[J].Physical Review Letters,1987,58:2486-2489.
[5]Yablonovitch E,Gmtter T J.Photonic band structure:the face-centered-cubic case[J].Physical Review Letters,1989,63:1950-1953.
[6]Yablonovitch E,Gmitter T J,Leung M L.Photonic band-structure-the face-centered-cubic case employing nonspherical atoms[J].Physical Review Letters,1991,67:2295-2298.
[7]付云起,袁乃昌,温熙森.微波光子晶体天线技术[M].国防工业出版社:北京,2006.
[8]Kushwaha M S,Halevi P,Martinez G,Dobrzynski L,Djafarirouhani B.Acoustic band structure of periodic elastic composites[J].Physical Review Letters,1993,71:2022-2025.
[9]Brekhovskikh L M.Waves in layered media[M].Academic:New York,1960.
[10]Ewing W M,Jardetsky W S,Press F.Elastic waves in layered media[M].McGraw-Hill:New York,1967.
[11]Guz' A N,Shulga N A.Dynamics of laminated and fibrous composites[J].Applied Mechanics Review,1992,45:35-60.
[12]Angel Y C,Achenbach J D.Harmonic waves in an elasic solid containg a doubly periodic array of cracks[J].Wave Motion,1987,9:377-385.
[13]Sigalas M M,Economou E N.Elastic and acoustic wave band structure[J].Journal of Sound and Vibration,1992,158:377-382.
[14]Martinez-Sala R,Sancho J,Sanchez J V,Gomez V,Llinares J,Meseguer F.Sound-attenuation by sculpture[J].Nature,1995,378:241-241.
[15]温熙森.光子/声子晶体理论与技术[M].科学出版社:北京,2006.
[16]黄昆著,韩汝琦改编.固体物理学[M].高等教育出版社:北京,1988.
[17]Kafesaki M,Penciu R S,Economou E N.Air bubbles in water:a strongly multiple scattering medium for acoustic waves[J].Physical Review Letters,2000,84:6050-6054.
[18]Kuang W M,Hou Z L,Liu Y Y.The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals[J].Physics Letters A,2004,332:481-490.
[19]温维佳,沈平.局域共振的光子、声子功能材料[J].物理,2004,33:106-110.
[20]Liu Z Y,Zhang X X,Mao Y W,Zhu Z Z,Yang Z Y,Chan C T,Sheng P.Locally resonant sonic materials[J].Science,2000,289:1734-1736.
[21]Wang G,Wen X S,Wen J H,Shao L H,Liu Z Y.Two-dimensional locally resonant phononic crystals with binary structures[J].Physical Review Letters,2004,93:154302.
[22]Vasseur J O,Deymier P A,Beaugeois M,Pennec Y,Djafari-Rouhani B,Prevost D.Experimental observation of resonant filtering in a two-dimensional phononic crystal waveguide[J].Z.Kristallogr,2005,220:829-835.
[23]Kushwaha M S,Halevi P,Martine,Dobrzyuski L,Djafari-Rouhani.Theory of acoustic band structure of periodic elastic composites[J].Physical Review B,1994,49:2313-2322.
[24]Miklowitz J.The Theory of Elastic Waves and Waveguides[M].North-Holland Publishing Company:.New York,1978.
[25]Sigalas M M,Soukoulis C M.Elastic-wave propagation through disordered and/or absorptive layered systems[J].Physical Review B,1995,51:2780-2789.
[26]Lin L L,Li Z Y,Ho K M.Lattice symmetry applied in transfer-matrix methods for photonic crystals[J].Journal of Applied Physics,2003,94:811-821.
[27]Li Z Y,Lin L L.Photonic band structures solved by a plane-wave-based transfer-matrix method[J].Physical Review E,2003,67:046607.
[28]Hou Z L,Kuang W M,Liu Y Y.Transmission property anslysis of two-dimensional phononic crystal[J].Physics Letters A,2004,333:172-180.
[29]Hou Z L,Fu X J,Liu Y Y.Calculation method to study the transmission properties of phononic crystals[J].Physical Revew B,2004,70:014304.
[30]Kafesaki M,Economou E N,Multiple-scattering theory for three-dimensional periodic acoustic composites[J].Physical Review B,1999,60:11993.
[31]成礼智,王红霞,罗永.小波的理论与应用[M].科学出版社,北京,2005.
[32]Zygmund A.Trigonometric series[M].Cambridge University Press:Cambridge,UK,1977.
[33]Li L F.Use of Fourier series in the analysis of discontinuous periodic structures[J].Journal of the Optical Society of AmericaA,1996,13:1870-1876.
[34]Cao Y J,Hou Z L,Liu Y Y.Convergence problem of plane-wave expansion method for phononic crystals[J].Physics Letters A,2004,327:247-253.
[35]闫志忠.基于小波理论的二维声子晶体带隙结构分析[D].北京交通大学博士学位论文,2007.
[36]Mallat S.A Wavelet Tour of Signal Processing[J].Academic Press:New York,1998.
[37]Daubechies I.The wavelet transform,time-frequency localization and signal analysis[J].IEEE Transactions on Information Theory,1990,36:961-1005.
[38]Danbechies I.Ten Lectures on Wavelets[M].SIAM Publ,Philadelphia,1992.
[39]Garcia-Pablos D,Sigalas M,Montero de Espinoza F R,Tones M,Kafesaki M,Gareia N.Theory and experiments on elastic band gaps[J].Physical Review Letters,2000,84:4349-4352.
[40]Tanaka,Y,Tomoyasu Y,Tamura S.Band structure of acoustic waves in phononic lattices:Two-dimensional composites with large acoustic mismatch[J].Physical Review B,2000,62:7378-7392.
[41]Vasseur J O,Deymier P A,Chenni B,Djafari-Rouhani B,Dobrzynski L,Prevost D.Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals[J].Phyiscal Review Letters,2001,86:3012.
[42]刘丰铭.二维声子晶体中的FDTD理论计算方法及其应用实例[D].武汉大学硕士学位论文,2005.
[43]Liu Z Y,Chan C T,Sheng P,Goertzen A L,Page J H.Elastic wave scattering by periodic structures of spherical objects:Theory and experiment[J].Physical Review B,2000,62:2446-2457.
[44]Mei J,Liu Z Y,Shi J,Tian D.Theory for elastic wave scattering by a two-dimensional periodical array of cylinders:An ideal approach for band-structure calculations[J].Physical Review B,2003,67:245107.
[45]Goffaux C,S(?)nchez-Dehesa J.Two-dimensional phononic crystals studied using a variational method:Application to lattices of locally resonant materials[J].Physical Review B,2003,67:144301.
[46]Khelif A,Aoubiza B,Mohammadi S,Adibi A,Laude V.Complete band gaps in two-dimensional phononic crystal slabs[J].Physical Review B,2006,74:046610.
[47]魏巍.层状介质中声子频带及瞬态弹性波[M].北京大学硕士学位论文,2005.
[48]Wang G,Wen J H,Liu Y Y.Lumped-mass method for the study of band structure in two-dimensional phononic crystals[J].Physical Review B,2004,69:184302.
[49]申光宪,肖宏,陈一鸣.边界元法[M].机械工业出版社:北京,1998.
[50]Vines R E,Wolfe J P.Scanning phononic lattices with surface acoustic waves[J].Physica B,1999,263-264:567-570.
[51]Every A G,Vines R E,Wolfe J P.Line-focus probe exciation of Scholte acoustic waves at the liquid-loaded surfaces of periodic structures[J].Physical Review B,1999,60:11755-11760.
[52]Every A G,Vines R E,Wolfe J P.Observation of scholte-like waves on the liquid-loaded surfaces of periodic structures[J].Ultrasonics,2000,38:761-766.
[53]Torres M,Montero de Espinosa F R,Garc(?)a-Pablos D,Garc(?)a N.Sonic band gaps in finite elastic media:surface states and localization phenomena in linear and point defects[J].Physical Review Letters,1999,82:3054-3057.
[54]Tanaka Y,Tamura S.Two-dimensional phononic crystals:surface acoustic waves[J].Physica B,1999,263-264:77-80.
[55]Gao J,Zou X Y,Cheng J C,Li B W.Band gaps of lower-order Lamb wave in thin plate with oen-dimensional phononic crystal layer:effect of substrate[J].Applied Physics Letters,2008,92:023510.
[56]Hsu J C,Wu T T.Lamb waves in binary locally resonant phononic plates with two-dimensional lattices[J].Applied Physics Letters,2007,90:201904.
[57]Bonello B,Charles C,Ganot F.Lamb waves in plates covered by a two-dimensional phononic film[J].Applied Physics Letters,2007,90:021909.
[58]Zhao D G,Liu Z Y,Qiu C Y,He Z J,Cai F Y,Ke M Z.Surface acoustic waves in two-dimensional phononic crystals:dispersion relation and the eigenfield distribution of surface modes[J].Physical Review B,2007,76:144301.
[59]Sainidou R,Djafari-Rouhani,Vasseur J O.Surface acoustic waves in finite slabs of three-dimensional phononic crystals[J].Physical Review B,2008,77:094304.
[60]Ke M Z,He Z J,Peng S S,Liu Z Y,Shi J.Surface resonant-states-enhanced acoustic wave tunneling in two-dimensional phononic crystals[J].Physical Review Letters,2007,99:044301.
[61]Hu X H,Shen Y F,Liu X H,Fu R T,Zi J.Complete band gaps for liquid surface waves propagating over a periodically drilled bottom[J].Physical Review E,2003,68:066308.
[62]Wu T T,Huang Z G,Lin S.Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy[J].Physical Review B,2004,69:094301.
[63]Laude V,Wilm M,Benchabane S,Khelif A.Full band gap for surface acoustic waves in a piezoelectric phononic crystal[J].Physical Review E,2005,71:036607.
[64]Benchabane S,Khelif A,Ranch J Y,Robert L,Laude V.Evidence for complete surface wave band gap in a piezoelectric phononic crystal[J].Physical Review E,2006,73:065601.
[65]Manzanares-Martinez B,Ramos-Mendieta F.Surface elastic waves in solid composites of two-dimensional periodicity[J].Physical Review B,2003,68:134303.
[66]Sigalas M M.Defect states of acoustic waves in a two-dimensional lattice of solid cylinders[J].Journal of Applied Mechanics,1998,84:3026-3030.
[67]Kafesaki M,Sigalas M M,Garcia N.Frequency modulation in the transmittivity of wave guides in elastic-wave band-gap materials[J].Physical Review Letters,2000,85:4044-4047.
[68]Khelif A,Djafari-Rouhani B,Vasseur J O,Deymier P A,Lambin Ph,Dobrzynski L.Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal[J].Physical Review B,2002,65:174308.
[69]Khelif A,Choujaa A,Djafari-Rouhani B,Wilm M,Ballandras S,Landa V.Traping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[J].Physical Reviw B,2003,68:214301.
[70]Khelif A,Choujaa A,Ballandras S,Djafari-Rouhani B,Laude V.Guiding and bending of acoustic waves in highly confined phononic crystal waveguides[J].Applied Physics Letters,2004,84:4400-4402.
[71]Pennec Y,Djafari-Rouhani B,Vasseur J O,Khelif A,Deymier P A.Tunable filtering and demultiplexing in phononic crystals with hollow cylinders[J].Physical Review E,2004,69:046608.
[72]Tones M,Montero de Espinosa F R.Ultrasonic band gaps and negative refraction[J].Ultrasonics,2004,42:787-790.
[73]Sun J H,Wu T T.Analyses of mode coupling in joined parallel phononic crystal waveguides[J].Physical Review B,2005,71:174303.
[74]Sun J H,Wu T T.Propagation of acoustic waves in phononic-crystal plates and waveguides using a finite-difference time-domain method[J].Physical Review B,2007,76:104304.
[75]Wu F G,Liu Z Y,Liu Y Y.Splitting and tuning characteristics of the point defect modes in two-dimensional phononic crystals[J].Physical Review E,2004,69:066609.
[76]赵言城,赵芳,苑立波.同质位错缺陷二维声子晶体带隙结构及传导模的研究[J].哈尔滨工程大学学报,2006,27:617-620.
[77]Toyakatsu M.Acoustic defect-mode waveguides fabricated in sonic crystal:numerical analyses by elastic finite-difference time-domain method[J].Japanese Journal of Applied Physics,2006,45:4440-4447.
[78]Vassenr J O,Deymier P A,Djafari-Rouhani B,Pennec Y,Hladky-Hennion A C.Absoult forbidden bands and waveguiding in two-dimensional phononic crystal plates[J].Physical Review B,2008,77:085415.
[79]Vasseur J O,Hladky-Hennion A C,Djafari-Rouhani B,Duval F,Dubus B,Pennec Y,Deymier P A.Waveguiding in two-dimensional piezoelectric phononic crystal plates[J].Journal of Applied Physics,2007,101:114904.
[80]Hsiao Fl,Khelif,Moubchir H,Choujaa A,Chen C C,Laude V.Waveguiding inside the complete band gaps ofa phononic crystal slab[J].Physical Review E,2007,76,056601.
[81]Qiu C Y,Liu Z Y,Shi J,Chan C T.Directional acoustic source based on the resonant cavity of two-dimensional phononic crystals[J].Applied Physics Letters,2005,86:224105.
[82]Wu F G,Zhong H L,Zhong S,Liu Z Y,Liu Y Y.Localized states of acoustic waves in three-dimensional periodic composites with point defects[J].The European Physical Journal B,2003,34:265-268.
[83]Chandra H,Deymier P A,Vassenr J O.Elastic wave propagation along waveguides in three-dimensional phononic crystals[J].Physical Review B,2004,70:054302.
[84]Psarobas I E,Stefanou N,Modinos A.Phononic crystals with planar defects[J].Physical Review B,2000,62:5536-5540.
[85]Wang X F,Kushwaha M S,Vasilopoulos P.Tunability of acoustic spectral gaps and transmission in periodically stubbed waveguides[J].Physical Review B,2001,65:035107.
[86]Hladky-Hennion A C,Vassenr J,Djafari-Rouhani B,Billy M.Sonic band gaps in one-dimensional phononic crystals with a symmetry stub[J].Physical Review B,2008,77,104304.
[87]Tanaka Y,Yano T,Tamura S.Surface guided waves in two-dimensional phononic crystals [J].Wave Motion,2007,44:501-512.
[88]Anderson P W.Absence of diffusion in certain random lattices[J].Physical Review,1958,109:1492-1505.
[89]Mott N F.Electrons in disordered structures[J].Advances in Physics,1967,16:49-144.
[90]John S,Stephen M.Localization in a disordered elastic medium near two dimensions[J].Physical Review B,1983,27:5592-5603.
[91]John S,Stephen M.Wave propagation and localization in a long-range correlated random potential[J].Physical Review B,1983,28:6358-6368.
[92]John S.Electromagnetic absorption in a disordered medium near a photon mobility edge [J].Physical Review Letters,1984,53:2169-2172.
[93]Ozbay E,Tuttil G,Sigalas M,Soukoulis C M,Ho K M.Defect structures in a layer-by-layer photonic band-gap crystal[J].Physical Review B,1995,51:13961-13965.
[94]Meade R D,Brommer K D,Rappo A M,Joannopoulos J D.Photonic band states in periodic dielectric materials[J].Physical Review B,1991,44:13772-13774.
[95]Yannopapas V,Modinos A,Stefanou N.Anderson localization of light in inverted opals [J].Physical Review B,2003,68:193205.
[96]Asatryan A A,Nicorovici N A,Botten L C,de Sterke C M,Robinson P A,McPhedran R C.Electromagnetic localization in dispersive stratified media with random loss and gain [J].Physical Review B,1998,57:13535-13549.
[97]Judge J,Pierre C,Mehmed O.Experimental investigation of mode localization and forced response amplitude magnification for a mistuned bladed disk[J].Journal of Engineering for Gas Turbines and Power-Transactions of the ASME,2001,123:940-950.
[98]Sigalas M M.Defect states of acoustic waves in a two-dimensional lattice of solid cylinders[J].Journal of Applied Physics,1998,84:3026-3030.
[99]Li F M,Wang Y S,Hu C,Huang W H.Localization of elastic waves in randomly disordered multi-coupled multi-span beams[J].Waves Random Media,2004,14:217-227.
[100]He S J,Maynard J D.Detailed measurements of inelastic scattering in Anderson localization[J].Physical Review Letters,1986,57:3171-3174.
[101]Genack A Z,Garcia N.Observation of photon localization in a three-dimensional disordered system[J].Physical Review Letters,1991,66:2264-2267.
[102]Hudges C H.Confinement of vibration by structural irregularity[J].Journal of Sound and Vibration,1982,82:411-424.
[103]李凤明,汪越胜,黄文虎,胡超.失谐周期结构中振动局部化问题的研究进展[J].力学进展,2005,35:498-512.
[104]Bouzit D,Pierre C.An experiment investigation of vibration localization in disordered multi-span beams[J].Journal of Sound and Vibration,1995,187:649-669.
[105]Bouzit D,Pierre C.Localization of vibration in disordered multi-span beams with damping[J].Journal of Sound and Vibration,1995,187:625-648.
[106]Kim D O,Lee I W.Mode localization in structures consisting of substructures and couplers[J].Engineering Structures,2000,22:39-48.
[107]Ariaratnam S T,Xie W C.Wave localization in randomly disordered nearly periodic long continuous beams[J].Journal of Sound and Vibration,1995,181:7-22.
[108]Xie W C.Buckling mode localization in randomly disordered multispan continu(?)us beams[J].AIAA Journal,1995,33:1142-1149.
[109]Ariaratnam S T,Xie W C.Buckling mode localization in randomly disordered continuous beam using a simplified model[J].Chaos,Solitons and Fractals,1996,8:1127-1144.
[110]Zhang D Z,Hu W,Zhang Y L,Li Z L,Cheng B Y,Yang G Z.Experimental verification of light localization for disordered multilayers in the visible-infrared spectrum[J].Physical Review B,1994,50:9810-9814.
[111]Vinogradov A P,Merzlikin A M.Band theory of light localization in one-dimensional disordered systems[J].Physical Review E,2004,70:026610.
[112]Vlasov Yu A,Kaliteevsld M A,Nikolaev V V.Different regimes of light localization in a disordered photonic crystal[J].Physical Review B,1999,60:1555-1562.
[113]Asatryan A A,Robinson P A,Botten L C,McPhedran R C,Nicorovici N A,Martijn de Sterke C.Effects of disorder on wave propagation in two-dimensional photonic crystals [J].Physical Review E,1999,60:006118.
[114]Topolancik J,Ilic B,Vollmer F.Experimental observation of strong photon localization in disordered photonic crystal waveguides[J].Physical Review Letters,2007,99:253901.
[115]Li F M,Xu M Q,Wang Y S.Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals[J].Solid State Communications,2007,141:296-301.
[116]Li X C,Lin Z Y,Liang H Y,Xiao Q W.Band gap and waveguide states in two-dimensional disordered phononic crystals[J].Chinese Physics Letters,2006,23:1830-1833.
[117]Han P,Chan C T,Zhang Z Q.Wave localization in one-dimensional random structures composed of single-negative metamaterials[J].Physical Review B,2008,77,115332.
[118]Bush R L.Localization lengths of electronic states in one-dimensional disordered potential chains[J].Physical Review B,1972,6:1182-1188.
[119]Nikoli(?) K,Mackirnnon A.Density of state and localization length in compositionally disordered quantum wires[J].Physical Review B,1993,47:6555-6565.
[120]Zhang D Z,Wei H,Zhang Y L,Li Z L,Cheng B Y,Yang G Z.Experimental verification of light localization for disordered multilayers in the visible-infrared spectrum[J].Physical Review B,1994,50:9810-9814.
[121]Sigalas M M,Soukoulis C M,Chan C T,Turner D.Localization of electromagnetic waves in two-dimensional disordered systems[J].Physical Review B,1996,53:8340-8348.
[122]Bliokh K Yu,Freilikher V D.Localization of transverse waves in randomly layered media at oblique incidence[J].Physical Review B,2004,70:245121.
[123]Scales J A,Van Vleck E S.Lyapunov exponents and localization in randomly layered media[J].Journal of Computational Physics,1997,133:27-42.
[124]Castanier M P,Pierre C.Predicting localization via Lyapunov exponent statistics[J].Journal of Sound and Vibration,1997,203:151-157.
[125]Xie W C,Ibrahim A.Buckling mode localization in rib-stiffened plates with misplaced stiffeners-a finite strip approach[J].Chaos,Solitons and Fractals,2000,11:1543-1558.
[126]Xie W C,Elishakoff I.Buckling mode localization in rib-stiffened plates with misplaced stiffeners-Kantorovich approach[J].Chaos,Solitons and Fractals,2000,11:1559-1574.
[127]李凤明.随机失谐周期结构中弹性波传播的局部化问题.北京交通大学博士后研究工作报告[D],2005.
[128]Li F M,Wang Y S,Hu C,Huang W H.Wave localization in randomly disordered periodic layered piezoelectric structures[J].Acta Mechanica Sinica,2006,22:559-567.
[129]Li F M,Wang Y S.Study on localization of plane elastic waves in disordered periodic 2-2piezoelectric composite structures[J].Journal of Sound and Vibration,2006,296:554-566.
[130]Li F M,Wang Y S.Study on wave localization in disordered periodic layered piezoelectric composite structures[J].International Journal of Solids and Structures,2005,42:6457-6474.
[131]周纪卿,朱因远.非线性振动[M].西安:西安交通大学出版社,1998.
[132]Kissel G J.Localization factor for multichannel disordered systems[J].Physical Review A,1991,44:1008-1014.
[133]Wolf A,Swift J B,Swinney H L,Vastano J A.Determining Lyapunov exponents from a time series[J].Physica D,1985,16:285-317.
[134]Castanier M P,Pierre C.Lyapunov exponents and localization phenomena in multi-coupled nearly periodic systems[J].Journal of Sound and Vibration,1995,183:493-515.
[135]Xie W C.Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners[J].Computers and Structures,1998,67:175-189.
[136]Bouzit D,Pierre C.Wave localization and conversion phenomena in multi-coupled multi-span beams[J].Chaos,Solitons and Fractals,2000,11:1575-1596.
[137]Cai G,Lin Y.Localization of wave propagation in disordered periodic structures[J].AIAA Journal,1990,29:450-456.
[138]Ruzzene M,Baz A.Attenuation and localization of wave propagation in periodic rods using shape memory inserts[J].Smart Materials and Structures,2000,9:805-816.
[139]Baz A.Active control of periodic structures[J].ASME,Journal of Vibration and Acoustics,2001,123:472-479.
[140]陈景良,陈向晖.特殊矩阵[M].清华大学出版社:北京,2000.
[141]Shechtman D,Blech I,Cratias D,Cahn J W.Metallic phase with long-range orientation order and translational symmetry[J].Physical Review Letters,1984,53:1951-1953.
[142]Levine D,Steinhardt P J.Quasicrystals:a new class of ordered structures[J].Physical Review Letters,1984,53:2477-2480.
[143]郭可信.五次对称性及Ti-Ni准晶相的发现与研究[J].物理,1989,18:344-346.
[144]郭可信.8次、12次对称及有关准晶的发现[J].物理,1991,20:11-14.
[145]陆洪文,费奔.二维准格点的算术结构[J].同济大学学报,2004,32:1100-1102.
[146]张利明,董闯.准晶材料性能及应用研究现状[J].材料导报,2000,14:22-24.
[147]陈敬中.准晶体的基本性质[J].地球科学-中国地质大学学报,1993,18(增刊):13-24.
[148]雷建林,王仁卉,王洲光,丁棣华.准晶切割投影图的计算与图示[J].计算物理,1997,14:47-53.
[149]陈敬中,万安娃,路湘豫,赵文霞,刘祥文.准晶结构几何理论初探[J].地球科学-中国地质大学学报,1991,16:173-180.
[150]陈敬中.准晶结构与Penrose拼图[J].地球科学-中国地质大学学报,1993,18(增刊):56-62.
[151]施倪承,闵乐泉,沈步明.准晶体的晶胞构成及准晶格的推导[J].中国科学 B,1991,11:1216-1223.
[152]胡承正,杨文革,王仁卉,丁棣华.准晶的对称性和物理性质[J].物理学进展,1997,17:345-366.
[153]冯志芳,张向东,王义全,李志远,程丙英,张道中.十二重准晶光子结构中的负折射与成像[J].物理,2006,35:9-13.
[154]李邦盛,郭俊清,李庆春.十二面体准晶形成机理的研究[J].电子显微学报,1993,2:166.
[155]刘新宝,宋广生,张振忠,杨根仓.块体单准晶材料制备的进展及展望[J].材料导 报,2000,14:14-20.
[156]Sutter-Widmer D,Deloudi S,Steurer W.Prediction of Bragg-scattering-induced band gaps in phononic quasicrystals[J].Physical Review B,2007,75:094304.
[157]范天佑.准晶数学弹性理论及应用[M].北京理工大学出版社:北京,1999.
[158]闵乐泉,刘钦圣,吴玉珍.具12次对称格点阵的衍射图[J].北京科技大学学报,1991,13:191-194.
[159]陈敬中.具有8次对称性的准晶结构模型[J].地球科学-中国地质大学学报,1993,18(增刊):105-113.
[160]陈敬中.具有12次对称性的准晶结构模型[J].地球科学-中国地质大学学报,1993,18(增刊):122-128.
[161]Steurer W,Sutter-Widmer D.Photonic and phononic quasicrystals[J].Journal of Physics D:Applied Physics,2007,40:R229-R247.
[162]Sutter-Widmer D,Deloudi S,Steurer W.Periodic average structures in phononic quasicrystals[J].Philosophical Magazine,2007,87:3095-3102.
[163]Steurer W,Haibach T.The periodic average structure of particular quasicrystals[J].Acta Crystallographica Section A,1999,55:48-57.
[164]蔡敏,姚源卫,屈晔.三元准周期半导体超晶格的电子态[J].华南理工大学学报(自然科学版),2001,29:27-30.
[165]朱家昆,胡承正,周详,龚平.准周期链状聚合物电子谱及特征[J].武汉大学学报(理学版),2003,49:369-372.
[166]朱敏,方云团,姚洁,沈廷根.一维准周期结构的光子晶体的带隙结构[J].激光杂志,2004,25:39-40.
[167]蒋美萍,王旭东,巢小刚,是度芳,陈光.准周期结构-维光子晶体的带隙特性与滤波特性[J].量子电子学报,2005,22:884-888.
[168]Hu A,Wen Z X,Jiang S S,Tong W T,Peng R W,Feng D.One-dimensional κ-component Fibonacci structures[J].Physical Review B,1993,48:829-835.
[169]Peng R W,Huang X Q,Qiu F,Wang Mu,Hu A,Jiang S S.Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers[J].Applied Physics Letters,2002,80:3063-3065.
[170]Peng R W,Liu Y M,Huang X Q,Qiu F,Wang Mu,Hu A,Jiang S S,Feng D,Ouyang L Z,Zou J.Dimerlike positional correlation and resonant transmission of electromagnetic waves in aperiodic dielectric multilayers[J].Physical Review B,2004,69:165109.
[171]Hu P,Ting C S.Electron transmission in a one-dimensional quasicrystal[J].Physical Review B,1986,34:8331-8334.
[172]Xiang T.Effect of superconducting correlation on the localization of quasiparticles in low dimensions[J].Physical Review B,1995,52:6204-6207.
[173]Gershenson M E,Khavin Yu B.Crossover from weak to strong localization in quasi-one-dimensional conductors[J].Physical Review Letters,1997,79:725-728.
[174]Avishai Y,Luck J M.Localization properties of quasi-one-dimensional conductor networks in a random magnetic field[J].Physical Review B,1994,49:8679-8688.
[175]Hjort M,Stafstr(o|¨)m.Localization in quasi-one-dimensional systems[J].Physical Review B,2000,62:5245-5250.
[176]Chan Y S,Chan C T,Liu Z Y.Photonic band gaps in two dimensional photonic quasicrystals[J].Physical Review Letters,1998,80:956-959.
[177]Jin C J,Cheng B Y,Man B Y,Li Z L,Zhang D Z.Band gap and wave guiding effect in a quasiperiodic photonic crystal[J].Applied Physics Letters,1999,27:1848-1850.
[178]Zoorob M E,Charlton M D B,Parker G J,Baumberg J J,Netti M C.Complete photonic bandgaps in 12-fold symmetric quasicrystals[J].Nature,2000,404:740-743.
[179]Velasco V R,P(?)rez-Alvarez R,Garcia-Moliner F.Some properties of the elastic waves in quasiregular heterostructures[J].Journal of Physics:Condensed Matter,2002,14:5933-5957.
[180]罗青.声波在一维准周期系统中的传播[J].南京师大学报(自然科学版),2002,25:110-115.
[181]曹永军,曹纯红,周培勤.一维准周期结构声子晶体透射性质的研究[J].物理学报,2006,55:6470-6474.
[182]方云团,王永顺,姜光.声波在球状准周期多层介质中的传播[J].应用声学,2004,23:23-28.
[183]Zhang X D.Universal non-near-field focus of acoustic waves through high-symmetry quasicrystals[J].Physical Review B,2007,75:024209.
[184]Lai Y,Zhang X D,Zhang Z Q.Large sonic band gaps in 12-fold quasicrystals[J].Journal of Applied Physics,2002,91:6191-6193.
[185]Lai Y,Zhang Z Q.Large enhancement of phononic gap in periodic and quasiperiodic elastic composites by using air inclusions[J].Z.Kristallogr.,2005,220:877-883.
[186]King P D C,Cox T J.Acoustic band gaps in periodically and quasiperiodically modulated waveguides[J].Journal of Applied Physics,2007,102:014902.
[187]Gao J,Cheng J C,Li B W.Propagation of Lamb waves in one-dimensional quasiperiodic composite thin plates:A split of phonon band gap[J].Applied Physics Letters,2007,90:111908.
[188]Aynaou H,Boudouti E H El,Djafari-Rouhani B,Akjouj A,Velasco V R.Propagation and localization of acoustic waves in Fibonacci phononic circuits[J].Journal of Physics:Condensed Matter,2005,17:4245-4262.
[189]Veselago V G.The electrodynamics of substances with simultaneously negative values of permittivity and permeablility[J].Soviet Physics USPEKI,1968,10:509-514[USP.Fiz.Nauk,1967,8:2854-2859].
[190]Shelby R A,Smith D R,Schultz S.Experimental verification of a negative index of refraction[J].Science,2001,292:77-79.
[191]Gralak B,Enoch S,Tayeb G.Anomalous refraction properties of photonic crystals[J].Journal of Optical Society of America,2000,17:1012-1020.
[192]Pendry J B.Negative refraction makes a perfect lens[J].Physical Review Letters,2000,85:3966-3969.
[193]Cervera F,Sanchis L,Sanchez-Perez J V,Martinez-Sala R,Rubio C,Meseguer F,Lopez C,Caballero D,Sanchez-Dehesa J.Refractive acoustic devices for airborne sound[J].Physical Review Letters,2002,88:023902.
[194]Gupta B C,Ye Z.Theoretical analysis of the focusing of acoustic waves by two-dimensional sonic crystals[J].Physical Review E,2003,67:046606.
[195]Zhang X D,Liu Z Y.Negative refraction of acoustic waves in two-dimensional phononic crystls[J].Apllied Physics Letters,2004,85:341-343.
[196]Ko M,Liu Z Y,Qiu C Y,Wang W,Shi J,Wen W J,Sheng P.Negative-refraction imaging with two-dimensional phononic crystals[J].Physical Review B,2005,72:064306.
[197]Yang S X,Page J H,Liu Z Y,Cowan M L,Chan C T,Sheng P.Focusing of sound in a 3D phononic crystal[J].Physical Review Letters,2004,93:024301.
[198]Sukhovich A,Jing L,Page J H.Negative refraction and focusing of ultrasound in two-dimensional phononic crystals[J].Physical Review B,2008,77:014301.
[199]Tarasenko O S,Tarasenko S V,Yurchenko V M.Effect of negative acoustic refraction in a one-dimensional phononic crystal[J].JETP Letters,2004,80:484-486.
[200]Li J,Liu Z Y,Qin C Y.Negative refraction imaging of acoustic waves by a two-dimensional three-component phononic crystal[J].Physical Review B,2006,73:054302.
[201]Feng L,Liu X P,Lu M H,Chen Y B,Chen Y F,Mao Y W,Zi J,Zhu Y Y,Zhu S N,Ming N B.Acoustic backward-wave negative refractions in the second band of a sonic crystal [J].Physical Review Letters,2006,96:014301.
[202]Fang N,Xi D J,Xu J Y,Ambati M,Srituravanish W,Sun C,Zhang X.Ultrasonic metamatcrials with negative modulus[J].Nature Materials,2006,5:452-456.
[203]Hu X H,Chan C T.Refraction of water waves by periodic cylinder arrays[J].Physical Review Letters,2005,95:154501.
[204]卢明辉,冯亮,刘小平,陈延斌,陈延峰,毛一薇,资剑,朱永元,祝世宁,闵乃本.声子晶体中第二能带的回波负折射[J].物理,2006,35:969-974.
[205]Alvarez A,Ye Z.Effects of fish school structures on acoustic scattering[J].ICES Journal of Marine Science,1999,56:361-369.
[206]Martinez-Sala R,Rubio C,Garcla-Raffi L M,S(?)nchez-P(?)rez J V,S(?)nchez-P(?)rez E A,Llinares J.Control of noise by tress arranged like sonic crystals[J].Journal of Sound and Vibration,2006,291:100-106.
[207]Benchabane S,Khelif A,Choujaa A,Djafari-Rouhani B,Laude V.Interaction of waveguide and localized modes in a phononic crystal[J].Europhysics Letters,2005,71:570-575.
[208]Zhang X,Liu Z Y,Liu Y Y,Wu F G.Defect states in 2D acoustic band-gap materials with bend-shaped linear defects[J].Solid State Communications,2004,130:67-71.
[209]Permec Y,Djafari-Rouhani B,Vasseur J O,Larabi H,Khelif A,Choujaa A,Benchabane S,Laude V.Acoustic channel drop tunneling in a phononic crystal[J].Applied Physics Letters,2005,87:261912.
[210]Pennec Y,Djafari-Rouhani B,Vasseur J O,Khelif A,Deymier P A.Tunable filtering and demultiplexing in phononie crystals with hollow cylinders[J].Physical Review E,2004,69:046608.
[211]Qiu C Y,Liu Z Y.Acoustic directional radiation and enhancement caused by band-edge states of two-dimensional phononie crystals[J].Applied Physics Letters,2006,89:063106.
[212]Ke M Z,Liu Z Y,Pang P,Wang W G.,Cheng Z G,Shi J,Zhao X Z.Highly directional acoustic wave radiation based on asymmetrical two-dimensional phononic crystal resonant cavity[J].Applied Physics Letters,2006,88:263505.
[213]Qiu C Y,Liu Z Y,Mei J,Shi J.Mode-selecting acoustic filter by using resonant tunneling of two-dimensional double phononic crystals[J].Applied Physics Letters,2005,87:104101.
[214]Diez A,Kakaranttzas G.Acoustic stop-bands in periodically microtapered optical fibers[J].Applied Physics Letter,2000,76:3481-3483.
[215]Cervera F,Sanchis L,S(?)nchez-P(?)rez J V,Martinez-Sala R,Rubio C,Meseguer F,L(?)pez C,Caballero D,S(?)nchez-Dehesa J.Refractive acoustic devices for airborne sound[J].Physical Review Letters,2002,88:023902.
[216]Hu X H,Chan C T,Zi J.Two-dimensional sonic crystals with Helmholtz resonators[J].Physical Review E,2005,71:055601(R).
[217]Jang J H,Ullal C K,Gorishnyy T,Tsukruk V V,Thomas E L.Mechanically tunable three-dimensional elastomeric network/air structures via interference lithography[J].Nano Letters,2006,6:740-743.
[218]Cheng W,Wang J J,Jonas U,Fytas G,Stefanou N.Observation and tuning of hypersonic bandgaps in colloidal crystals[J].Nature Materials,2006,5:830-836.
[219]Baz(?)n A,Torres M,Montero de Espinosa F R,Quintero-Torres R,Arag(o|¨)n J L.Conformal mapping of ultrasonic crystals:Confining ultrasound and cochlearlike waveguiding[J].Applied Physics Letters,2007,90:094101.
[220]Tommaseo G,Petekidis G,Steffen W,Fytas G,Schofield A B,Stefanou N.Hypersonic acoustic excitations in binary colloidal crystals:big versus small hard sphere control[J].The Journal of Chemical Physics,2007,126:014707.
[221]Yilmaz C,Hulbert G M,Kikuchi N.Phononic band gaps induced by inertial amplification in periodic media[J].Physical Review B,2007,76:054309.
[222]Halkj(?)r S,Sigmund O,Jensen J.Inverse design of phononic crystals by topology optimization[J].Z.Kristallogr.,2005,220:895-905.
[223]钟会林,吴福根,姚立宁.遗传算法在二维声子晶体带隙优化中的应用[J].物理学报,2006,55:275-280.
[224]Wang G,Yu D L,Wen J H,Liu Y Z,Wen X S.One-dimensional phononic crystals with locally resonant structures[J].Physics Letters A,2004,327:512-521.
[225]Sepehrinia R,Bahraminasab A,Sahimi M,Tabar M R R.Dynamic renormalization group analysis of propagation of elastic waves in two-dimensional heterogeneous media[J].Physical Review B,2008,77:014203.
[226]张欣.声子晶体中声波或弹性波带隙的研究[D].华南理工大学博士学位论文,2004.
[227]曹永军.弹性波在二维声子晶体中传播特性的研究[D].华南理工大学博士学位论文,2005.
[228]Lu Y,Peng R W,Wang Z,Tang Z H,Huang X Q,Wang M,Qiu Y,Hu A Jiang S S,Feng D.Resonant transmission of light waves in dielectric heterostructures[J].Journal of Applied Physics,2005,97:123106.
[229]Yang S X,Page J H,Liu Z Y,Cowan M L,Chart C T,Sheng P.Ultrasound tunneling through 3D Phononic crystals[J].Physical Review Letters,2002,88:104301.
[230]Wu F G,Liu Z Y,Liu Y Y.Acoustic band gaps created by rotating square rods in a two-dimensional lattice[J].Physical Review E,2002,66:046628.
[231]Wu T T,Hsu C H,Sun J H.Design of a highly magnified directional acoustic source based on the resonant cavity of two-dimensional phononic crystals[J].Applied Physics Letters,2006,89:171912.
[232]Tanaka Y,Tamura S.Acoustic stop bands of surface and bulk modes in two-dimensional phononic lattices consisting of aluminum and a polymer[J].Physical Review B,1999,60:13294-13297.
[233]Zhang X,Liu Z Y,Liu Y Y,Wu F G.Elastic wave band gaps for three-dimensional phononic crystals with two structural units[J].Physical Letters A,2003,313:455-460.
[234]Zhao H G,Liu Y Z,Yu D L,Wang G,Wen J H,Wen X S.Absorptive properties of three-dimensional phononic crystal[J].Journal of Sound and Vibration,2007,303:185-194.
[235]Wang G,Wen J H,Han X Y.Finite difference time domain method for the study of band gap in two-dimensional phononie crystals[J].Acta Physica Sinica-Overseas Edition,2003,52:1943-1947.
[236]Zhang X,Liu Y Y,Wu F G,Liu Z Y.Large two-dimensional band gaps in three-component phononic crystals[J].Physics Letters A,2003,317:144-149.
[237]Hou Z L,Fu X J,Liu Y Y.Acoustic wave in a two-dimensional composite medium with anisotropie inclusions[J].Physics Letters A,2003,317:127-134.
[238]Hou Z L,Wu F G,Liu Y Y.Phononie crystals containing piezoelectrie material[J].Solid State Communications,2004,130:745-749.
[239]吴福根.声子晶体中的带隙及缺陷态的研究[D].华南理工大学博士学位论文,2001.
[240]Cao Y J,Hou Z L,Liu Y Y.Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals[J].Solid State Communications,2004,132:539-543.
[241]Wen J H,Yu D L,Wang G,Zhao H G,Liu Y Z,Wen X S.The directional propagation characteristics of elastic wave in two-dimensional thin plate phononic crystals[J].Physics Letters A,2007,364:323-328.
[242]Hou Z L,Kuang W,Liu Y Y.Transmission property analysis of a two-dimensional phononic crystal[J].Physics Letters A,2004,333:172-180.
[243]Li J,Chan C T.Double-negative acoustic metamaterial[J].Physical Review E,2004,70:055602.
[244]Goffaux C,Vigneron J P.Theoretical study of a tunable phononic band gap system[J].Physical Review B,2001,64:075118.
[245]齐共金.二维声子晶体的带隙研究[D].国防科技大学硕士学位论文,2002.
[246]Sigalas M,Kushwaha M S,Economou E N,Kafesaki M,Psarobas I E,Steurer W.Classical vibrational modes in phononie lattices:theory and experiment[J].Z.Kristallogr.,2005,220:765-809.
[247]Huang Z G,Wu T T.Analysis of wave propagation in phonunic crystals with channel using the plane-wave expansion and supercell techniques[J].IEEE Ultrasonics Symposium,2005,77-80.
[248]Yu D L,Liu Y Z,Wang G,Cai L,Qiu J.Low frequency torsional vibration gaps in the shaft with locally resonant structures[J].Physics Letters A,2006,348:410-415.
[249]Wang G,Liu Y Z,Wen J H,Yu D L.Formation mechanism of the low-frequency locally resonant band gap in the two-dimensional ternary phononic crystals[J].Chinese Physics,2006,15:407-411.
[250]Miyashita T.Acoustic defect-mode waveguides fabricated in sonic crystal:Numerical analyses by elastic finite-difference time-domain method[J].Journal of Applied Physics,2006,45:4440-4447.
[251]Wang G,Shao L H,Liu Y Z,Wen J H.Accurate evaluation of lowest band gaps in ternary locally resonant phononic crystals[J].Chinese Physics,2006,15:1843-1848.
[252]Yao Y W,Hou Z,Cao Y J,Liu Y Y.An improved method of eigen-mode matching theory in two-dimensional phononic crystal[J].Physica B,2007,388:75-81.
[253]Bonello B,Charles C,Ganot F.Lamb waves in plates covered by a two-dimensional phononic film[J].Appllied Physics Letters,2007,90:021909.
[254]Chen H Y,Luo X D,Ma H N.Scattering of elastic waves by elastic spheres in a NaCl-type phononic crystal[J].Physical Review B,2007,75:024306.
[255]Liu Y Z,Yu D L,Li L,Zhao H G,Wen J H,Wen X S.Design guidelines for flexural wave attenuation of slender beams with local resonators[J].Physics Letters A,2007,362:344-347.
[256]Yao Y W,Hou Z L,Liu Y Y.The two-dimensional phononic band gaps tuned by the position of the additional rod[J].Physics Letters A,2007,362:494-499.
[2]Esaki L,Tsu.Superlattice and negative differential conductivity in semiconductors[M].IBM J Res Develop,1970.
[3]Yablonovitch E.Inhibited spontaneous emission in solid state physics and electronics[J].Physical Review Letters,1987,58:2059-2062.
[4]John S.Strong localization of photons in certain disordered dielectric superlattices[J].Physical Review Letters,1987,58:2486-2489.
[5]Yablonovitch E,Gmtter T J.Photonic band structure:the face-centered-cubic case[J].Physical Review Letters,1989,63:1950-1953.
[6]Yablonovitch E,Gmitter T J,Leung M L.Photonic band-structure-the face-centered-cubic case employing nonspherical atoms[J].Physical Review Letters,1991,67:2295-2298.
[7]付云起,袁乃昌,温熙森.微波光子晶体天线技术[M].国防工业出版社:北京,2006.
[8]Kushwaha M S,Halevi P,Martinez G,Dobrzynski L,Djafarirouhani B.Acoustic band structure of periodic elastic composites[J].Physical Review Letters,1993,71:2022-2025.
[9]Brekhovskikh L M.Waves in layered media[M].Academic:New York,1960.
[10]Ewing W M,Jardetsky W S,Press F.Elastic waves in layered media[M].McGraw-Hill:New York,1967.
[11]Guz' A N,Shulga N A.Dynamics of laminated and fibrous composites[J].Applied Mechanics Review,1992,45:35-60.
[12]Angel Y C,Achenbach J D.Harmonic waves in an elasic solid containg a doubly periodic array of cracks[J].Wave Motion,1987,9:377-385.
[13]Sigalas M M,Economou E N.Elastic and acoustic wave band structure[J].Journal of Sound and Vibration,1992,158:377-382.
[14]Martinez-Sala R,Sancho J,Sanchez J V,Gomez V,Llinares J,Meseguer F.Sound-attenuation by sculpture[J].Nature,1995,378:241-241.
[15]温熙森.光子/声子晶体理论与技术[M].科学出版社:北京,2006.
[16]黄昆著,韩汝琦改编.固体物理学[M].高等教育出版社:北京,1988.
[17]Kafesaki M,Penciu R S,Economou E N.Air bubbles in water:a strongly multiple scattering medium for acoustic waves[J].Physical Review Letters,2000,84:6050-6054.
[18]Kuang W M,Hou Z L,Liu Y Y.The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals[J].Physics Letters A,2004,332:481-490.
[19]温维佳,沈平.局域共振的光子、声子功能材料[J].物理,2004,33:106-110.
[20]Liu Z Y,Zhang X X,Mao Y W,Zhu Z Z,Yang Z Y,Chan C T,Sheng P.Locally resonant sonic materials[J].Science,2000,289:1734-1736.
[21]Wang G,Wen X S,Wen J H,Shao L H,Liu Z Y.Two-dimensional locally resonant phononic crystals with binary structures[J].Physical Review Letters,2004,93:154302.
[22]Vasseur J O,Deymier P A,Beaugeois M,Pennec Y,Djafari-Rouhani B,Prevost D.Experimental observation of resonant filtering in a two-dimensional phononic crystal waveguide[J].Z.Kristallogr,2005,220:829-835.
[23]Kushwaha M S,Halevi P,Martine,Dobrzyuski L,Djafari-Rouhani.Theory of acoustic band structure of periodic elastic composites[J].Physical Review B,1994,49:2313-2322.
[24]Miklowitz J.The Theory of Elastic Waves and Waveguides[M].North-Holland Publishing Company:.New York,1978.
[25]Sigalas M M,Soukoulis C M.Elastic-wave propagation through disordered and/or absorptive layered systems[J].Physical Review B,1995,51:2780-2789.
[26]Lin L L,Li Z Y,Ho K M.Lattice symmetry applied in transfer-matrix methods for photonic crystals[J].Journal of Applied Physics,2003,94:811-821.
[27]Li Z Y,Lin L L.Photonic band structures solved by a plane-wave-based transfer-matrix method[J].Physical Review E,2003,67:046607.
[28]Hou Z L,Kuang W M,Liu Y Y.Transmission property anslysis of two-dimensional phononic crystal[J].Physics Letters A,2004,333:172-180.
[29]Hou Z L,Fu X J,Liu Y Y.Calculation method to study the transmission properties of phononic crystals[J].Physical Revew B,2004,70:014304.
[30]Kafesaki M,Economou E N,Multiple-scattering theory for three-dimensional periodic acoustic composites[J].Physical Review B,1999,60:11993.
[31]成礼智,王红霞,罗永.小波的理论与应用[M].科学出版社,北京,2005.
[32]Zygmund A.Trigonometric series[M].Cambridge University Press:Cambridge,UK,1977.
[33]Li L F.Use of Fourier series in the analysis of discontinuous periodic structures[J].Journal of the Optical Society of AmericaA,1996,13:1870-1876.
[34]Cao Y J,Hou Z L,Liu Y Y.Convergence problem of plane-wave expansion method for phononic crystals[J].Physics Letters A,2004,327:247-253.
[35]闫志忠.基于小波理论的二维声子晶体带隙结构分析[D].北京交通大学博士学位论文,2007.
[36]Mallat S.A Wavelet Tour of Signal Processing[J].Academic Press:New York,1998.
[37]Daubechies I.The wavelet transform,time-frequency localization and signal analysis[J].IEEE Transactions on Information Theory,1990,36:961-1005.
[38]Danbechies I.Ten Lectures on Wavelets[M].SIAM Publ,Philadelphia,1992.
[39]Garcia-Pablos D,Sigalas M,Montero de Espinoza F R,Tones M,Kafesaki M,Gareia N.Theory and experiments on elastic band gaps[J].Physical Review Letters,2000,84:4349-4352.
[40]Tanaka,Y,Tomoyasu Y,Tamura S.Band structure of acoustic waves in phononic lattices:Two-dimensional composites with large acoustic mismatch[J].Physical Review B,2000,62:7378-7392.
[41]Vasseur J O,Deymier P A,Chenni B,Djafari-Rouhani B,Dobrzynski L,Prevost D.Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals[J].Phyiscal Review Letters,2001,86:3012.
[42]刘丰铭.二维声子晶体中的FDTD理论计算方法及其应用实例[D].武汉大学硕士学位论文,2005.
[43]Liu Z Y,Chan C T,Sheng P,Goertzen A L,Page J H.Elastic wave scattering by periodic structures of spherical objects:Theory and experiment[J].Physical Review B,2000,62:2446-2457.
[44]Mei J,Liu Z Y,Shi J,Tian D.Theory for elastic wave scattering by a two-dimensional periodical array of cylinders:An ideal approach for band-structure calculations[J].Physical Review B,2003,67:245107.
[45]Goffaux C,S(?)nchez-Dehesa J.Two-dimensional phononic crystals studied using a variational method:Application to lattices of locally resonant materials[J].Physical Review B,2003,67:144301.
[46]Khelif A,Aoubiza B,Mohammadi S,Adibi A,Laude V.Complete band gaps in two-dimensional phononic crystal slabs[J].Physical Review B,2006,74:046610.
[47]魏巍.层状介质中声子频带及瞬态弹性波[M].北京大学硕士学位论文,2005.
[48]Wang G,Wen J H,Liu Y Y.Lumped-mass method for the study of band structure in two-dimensional phononic crystals[J].Physical Review B,2004,69:184302.
[49]申光宪,肖宏,陈一鸣.边界元法[M].机械工业出版社:北京,1998.
[50]Vines R E,Wolfe J P.Scanning phononic lattices with surface acoustic waves[J].Physica B,1999,263-264:567-570.
[51]Every A G,Vines R E,Wolfe J P.Line-focus probe exciation of Scholte acoustic waves at the liquid-loaded surfaces of periodic structures[J].Physical Review B,1999,60:11755-11760.
[52]Every A G,Vines R E,Wolfe J P.Observation of scholte-like waves on the liquid-loaded surfaces of periodic structures[J].Ultrasonics,2000,38:761-766.
[53]Torres M,Montero de Espinosa F R,Garc(?)a-Pablos D,Garc(?)a N.Sonic band gaps in finite elastic media:surface states and localization phenomena in linear and point defects[J].Physical Review Letters,1999,82:3054-3057.
[54]Tanaka Y,Tamura S.Two-dimensional phononic crystals:surface acoustic waves[J].Physica B,1999,263-264:77-80.
[55]Gao J,Zou X Y,Cheng J C,Li B W.Band gaps of lower-order Lamb wave in thin plate with oen-dimensional phononic crystal layer:effect of substrate[J].Applied Physics Letters,2008,92:023510.
[56]Hsu J C,Wu T T.Lamb waves in binary locally resonant phononic plates with two-dimensional lattices[J].Applied Physics Letters,2007,90:201904.
[57]Bonello B,Charles C,Ganot F.Lamb waves in plates covered by a two-dimensional phononic film[J].Applied Physics Letters,2007,90:021909.
[58]Zhao D G,Liu Z Y,Qiu C Y,He Z J,Cai F Y,Ke M Z.Surface acoustic waves in two-dimensional phononic crystals:dispersion relation and the eigenfield distribution of surface modes[J].Physical Review B,2007,76:144301.
[59]Sainidou R,Djafari-Rouhani,Vasseur J O.Surface acoustic waves in finite slabs of three-dimensional phononic crystals[J].Physical Review B,2008,77:094304.
[60]Ke M Z,He Z J,Peng S S,Liu Z Y,Shi J.Surface resonant-states-enhanced acoustic wave tunneling in two-dimensional phononic crystals[J].Physical Review Letters,2007,99:044301.
[61]Hu X H,Shen Y F,Liu X H,Fu R T,Zi J.Complete band gaps for liquid surface waves propagating over a periodically drilled bottom[J].Physical Review E,2003,68:066308.
[62]Wu T T,Huang Z G,Lin S.Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy[J].Physical Review B,2004,69:094301.
[63]Laude V,Wilm M,Benchabane S,Khelif A.Full band gap for surface acoustic waves in a piezoelectric phononic crystal[J].Physical Review E,2005,71:036607.
[64]Benchabane S,Khelif A,Ranch J Y,Robert L,Laude V.Evidence for complete surface wave band gap in a piezoelectric phononic crystal[J].Physical Review E,2006,73:065601.
[65]Manzanares-Martinez B,Ramos-Mendieta F.Surface elastic waves in solid composites of two-dimensional periodicity[J].Physical Review B,2003,68:134303.
[66]Sigalas M M.Defect states of acoustic waves in a two-dimensional lattice of solid cylinders[J].Journal of Applied Mechanics,1998,84:3026-3030.
[67]Kafesaki M,Sigalas M M,Garcia N.Frequency modulation in the transmittivity of wave guides in elastic-wave band-gap materials[J].Physical Review Letters,2000,85:4044-4047.
[68]Khelif A,Djafari-Rouhani B,Vasseur J O,Deymier P A,Lambin Ph,Dobrzynski L.Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal[J].Physical Review B,2002,65:174308.
[69]Khelif A,Choujaa A,Djafari-Rouhani B,Wilm M,Ballandras S,Landa V.Traping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[J].Physical Reviw B,2003,68:214301.
[70]Khelif A,Choujaa A,Ballandras S,Djafari-Rouhani B,Laude V.Guiding and bending of acoustic waves in highly confined phononic crystal waveguides[J].Applied Physics Letters,2004,84:4400-4402.
[71]Pennec Y,Djafari-Rouhani B,Vasseur J O,Khelif A,Deymier P A.Tunable filtering and demultiplexing in phononic crystals with hollow cylinders[J].Physical Review E,2004,69:046608.
[72]Tones M,Montero de Espinosa F R.Ultrasonic band gaps and negative refraction[J].Ultrasonics,2004,42:787-790.
[73]Sun J H,Wu T T.Analyses of mode coupling in joined parallel phononic crystal waveguides[J].Physical Review B,2005,71:174303.
[74]Sun J H,Wu T T.Propagation of acoustic waves in phononic-crystal plates and waveguides using a finite-difference time-domain method[J].Physical Review B,2007,76:104304.
[75]Wu F G,Liu Z Y,Liu Y Y.Splitting and tuning characteristics of the point defect modes in two-dimensional phononic crystals[J].Physical Review E,2004,69:066609.
[76]赵言城,赵芳,苑立波.同质位错缺陷二维声子晶体带隙结构及传导模的研究[J].哈尔滨工程大学学报,2006,27:617-620.
[77]Toyakatsu M.Acoustic defect-mode waveguides fabricated in sonic crystal:numerical analyses by elastic finite-difference time-domain method[J].Japanese Journal of Applied Physics,2006,45:4440-4447.
[78]Vassenr J O,Deymier P A,Djafari-Rouhani B,Pennec Y,Hladky-Hennion A C.Absoult forbidden bands and waveguiding in two-dimensional phononic crystal plates[J].Physical Review B,2008,77:085415.
[79]Vasseur J O,Hladky-Hennion A C,Djafari-Rouhani B,Duval F,Dubus B,Pennec Y,Deymier P A.Waveguiding in two-dimensional piezoelectric phononic crystal plates[J].Journal of Applied Physics,2007,101:114904.
[80]Hsiao Fl,Khelif,Moubchir H,Choujaa A,Chen C C,Laude V.Waveguiding inside the complete band gaps ofa phononic crystal slab[J].Physical Review E,2007,76,056601.
[81]Qiu C Y,Liu Z Y,Shi J,Chan C T.Directional acoustic source based on the resonant cavity of two-dimensional phononic crystals[J].Applied Physics Letters,2005,86:224105.
[82]Wu F G,Zhong H L,Zhong S,Liu Z Y,Liu Y Y.Localized states of acoustic waves in three-dimensional periodic composites with point defects[J].The European Physical Journal B,2003,34:265-268.
[83]Chandra H,Deymier P A,Vassenr J O.Elastic wave propagation along waveguides in three-dimensional phononic crystals[J].Physical Review B,2004,70:054302.
[84]Psarobas I E,Stefanou N,Modinos A.Phononic crystals with planar defects[J].Physical Review B,2000,62:5536-5540.
[85]Wang X F,Kushwaha M S,Vasilopoulos P.Tunability of acoustic spectral gaps and transmission in periodically stubbed waveguides[J].Physical Review B,2001,65:035107.
[86]Hladky-Hennion A C,Vassenr J,Djafari-Rouhani B,Billy M.Sonic band gaps in one-dimensional phononic crystals with a symmetry stub[J].Physical Review B,2008,77,104304.
[87]Tanaka Y,Yano T,Tamura S.Surface guided waves in two-dimensional phononic crystals [J].Wave Motion,2007,44:501-512.
[88]Anderson P W.Absence of diffusion in certain random lattices[J].Physical Review,1958,109:1492-1505.
[89]Mott N F.Electrons in disordered structures[J].Advances in Physics,1967,16:49-144.
[90]John S,Stephen M.Localization in a disordered elastic medium near two dimensions[J].Physical Review B,1983,27:5592-5603.
[91]John S,Stephen M.Wave propagation and localization in a long-range correlated random potential[J].Physical Review B,1983,28:6358-6368.
[92]John S.Electromagnetic absorption in a disordered medium near a photon mobility edge [J].Physical Review Letters,1984,53:2169-2172.
[93]Ozbay E,Tuttil G,Sigalas M,Soukoulis C M,Ho K M.Defect structures in a layer-by-layer photonic band-gap crystal[J].Physical Review B,1995,51:13961-13965.
[94]Meade R D,Brommer K D,Rappo A M,Joannopoulos J D.Photonic band states in periodic dielectric materials[J].Physical Review B,1991,44:13772-13774.
[95]Yannopapas V,Modinos A,Stefanou N.Anderson localization of light in inverted opals [J].Physical Review B,2003,68:193205.
[96]Asatryan A A,Nicorovici N A,Botten L C,de Sterke C M,Robinson P A,McPhedran R C.Electromagnetic localization in dispersive stratified media with random loss and gain [J].Physical Review B,1998,57:13535-13549.
[97]Judge J,Pierre C,Mehmed O.Experimental investigation of mode localization and forced response amplitude magnification for a mistuned bladed disk[J].Journal of Engineering for Gas Turbines and Power-Transactions of the ASME,2001,123:940-950.
[98]Sigalas M M.Defect states of acoustic waves in a two-dimensional lattice of solid cylinders[J].Journal of Applied Physics,1998,84:3026-3030.
[99]Li F M,Wang Y S,Hu C,Huang W H.Localization of elastic waves in randomly disordered multi-coupled multi-span beams[J].Waves Random Media,2004,14:217-227.
[100]He S J,Maynard J D.Detailed measurements of inelastic scattering in Anderson localization[J].Physical Review Letters,1986,57:3171-3174.
[101]Genack A Z,Garcia N.Observation of photon localization in a three-dimensional disordered system[J].Physical Review Letters,1991,66:2264-2267.
[102]Hudges C H.Confinement of vibration by structural irregularity[J].Journal of Sound and Vibration,1982,82:411-424.
[103]李凤明,汪越胜,黄文虎,胡超.失谐周期结构中振动局部化问题的研究进展[J].力学进展,2005,35:498-512.
[104]Bouzit D,Pierre C.An experiment investigation of vibration localization in disordered multi-span beams[J].Journal of Sound and Vibration,1995,187:649-669.
[105]Bouzit D,Pierre C.Localization of vibration in disordered multi-span beams with damping[J].Journal of Sound and Vibration,1995,187:625-648.
[106]Kim D O,Lee I W.Mode localization in structures consisting of substructures and couplers[J].Engineering Structures,2000,22:39-48.
[107]Ariaratnam S T,Xie W C.Wave localization in randomly disordered nearly periodic long continuous beams[J].Journal of Sound and Vibration,1995,181:7-22.
[108]Xie W C.Buckling mode localization in randomly disordered multispan continu(?)us beams[J].AIAA Journal,1995,33:1142-1149.
[109]Ariaratnam S T,Xie W C.Buckling mode localization in randomly disordered continuous beam using a simplified model[J].Chaos,Solitons and Fractals,1996,8:1127-1144.
[110]Zhang D Z,Hu W,Zhang Y L,Li Z L,Cheng B Y,Yang G Z.Experimental verification of light localization for disordered multilayers in the visible-infrared spectrum[J].Physical Review B,1994,50:9810-9814.
[111]Vinogradov A P,Merzlikin A M.Band theory of light localization in one-dimensional disordered systems[J].Physical Review E,2004,70:026610.
[112]Vlasov Yu A,Kaliteevsld M A,Nikolaev V V.Different regimes of light localization in a disordered photonic crystal[J].Physical Review B,1999,60:1555-1562.
[113]Asatryan A A,Robinson P A,Botten L C,McPhedran R C,Nicorovici N A,Martijn de Sterke C.Effects of disorder on wave propagation in two-dimensional photonic crystals [J].Physical Review E,1999,60:006118.
[114]Topolancik J,Ilic B,Vollmer F.Experimental observation of strong photon localization in disordered photonic crystal waveguides[J].Physical Review Letters,2007,99:253901.
[115]Li F M,Xu M Q,Wang Y S.Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals[J].Solid State Communications,2007,141:296-301.
[116]Li X C,Lin Z Y,Liang H Y,Xiao Q W.Band gap and waveguide states in two-dimensional disordered phononic crystals[J].Chinese Physics Letters,2006,23:1830-1833.
[117]Han P,Chan C T,Zhang Z Q.Wave localization in one-dimensional random structures composed of single-negative metamaterials[J].Physical Review B,2008,77,115332.
[118]Bush R L.Localization lengths of electronic states in one-dimensional disordered potential chains[J].Physical Review B,1972,6:1182-1188.
[119]Nikoli(?) K,Mackirnnon A.Density of state and localization length in compositionally disordered quantum wires[J].Physical Review B,1993,47:6555-6565.
[120]Zhang D Z,Wei H,Zhang Y L,Li Z L,Cheng B Y,Yang G Z.Experimental verification of light localization for disordered multilayers in the visible-infrared spectrum[J].Physical Review B,1994,50:9810-9814.
[121]Sigalas M M,Soukoulis C M,Chan C T,Turner D.Localization of electromagnetic waves in two-dimensional disordered systems[J].Physical Review B,1996,53:8340-8348.
[122]Bliokh K Yu,Freilikher V D.Localization of transverse waves in randomly layered media at oblique incidence[J].Physical Review B,2004,70:245121.
[123]Scales J A,Van Vleck E S.Lyapunov exponents and localization in randomly layered media[J].Journal of Computational Physics,1997,133:27-42.
[124]Castanier M P,Pierre C.Predicting localization via Lyapunov exponent statistics[J].Journal of Sound and Vibration,1997,203:151-157.
[125]Xie W C,Ibrahim A.Buckling mode localization in rib-stiffened plates with misplaced stiffeners-a finite strip approach[J].Chaos,Solitons and Fractals,2000,11:1543-1558.
[126]Xie W C,Elishakoff I.Buckling mode localization in rib-stiffened plates with misplaced stiffeners-Kantorovich approach[J].Chaos,Solitons and Fractals,2000,11:1559-1574.
[127]李凤明.随机失谐周期结构中弹性波传播的局部化问题.北京交通大学博士后研究工作报告[D],2005.
[128]Li F M,Wang Y S,Hu C,Huang W H.Wave localization in randomly disordered periodic layered piezoelectric structures[J].Acta Mechanica Sinica,2006,22:559-567.
[129]Li F M,Wang Y S.Study on localization of plane elastic waves in disordered periodic 2-2piezoelectric composite structures[J].Journal of Sound and Vibration,2006,296:554-566.
[130]Li F M,Wang Y S.Study on wave localization in disordered periodic layered piezoelectric composite structures[J].International Journal of Solids and Structures,2005,42:6457-6474.
[131]周纪卿,朱因远.非线性振动[M].西安:西安交通大学出版社,1998.
[132]Kissel G J.Localization factor for multichannel disordered systems[J].Physical Review A,1991,44:1008-1014.
[133]Wolf A,Swift J B,Swinney H L,Vastano J A.Determining Lyapunov exponents from a time series[J].Physica D,1985,16:285-317.
[134]Castanier M P,Pierre C.Lyapunov exponents and localization phenomena in multi-coupled nearly periodic systems[J].Journal of Sound and Vibration,1995,183:493-515.
[135]Xie W C.Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners[J].Computers and Structures,1998,67:175-189.
[136]Bouzit D,Pierre C.Wave localization and conversion phenomena in multi-coupled multi-span beams[J].Chaos,Solitons and Fractals,2000,11:1575-1596.
[137]Cai G,Lin Y.Localization of wave propagation in disordered periodic structures[J].AIAA Journal,1990,29:450-456.
[138]Ruzzene M,Baz A.Attenuation and localization of wave propagation in periodic rods using shape memory inserts[J].Smart Materials and Structures,2000,9:805-816.
[139]Baz A.Active control of periodic structures[J].ASME,Journal of Vibration and Acoustics,2001,123:472-479.
[140]陈景良,陈向晖.特殊矩阵[M].清华大学出版社:北京,2000.
[141]Shechtman D,Blech I,Cratias D,Cahn J W.Metallic phase with long-range orientation order and translational symmetry[J].Physical Review Letters,1984,53:1951-1953.
[142]Levine D,Steinhardt P J.Quasicrystals:a new class of ordered structures[J].Physical Review Letters,1984,53:2477-2480.
[143]郭可信.五次对称性及Ti-Ni准晶相的发现与研究[J].物理,1989,18:344-346.
[144]郭可信.8次、12次对称及有关准晶的发现[J].物理,1991,20:11-14.
[145]陆洪文,费奔.二维准格点的算术结构[J].同济大学学报,2004,32:1100-1102.
[146]张利明,董闯.准晶材料性能及应用研究现状[J].材料导报,2000,14:22-24.
[147]陈敬中.准晶体的基本性质[J].地球科学-中国地质大学学报,1993,18(增刊):13-24.
[148]雷建林,王仁卉,王洲光,丁棣华.准晶切割投影图的计算与图示[J].计算物理,1997,14:47-53.
[149]陈敬中,万安娃,路湘豫,赵文霞,刘祥文.准晶结构几何理论初探[J].地球科学-中国地质大学学报,1991,16:173-180.
[150]陈敬中.准晶结构与Penrose拼图[J].地球科学-中国地质大学学报,1993,18(增刊):56-62.
[151]施倪承,闵乐泉,沈步明.准晶体的晶胞构成及准晶格的推导[J].中国科学 B,1991,11:1216-1223.
[152]胡承正,杨文革,王仁卉,丁棣华.准晶的对称性和物理性质[J].物理学进展,1997,17:345-366.
[153]冯志芳,张向东,王义全,李志远,程丙英,张道中.十二重准晶光子结构中的负折射与成像[J].物理,2006,35:9-13.
[154]李邦盛,郭俊清,李庆春.十二面体准晶形成机理的研究[J].电子显微学报,1993,2:166.
[155]刘新宝,宋广生,张振忠,杨根仓.块体单准晶材料制备的进展及展望[J].材料导 报,2000,14:14-20.
[156]Sutter-Widmer D,Deloudi S,Steurer W.Prediction of Bragg-scattering-induced band gaps in phononic quasicrystals[J].Physical Review B,2007,75:094304.
[157]范天佑.准晶数学弹性理论及应用[M].北京理工大学出版社:北京,1999.
[158]闵乐泉,刘钦圣,吴玉珍.具12次对称格点阵的衍射图[J].北京科技大学学报,1991,13:191-194.
[159]陈敬中.具有8次对称性的准晶结构模型[J].地球科学-中国地质大学学报,1993,18(增刊):105-113.
[160]陈敬中.具有12次对称性的准晶结构模型[J].地球科学-中国地质大学学报,1993,18(增刊):122-128.
[161]Steurer W,Sutter-Widmer D.Photonic and phononic quasicrystals[J].Journal of Physics D:Applied Physics,2007,40:R229-R247.
[162]Sutter-Widmer D,Deloudi S,Steurer W.Periodic average structures in phononic quasicrystals[J].Philosophical Magazine,2007,87:3095-3102.
[163]Steurer W,Haibach T.The periodic average structure of particular quasicrystals[J].Acta Crystallographica Section A,1999,55:48-57.
[164]蔡敏,姚源卫,屈晔.三元准周期半导体超晶格的电子态[J].华南理工大学学报(自然科学版),2001,29:27-30.
[165]朱家昆,胡承正,周详,龚平.准周期链状聚合物电子谱及特征[J].武汉大学学报(理学版),2003,49:369-372.
[166]朱敏,方云团,姚洁,沈廷根.一维准周期结构的光子晶体的带隙结构[J].激光杂志,2004,25:39-40.
[167]蒋美萍,王旭东,巢小刚,是度芳,陈光.准周期结构-维光子晶体的带隙特性与滤波特性[J].量子电子学报,2005,22:884-888.
[168]Hu A,Wen Z X,Jiang S S,Tong W T,Peng R W,Feng D.One-dimensional κ-component Fibonacci structures[J].Physical Review B,1993,48:829-835.
[169]Peng R W,Huang X Q,Qiu F,Wang Mu,Hu A,Jiang S S.Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers[J].Applied Physics Letters,2002,80:3063-3065.
[170]Peng R W,Liu Y M,Huang X Q,Qiu F,Wang Mu,Hu A,Jiang S S,Feng D,Ouyang L Z,Zou J.Dimerlike positional correlation and resonant transmission of electromagnetic waves in aperiodic dielectric multilayers[J].Physical Review B,2004,69:165109.
[171]Hu P,Ting C S.Electron transmission in a one-dimensional quasicrystal[J].Physical Review B,1986,34:8331-8334.
[172]Xiang T.Effect of superconducting correlation on the localization of quasiparticles in low dimensions[J].Physical Review B,1995,52:6204-6207.
[173]Gershenson M E,Khavin Yu B.Crossover from weak to strong localization in quasi-one-dimensional conductors[J].Physical Review Letters,1997,79:725-728.
[174]Avishai Y,Luck J M.Localization properties of quasi-one-dimensional conductor networks in a random magnetic field[J].Physical Review B,1994,49:8679-8688.
[175]Hjort M,Stafstr(o|¨)m.Localization in quasi-one-dimensional systems[J].Physical Review B,2000,62:5245-5250.
[176]Chan Y S,Chan C T,Liu Z Y.Photonic band gaps in two dimensional photonic quasicrystals[J].Physical Review Letters,1998,80:956-959.
[177]Jin C J,Cheng B Y,Man B Y,Li Z L,Zhang D Z.Band gap and wave guiding effect in a quasiperiodic photonic crystal[J].Applied Physics Letters,1999,27:1848-1850.
[178]Zoorob M E,Charlton M D B,Parker G J,Baumberg J J,Netti M C.Complete photonic bandgaps in 12-fold symmetric quasicrystals[J].Nature,2000,404:740-743.
[179]Velasco V R,P(?)rez-Alvarez R,Garcia-Moliner F.Some properties of the elastic waves in quasiregular heterostructures[J].Journal of Physics:Condensed Matter,2002,14:5933-5957.
[180]罗青.声波在一维准周期系统中的传播[J].南京师大学报(自然科学版),2002,25:110-115.
[181]曹永军,曹纯红,周培勤.一维准周期结构声子晶体透射性质的研究[J].物理学报,2006,55:6470-6474.
[182]方云团,王永顺,姜光.声波在球状准周期多层介质中的传播[J].应用声学,2004,23:23-28.
[183]Zhang X D.Universal non-near-field focus of acoustic waves through high-symmetry quasicrystals[J].Physical Review B,2007,75:024209.
[184]Lai Y,Zhang X D,Zhang Z Q.Large sonic band gaps in 12-fold quasicrystals[J].Journal of Applied Physics,2002,91:6191-6193.
[185]Lai Y,Zhang Z Q.Large enhancement of phononic gap in periodic and quasiperiodic elastic composites by using air inclusions[J].Z.Kristallogr.,2005,220:877-883.
[186]King P D C,Cox T J.Acoustic band gaps in periodically and quasiperiodically modulated waveguides[J].Journal of Applied Physics,2007,102:014902.
[187]Gao J,Cheng J C,Li B W.Propagation of Lamb waves in one-dimensional quasiperiodic composite thin plates:A split of phonon band gap[J].Applied Physics Letters,2007,90:111908.
[188]Aynaou H,Boudouti E H El,Djafari-Rouhani B,Akjouj A,Velasco V R.Propagation and localization of acoustic waves in Fibonacci phononic circuits[J].Journal of Physics:Condensed Matter,2005,17:4245-4262.
[189]Veselago V G.The electrodynamics of substances with simultaneously negative values of permittivity and permeablility[J].Soviet Physics USPEKI,1968,10:509-514[USP.Fiz.Nauk,1967,8:2854-2859].
[190]Shelby R A,Smith D R,Schultz S.Experimental verification of a negative index of refraction[J].Science,2001,292:77-79.
[191]Gralak B,Enoch S,Tayeb G.Anomalous refraction properties of photonic crystals[J].Journal of Optical Society of America,2000,17:1012-1020.
[192]Pendry J B.Negative refraction makes a perfect lens[J].Physical Review Letters,2000,85:3966-3969.
[193]Cervera F,Sanchis L,Sanchez-Perez J V,Martinez-Sala R,Rubio C,Meseguer F,Lopez C,Caballero D,Sanchez-Dehesa J.Refractive acoustic devices for airborne sound[J].Physical Review Letters,2002,88:023902.
[194]Gupta B C,Ye Z.Theoretical analysis of the focusing of acoustic waves by two-dimensional sonic crystals[J].Physical Review E,2003,67:046606.
[195]Zhang X D,Liu Z Y.Negative refraction of acoustic waves in two-dimensional phononic crystls[J].Apllied Physics Letters,2004,85:341-343.
[196]Ko M,Liu Z Y,Qiu C Y,Wang W,Shi J,Wen W J,Sheng P.Negative-refraction imaging with two-dimensional phononic crystals[J].Physical Review B,2005,72:064306.
[197]Yang S X,Page J H,Liu Z Y,Cowan M L,Chan C T,Sheng P.Focusing of sound in a 3D phononic crystal[J].Physical Review Letters,2004,93:024301.
[198]Sukhovich A,Jing L,Page J H.Negative refraction and focusing of ultrasound in two-dimensional phononic crystals[J].Physical Review B,2008,77:014301.
[199]Tarasenko O S,Tarasenko S V,Yurchenko V M.Effect of negative acoustic refraction in a one-dimensional phononic crystal[J].JETP Letters,2004,80:484-486.
[200]Li J,Liu Z Y,Qin C Y.Negative refraction imaging of acoustic waves by a two-dimensional three-component phononic crystal[J].Physical Review B,2006,73:054302.
[201]Feng L,Liu X P,Lu M H,Chen Y B,Chen Y F,Mao Y W,Zi J,Zhu Y Y,Zhu S N,Ming N B.Acoustic backward-wave negative refractions in the second band of a sonic crystal [J].Physical Review Letters,2006,96:014301.
[202]Fang N,Xi D J,Xu J Y,Ambati M,Srituravanish W,Sun C,Zhang X.Ultrasonic metamatcrials with negative modulus[J].Nature Materials,2006,5:452-456.
[203]Hu X H,Chan C T.Refraction of water waves by periodic cylinder arrays[J].Physical Review Letters,2005,95:154501.
[204]卢明辉,冯亮,刘小平,陈延斌,陈延峰,毛一薇,资剑,朱永元,祝世宁,闵乃本.声子晶体中第二能带的回波负折射[J].物理,2006,35:969-974.
[205]Alvarez A,Ye Z.Effects of fish school structures on acoustic scattering[J].ICES Journal of Marine Science,1999,56:361-369.
[206]Martinez-Sala R,Rubio C,Garcla-Raffi L M,S(?)nchez-P(?)rez J V,S(?)nchez-P(?)rez E A,Llinares J.Control of noise by tress arranged like sonic crystals[J].Journal of Sound and Vibration,2006,291:100-106.
[207]Benchabane S,Khelif A,Choujaa A,Djafari-Rouhani B,Laude V.Interaction of waveguide and localized modes in a phononic crystal[J].Europhysics Letters,2005,71:570-575.
[208]Zhang X,Liu Z Y,Liu Y Y,Wu F G.Defect states in 2D acoustic band-gap materials with bend-shaped linear defects[J].Solid State Communications,2004,130:67-71.
[209]Permec Y,Djafari-Rouhani B,Vasseur J O,Larabi H,Khelif A,Choujaa A,Benchabane S,Laude V.Acoustic channel drop tunneling in a phononic crystal[J].Applied Physics Letters,2005,87:261912.
[210]Pennec Y,Djafari-Rouhani B,Vasseur J O,Khelif A,Deymier P A.Tunable filtering and demultiplexing in phononie crystals with hollow cylinders[J].Physical Review E,2004,69:046608.
[211]Qiu C Y,Liu Z Y.Acoustic directional radiation and enhancement caused by band-edge states of two-dimensional phononie crystals[J].Applied Physics Letters,2006,89:063106.
[212]Ke M Z,Liu Z Y,Pang P,Wang W G.,Cheng Z G,Shi J,Zhao X Z.Highly directional acoustic wave radiation based on asymmetrical two-dimensional phononic crystal resonant cavity[J].Applied Physics Letters,2006,88:263505.
[213]Qiu C Y,Liu Z Y,Mei J,Shi J.Mode-selecting acoustic filter by using resonant tunneling of two-dimensional double phononic crystals[J].Applied Physics Letters,2005,87:104101.
[214]Diez A,Kakaranttzas G.Acoustic stop-bands in periodically microtapered optical fibers[J].Applied Physics Letter,2000,76:3481-3483.
[215]Cervera F,Sanchis L,S(?)nchez-P(?)rez J V,Martinez-Sala R,Rubio C,Meseguer F,L(?)pez C,Caballero D,S(?)nchez-Dehesa J.Refractive acoustic devices for airborne sound[J].Physical Review Letters,2002,88:023902.
[216]Hu X H,Chan C T,Zi J.Two-dimensional sonic crystals with Helmholtz resonators[J].Physical Review E,2005,71:055601(R).
[217]Jang J H,Ullal C K,Gorishnyy T,Tsukruk V V,Thomas E L.Mechanically tunable three-dimensional elastomeric network/air structures via interference lithography[J].Nano Letters,2006,6:740-743.
[218]Cheng W,Wang J J,Jonas U,Fytas G,Stefanou N.Observation and tuning of hypersonic bandgaps in colloidal crystals[J].Nature Materials,2006,5:830-836.
[219]Baz(?)n A,Torres M,Montero de Espinosa F R,Quintero-Torres R,Arag(o|¨)n J L.Conformal mapping of ultrasonic crystals:Confining ultrasound and cochlearlike waveguiding[J].Applied Physics Letters,2007,90:094101.
[220]Tommaseo G,Petekidis G,Steffen W,Fytas G,Schofield A B,Stefanou N.Hypersonic acoustic excitations in binary colloidal crystals:big versus small hard sphere control[J].The Journal of Chemical Physics,2007,126:014707.
[221]Yilmaz C,Hulbert G M,Kikuchi N.Phononic band gaps induced by inertial amplification in periodic media[J].Physical Review B,2007,76:054309.
[222]Halkj(?)r S,Sigmund O,Jensen J.Inverse design of phononic crystals by topology optimization[J].Z.Kristallogr.,2005,220:895-905.
[223]钟会林,吴福根,姚立宁.遗传算法在二维声子晶体带隙优化中的应用[J].物理学报,2006,55:275-280.
[224]Wang G,Yu D L,Wen J H,Liu Y Z,Wen X S.One-dimensional phononic crystals with locally resonant structures[J].Physics Letters A,2004,327:512-521.
[225]Sepehrinia R,Bahraminasab A,Sahimi M,Tabar M R R.Dynamic renormalization group analysis of propagation of elastic waves in two-dimensional heterogeneous media[J].Physical Review B,2008,77:014203.
[226]张欣.声子晶体中声波或弹性波带隙的研究[D].华南理工大学博士学位论文,2004.
[227]曹永军.弹性波在二维声子晶体中传播特性的研究[D].华南理工大学博士学位论文,2005.
[228]Lu Y,Peng R W,Wang Z,Tang Z H,Huang X Q,Wang M,Qiu Y,Hu A Jiang S S,Feng D.Resonant transmission of light waves in dielectric heterostructures[J].Journal of Applied Physics,2005,97:123106.
[229]Yang S X,Page J H,Liu Z Y,Cowan M L,Chart C T,Sheng P.Ultrasound tunneling through 3D Phononic crystals[J].Physical Review Letters,2002,88:104301.
[230]Wu F G,Liu Z Y,Liu Y Y.Acoustic band gaps created by rotating square rods in a two-dimensional lattice[J].Physical Review E,2002,66:046628.
[231]Wu T T,Hsu C H,Sun J H.Design of a highly magnified directional acoustic source based on the resonant cavity of two-dimensional phononic crystals[J].Applied Physics Letters,2006,89:171912.
[232]Tanaka Y,Tamura S.Acoustic stop bands of surface and bulk modes in two-dimensional phononic lattices consisting of aluminum and a polymer[J].Physical Review B,1999,60:13294-13297.
[233]Zhang X,Liu Z Y,Liu Y Y,Wu F G.Elastic wave band gaps for three-dimensional phononic crystals with two structural units[J].Physical Letters A,2003,313:455-460.
[234]Zhao H G,Liu Y Z,Yu D L,Wang G,Wen J H,Wen X S.Absorptive properties of three-dimensional phononic crystal[J].Journal of Sound and Vibration,2007,303:185-194.
[235]Wang G,Wen J H,Han X Y.Finite difference time domain method for the study of band gap in two-dimensional phononie crystals[J].Acta Physica Sinica-Overseas Edition,2003,52:1943-1947.
[236]Zhang X,Liu Y Y,Wu F G,Liu Z Y.Large two-dimensional band gaps in three-component phononic crystals[J].Physics Letters A,2003,317:144-149.
[237]Hou Z L,Fu X J,Liu Y Y.Acoustic wave in a two-dimensional composite medium with anisotropie inclusions[J].Physics Letters A,2003,317:127-134.
[238]Hou Z L,Wu F G,Liu Y Y.Phononie crystals containing piezoelectrie material[J].Solid State Communications,2004,130:745-749.
[239]吴福根.声子晶体中的带隙及缺陷态的研究[D].华南理工大学博士学位论文,2001.
[240]Cao Y J,Hou Z L,Liu Y Y.Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals[J].Solid State Communications,2004,132:539-543.
[241]Wen J H,Yu D L,Wang G,Zhao H G,Liu Y Z,Wen X S.The directional propagation characteristics of elastic wave in two-dimensional thin plate phononic crystals[J].Physics Letters A,2007,364:323-328.
[242]Hou Z L,Kuang W,Liu Y Y.Transmission property analysis of a two-dimensional phononic crystal[J].Physics Letters A,2004,333:172-180.
[243]Li J,Chan C T.Double-negative acoustic metamaterial[J].Physical Review E,2004,70:055602.
[244]Goffaux C,Vigneron J P.Theoretical study of a tunable phononic band gap system[J].Physical Review B,2001,64:075118.
[245]齐共金.二维声子晶体的带隙研究[D].国防科技大学硕士学位论文,2002.
[246]Sigalas M,Kushwaha M S,Economou E N,Kafesaki M,Psarobas I E,Steurer W.Classical vibrational modes in phononie lattices:theory and experiment[J].Z.Kristallogr.,2005,220:765-809.
[247]Huang Z G,Wu T T.Analysis of wave propagation in phonunic crystals with channel using the plane-wave expansion and supercell techniques[J].IEEE Ultrasonics Symposium,2005,77-80.
[248]Yu D L,Liu Y Z,Wang G,Cai L,Qiu J.Low frequency torsional vibration gaps in the shaft with locally resonant structures[J].Physics Letters A,2006,348:410-415.
[249]Wang G,Liu Y Z,Wen J H,Yu D L.Formation mechanism of the low-frequency locally resonant band gap in the two-dimensional ternary phononic crystals[J].Chinese Physics,2006,15:407-411.
[250]Miyashita T.Acoustic defect-mode waveguides fabricated in sonic crystal:Numerical analyses by elastic finite-difference time-domain method[J].Journal of Applied Physics,2006,45:4440-4447.
[251]Wang G,Shao L H,Liu Y Z,Wen J H.Accurate evaluation of lowest band gaps in ternary locally resonant phononic crystals[J].Chinese Physics,2006,15:1843-1848.
[252]Yao Y W,Hou Z,Cao Y J,Liu Y Y.An improved method of eigen-mode matching theory in two-dimensional phononic crystal[J].Physica B,2007,388:75-81.
[253]Bonello B,Charles C,Ganot F.Lamb waves in plates covered by a two-dimensional phononic film[J].Appllied Physics Letters,2007,90:021909.
[254]Chen H Y,Luo X D,Ma H N.Scattering of elastic waves by elastic spheres in a NaCl-type phononic crystal[J].Physical Review B,2007,75:024306.
[255]Liu Y Z,Yu D L,Li L,Zhao H G,Wen J H,Wen X S.Design guidelines for flexural wave attenuation of slender beams with local resonators[J].Physics Letters A,2007,362:344-347.
[256]Yao Y W,Hou Z L,Liu Y Y.The two-dimensional phononic band gaps tuned by the position of the additional rod[J].Physics Letters A,2007,362:494-499.