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Propagation of acoustic waves in the woodpile sonic crystal with a defect
Applied Acoustics 73 (2012) 312–322
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Propagation of acoustic waves in the woodpile sonic crystal with a defect Liang-Yu Wu, Lien-Wen Chen ⇑ Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
a r t i c l e
i n f o
Article history: Received 10 May 2011 Received in revised form 1 October 2011 Accepted 3 October 2011 Available online 25 October 2011 Keywords: Sonic crystal Woodpile structure Defect band Defect mode
a b s t r a c t Acoustic wave propagation in a woodpile sonic crystal with a defect is studied theoretically and experimentally. The woodpile sonic crystal is composed of polymethyl methacrylate square rods which orthogonally stacked together, and it is embedded in air background. Defects are created by varying the width and positions of the middle rods in the periodic structure. Defect bands and transmission spectra are calculated by using the finite element method with the periodic boundary condition and the Bloch–Floquet theorem. Frequencies of defect bands are strongly dependent on the width and positions of the middle rods in the periodic structure. The experimental transmission spectra of the woodpile sonic crystals with a defect are also presented and compared with the numerical results. The defect mode properties of the woodpile sonic crystal with a defect can be applied to design novel acoustic devices for filtering sound and trapping sound in defects. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The acoustic and elastic wave propagation in periodic composite media (called sonic crystal or phononic crystal) has received much experimental and theoretical attention in recent years. Phononic crystals can exhibit acoustic or elastic band gaps in which acoustic wave and vibration are forbidden in any direction. The band gap phenomena can be used as filters, transducers, and for creation of vibration-free environments. The width of band gaps is determined by the contrast of elastic constants of constructed elements, the filling ratio and the lattice. Most of the studying works have been focused on the two-dimensional (2D) sonic crystals. The band structures and band gap width of sonic crystals consisting of lattices with various symmetry and scatterers of different shapes, orientations, and sizes have been investigated [1–5]. The acoustic band gaps have been experimentally presented for the rigid rods in air [6,7]. Martínez-Sala et al. [6] have measured the attenuation of sound in a minimalist sculpture consisting of steel cylinders in air; Sánchez-Pérez et al. [7] have demonstrated an experimental analysis of the acoustic transmission of a 2D periodic array of rigid cylinders in air with the square and triangular lattices. Moreover, one interesting property of these periodic structures is the possibility of generating defects that confine elastic or acoustic waves in localized modes [8–14]. As the periodicity of the sonic crystals is broken locally, the defect modes can be found within the acoustic band gaps, which are strongly localized around the ⇑ Corresponding author. Tel.: +886 6 2757575x62143; fax: +886 6 2352973. E-mail address: [email protected] (L.-W. Chen). 0003-682X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2011.10.002
defects. Point defects and line defects can act as resonant cavities and waveguides, respectively. An acoustic wave in range of gap frequencies is trapped in the point defect or propagates along the line defect. The defect modes in a 2D sonic crystal have been reported and discussed [8–10]. The point defect band can sweep inside the band gap as the defect radius changes. Khelif et al. [11] theoretically and experimentally have observed a full band gap and localized defect modes in an acoustic band gap in a sonic crystal that comprised steel cylinders and water from the transmission spectra. Moreover, Wu et al. [12,13] experimentally have measured the spectra and pressures in the defect of the sonic crystal, which is composed of polymethyl methacrylate (PMMA) cylinders and air, to observe the localization of the acoustic waves in the defect at the resonant frequency. Wu and Chen [14] have also studied the wave propagation of the 2D sonic crystal with a local resonant defect theoretically and experimentally. Three-dimensional (3D) sonic crystals have also been investigated by several authors. Kafesaki et al. [15] and Kushwaha et al. [16,17] have investigated band structures and acoustic band gaps of 3D sonic crystals with the face-centered-cubic, body-centeredcubic and simple cubic lattice for solid–solid, gas–liquid and liquid–liquid systems. Sainidou et al. [18] have studied the absolute band gaps in a 3D sonic crystal consisting of steel spheres in polyester, and reported how the width of band gaps depends on the lattice geometry. Kuang et al. [19] have reported the band structures of 3D solid sonic crystals consisting of four different scatterers (spherical, cubic, rhombic dodecahedral and truncated octahedral). Moreover, acoustic point defect states in 3D simplecubic periodic structure composed of water sphere in mercury background have also been investigated [20].
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The woodpile structure is one of the most popular 3D periodic structures in photonic crystals [21–26]. The woodpile structure, which is also called a layer-by-layer structure, differs from the other geometries by its simpler design. A 3D woodpile sonic crystal structure is built by blocks or rods that are orthogonally stacked together. It means that the adjacent layers are perpendicular to each other. The rods may touch each other, may overlap or be detached to a certain extent. The shape of the rods can be spherical, cuboidal, or ellipsoidal, etc. Between every other layer, the rods can be either shifted relative to each other by one-half of a period, or aligned without shift. These structures possess face-centered-tetragonal and body-centered-tetragonal lattice symmetries correspondingly. The woodpile sonic crystal has been proposed to demonstrate the self-collimation of the ultrasonic wave [27]. Jiang et al. [28] have also studied an acoustic woodpile structure fabricated by locally resonant materials, and the result shows that this structure has a strong sound absorbing capability in a wide frequency range. And then, in our previous researches, the acoustic band gaps of a 3D woodpile sonic crystal have been investigated theoretically and experimentally [29]. The purpose of this paper is to study the acoustic wave propagation in woodpile sonic crystals with a defect. The woodpile structure can be considered as a one-dimensional (1D) periodic structure as acoustic wave propagate along a single direction. A traditional 1D sonic crystal is composed of the layered structure [30,31]. The defect can be introduced into a 1D layered sonic crystal by varying the thickness or material of the middle layer in the periodic structure. The woodpile structure is more complex than the layered structure, so it has more geometric parameters to control the locations of band gaps and defect modes. Two kinds of defects are proposed in the woodpile structure. The first one is the rod defect, which is created by varying width of the middle rods in the periodic structure; the second one is the shift defect,
Fig. 1. (a) Schematic of a 3D woodpile structure with the body-centered-tetragonal lattice. The structure is composed of rectangular rods that are orthogonally stacked together. l1 and l2 are the width of rods, and a0 is the lattice constant. (b) 7 1 1 supercell of the C–X direction. (c) 1 1 7 supercell of the C–X0 direction.
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which is created by varying positions of the middle rods in the periodic structure. Defect bands and transmission spectra are obtained by using the finite element method (FEM) with the periodic boundary condition and the Bloch–Floquet theorem. Moreover, the transmission spectra of woodpile sonic crystals with the rod and shift defects are measured and compared with the numerical results. 2. Model and numerical method A woodpile sonic crystal composed of solid rod layers with a structural period of two layers in the z direction is considered to study, which is shown in Fig. 1a. This woodpile structure is of body-centered-tetragonal lattice. This structure also can be stacked by simple cubic unit cells, so it can be considered as the simple cubic lattice [29]. a0 is the lattice constant. l1 and l2 are the width of the rectangular rod. The z direction is called the C–X0 direction, and the x and y directions are called the C–X direction. We vary the width or shift the position of one of rods in the infinite woodpile structure to destroy its periodicity. This imperfect region is called as a defect. The C–X and C–X0 directions of the woodpile structure with the simple cubic lattice is proposed to study wave propagation in the woodpile sonic crystal with a defect. The woodpile structure can be regarded as a 1D periodic structure as wave prop-
Fig. 2. (a) Woodpile sonic crystal with a rod defect along the C–X direction. The width of the middle rod is varied to create a rod defect, where l is the width of the middle rod. (b) Defect mode frequencies in the first and second band gaps of the C– X direction as a function of the width of the middle rod. The inset is the zoom of the highest defect band in the second band gap in the region 0 5 l/a0 5 0.2, where is indicated by a rectangular frame. Solid lines represent edges of the first and second band gaps of the C–X direction for the perfect woodpile structure.
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agation along a single direction (C–X or C–X0 direction) is considered. In order to analyze the non-ideal periodical structure of the sonic crystal, the supercell method is adopted. The supercell structures of the C–X and C–X0 directions are shown in Fig. 1b and c, respectively. The supercell structure consists of many unit cells. Both supercell structures have seven periods, and a defect can be introduced into the middle of the supercell. Both of them are acted as unit cells in our calculations. The size of the supercells should be large enough since we must guarantee that the coupling effect between neighboring supercells can be ignored. The structure is surrounded by fluid background. The woodpile sonic crystals can be analyzed by using a commercial program, COMSOL MultiphysicsÒ, based on the FEM [32]. As the acoustic impedance of fluid background is much smaller than that of solid rods, the total longitudinal waves propagating in fluid will be almost reflected by solid rods. The wave propagation in such sonic crystal is predominant in the fluid. So, the transverse waves in solid rods can be ignored. In order to achieve this approximation, we can consider the solid rods as fluid with very high stiffness and specific mass. The structure is theoretically assumed to be infinite and periodic in the x, y, and z directions. Considering the periodic boundary conditions above allows us to reduce the model to a single unit cell. According to the Bloch–Floquet theorem, the relation between the
pressure distribution p for nodes lying on the boundary of the unit cell can be expressed as
pðx þ a1 ; y; zÞ ¼ pðx; y; zÞ expðiK 1 a1 Þ; pðx; y þ a2 ; zÞ ¼ pðx; y; zÞ expðiK 2 a2 Þ;
ð1Þ
pðx; y; z þ a3 Þ ¼ pðx; y; zÞ expðiK 3 a3 Þ; where a1, a2, and a3 are the basis vectors of the periodic structure, and K = (K1, K2, K3) is the Bloch wave vector. A phase relation is applied on the lateral faces of the unit cell, defining boundary conditions between adjacent units. This phase relation is related to the wave number of the incident wave in the periodic structure. The analysis of eigenfrequencies and the corresponding eigenvectors can be performed by giving Bloch wave vectors. The eigenvectors are related to the pressure distribution of the modes. And then, eigenfrequencies are assembled to build the band structures along the C–X and C–X0 directions of the first Brillouin zone. Defect bands of supercell structures can be found in the band structure. Furthermore, the woodpile sonic crystals can be studied by calculating the transmission spectra. We stack several unit cells to construct the periodic structure with and without a defect along a specific direction (ex. C–X or C–X0 direction). And then, the periodic boundary conditions are set on other directions. The acoustic wave at the pass band or band gap frequencies is incident into the
Fig. 3. Calculated transmission spectra of the woodpile sonic crystal with a rod defect along the C–X direction. The values of l/a0 for (a), (b) and (c) are 0, 0.2 and 0.8, respectively. Arrows indicate the defect modes in the band gaps.
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periodic structure and propagates in the structure along the specific direction. The power ratio of the transmitted wave to incident wave represents the transmission coefficient. wo and wi denote the transmitted power at the outlet of the periodic structure and the incident power at the inlet of the periodic structure, respectively. wo and wi are defined as [32]
wo ¼
Z X
jpo j2 dA and wi ¼ 2qair cair
Z X
jpi j2 dA; 2qair cair
ð2Þ
where po and pi are the pressure at the outlet and the inlet, respectively, and X is the area of the outlet and the inlet. The transmission coefficient can be expressed as
T ¼ wo =wi
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rod is square (l1 = l2) and the ratio of l1/a0 is 0.5. We have previously studied that the first and second band gaps of the C–X direction extend from 0.284 to 0.575 xa0/2pcair and from 0.628 to 0.831 xa0/2pcair, respectively, and the band gap of the C–X0 direction extends from 0.628 to 1 xa0/2pcair [29]. The supercells shown in Fig. 1b and c are adopted to calculate the band structures and defect bands. Band structure calculated by the FEM calculation with a periodic boundary condition is used to analyze the infinite structure. The FEM calculation is also adopted to calculate the transmission spectra of the finite structure. In this paper, the defect is located at the middle of periodic structure for all transmission calculation works, and both periodic structures along the C–X and C– X0 directions have seven periods.
ð3Þ
The transmission coefficients are functions of the frequency. The transmission spectrum can be built by assembling transmission coefficients of the different frequencies. 3. Numerical results A woodpile sonic crystal composed of PMMA rods in air background is proposed to study. The material parameters are density qPMMA = 1180 kg/m3 and qair = 1.2 kg/m3, and sound of speed cPMMA = 2700 kg/m3 and cair = 343 kg/m3. The cross-section of the
Fig. 4. (a) Woodpile sonic crystal with a shift defect along the C–X direction. The position of the middle rod is shifted to create a shift defect, where d is the shift distance of the middle rod from the original position. (b) Defect mode frequencies in the first and second band gaps of the C–X direction as a function of shift distance of the middle rod. Solid lines represent edges of the first and second band gaps of the C–X direction for the perfect woodpile structure.
3.1. C–X direction 3.1.1. Rod defect We vary the width of the middle rod to create a rod defect in the C–X direction as shown in Fig. 2a, where l is the width of the middle rod. The dependences of the defect band frequencies in the first two band gaps of the C–X direction on the value of l/a0 are presented in Fig. 2b. Solid lines represent the first and second band gap edges of the C–X direction of the perfect woodpile structure. The inset in Fig. 2b is the zoom of the highest defect band in the second band gap in the region 0 5 l/a0 5 0.2. From Fig. 2b, it reveals that there are two defect bands in the first band gap and three defect bands in the second band gap as the middle rod does not exist (l/a0 = 0). The defect bands in the first band gap and the lowest defect band in the second band gap disappear in the region 0.1 5 l/ a0 5 0.4. Note that the highest defect band in the second band gap becomes two defect bands as the width of a rod defect exists. Both defect bands move toward and immerge into the upper edge of the second band gap as l/a0 increases from 0.1 to 0.2. The middle defect band in the second band gap moves toward and immerges into the lower edge of the second band gap as l/a0 increases from 0 to 0.5. There is no defect mode at l/a0 = 0.5 since it is a perfect periodic structure. For l/a0 > 0.6, the defect bands emerge from the upper edge of the first and second band gaps moving toward the lower band gap edge as l/a0 increases. On the other hand, an additional defect band emerges from the lower edge of the first band gap at l/a0 = 0.8 and moves toward the upper band gap edge as l/a0 increases. The transmission spectra of the periodic structure with a rod defect are shown in Fig. 3, where the values of l/a0 are 0, 0.2 and 0.8, respectively. Since their upper edge frequency of the second band gap is higher than that of the perfect periodic structure, the defect mode which is close to the upper edge of the second band gap can be clearly observed in Fig. 3a and b. The frequencies of defect modes are in good agreement with the band structures. From Fig. 3c, only one defect mode can be found in the first band gap since the lower defect mode is very close to the lower edge of the first band gap. We can see that the middle pass band disappears but the defect modes still correspond to the band structure. 3.1.2. Shift defect We shift the middle rod along the C–X direction to create a shift defect as shown in Fig. 4a, where d is the shift distance of the middle rod from the original position. Fig. 4b shows that the defect band frequencies in the first two band gaps of the C–X direction are dependent on the value of d/a0. The structure is a perfect periodic structure as d/a0 is 0. Variation tendency of the defect band frequencies for the positive and negative shift distance is symmetric. In the first and second band gaps, defect modes emerge from the lower and the upper edges of both band gaps as the absolute value of d/a0 increases. An additional defect band emerges from the upper edge of the second band gap at |d/a0| = 0.3 and moves to-
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Fig. 5. Calculated transmission spectra of the woodpile sonic crystal with a shift defect along the C–X direction. The values of d/a0 for (a), (b) and (c) are 0.2, 0.3 and 0.4, respectively. Arrows indicate the defect modes in the band gaps.
Fig. 6. (a) Woodpile sonic crystal with a rod defect along the C–X0 direction. The width of the middle rod is varied to create a rod defect, where l is the width of the middle rod. (b) Defect mode frequencies of in the band gap of the C–X0 direction as a function of the width of the middle rod. Solid lines represent edges of the band gap of the C–X0 direction for the perfect woodpile structure.
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Fig. 7. Calculated transmission spectra of the woodpile sonic crystal with a rod defect along the C–X0 direction. The values of l/a0 for (a), (b) and (c) are 0, 0.3 and 0.8, respectively. Arrows indicate the defect modes in the band gaps.
Fig. 8. Woodpile sonic crystal with a shift defect along the C–X0 direction. The position of the middle rod is shifted to create a shift defect, where d is the shift distance of the middle rod from the original position.
ward the lower band gap edge as |l/a0| increases. Note that this defect band is very close to the upper edge of the second band gap.
The transmission spectra of the periodic structure with a shift defect are shown in Fig. 5, where the values of d/a0 are 0.2, 0.3
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and 0.4, respectively. Similarly, their upper edge frequency of the second band gap is also higher than that of perfect periodic structure, so we can obviously observe that there are three defect modes in the second band gap. The transmission spectrum of d/a0 = 0.2 is different from the result of band structure of d/a0 = 0.2. Because the highest defect band shown in Fig. 5a is out of the second band gap of the perfect periodic structure, it cannot be observed in Fig. 4b. Frequencies of the highest defect modes are almost independent of the value of d/a0. Variation tendency of defect modes in the transmission spectra also has a good agreement with the results obtained from the band structures.
3.2. C–X0 direction 3.2.1. Rod defect Fig. 6a shows the rod defect in the C–X0 direction, where l is the width of the middle rod. Fig. 6b shows that the defect band frequencies in the band gap are dependent of the value of l/a0. It reveals that there is one defect band in the band gap as l/a0 = 0, and two defect bands exists in the band gap as l/a0 = 0.05. It is because another defect mode would occur as the width of rod defect exists. Both of defect bands immerge into the upper and lower edges of band gap respectively. For l/a0 = 0.5, the defect band cannot be found in the band gap.
Fig. 7 shows the transmission spectra of the periodic structure with a rod defect, where the values of l/a0 are 0, 0.3 and 0.8, respectively. The transmission spectra of l/a0 = 0 and 0.8 have a good agreement with the band structures. However, from Fig. 7b, it reveals that only one defect mode is found in the band gap. The higher defect band cannot found in the transmission spectra. The higher defect mode can be seen as the deaf defect mode [7,14,29]. The incident plane wave along the C–X0 direction does not excite this defect mode. So, the higher defect mode cannot be observed in the transmission spectrum of the C–X0 direction.
3.2.2. Shift defect The shift defect in the C–X0 direction is shown in Fig. 8, where d is the shift distance of the middle rod from the original position. A defect band emerges from the lower edge of the band gap as the middle rod is shifted. But, it is noticed that the defect mode is very close to the lower edge of the band gap and can be acted as the edge of the band gap. The transmission spectra of the periodic structure with a shift defect are shown in Fig. 9, where the values of d/a0 are 0.2, 0.35 and 0.5, respectively. From Fig. 9a and b, there exists no defect mode in the band gap. However, it is noticed that there is a defect mode in Fig. 9c. This defect mode is very close to the lower edge of band gap.
Fig. 9. Calculated transmission spectra of the woodpile sonic crystal with a shift defect along the C–X0 direction. The values of d/a0 for (a), (b) and (c) are 0.2, 0.35 and 0.5, respectively. An arrow indicates the defect mode in the band gaps.
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4. Experimental results and discussion The experimental modal is a woodpile structure consisting of PMMA square rods in air background. PMMA square rods with a width l1 = l2 = 2 cm are orthogonally stacked together and the lattice constant a0 is 4 cm. The experimental models of the C–X and C–X0 directions both have seven periods, and the defect is located at the middle of periodic structure. The speaker (Fostex: FF85K) and the microphone (Brüel & Kjær: 4190) are used as a sound source and a receiver, respectively. The speaker connected with the function generator (NF: WF-1945B) can either set to specific frequencies, or sweep through a variable range of the frequencies. In this experiment, the range of sweeping frequencies is from 200 Hz to 15 kHz. Sound signals can be detected by the microphone and are displayed in the digital sampling oscilloscope (Tek-
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tronix: TDS5032B). Spectra can also be obtained by using the spectrum analysis. In order to reduce reflected waves from the environment, sound-absorption sponges are used to enclose the sonic crystal structure. The sonic crystal is placed between the speaker and the microphone. The distance between the speaker and sonic crystal is about 22 cm; the distance between the microphone and the sonic crystal is about 10 cm. Both of the microphone and speaker are located at the center of woodpile structure, and the microphone is collinear to the speaker. 4.1. C–X direction Fig. 10 shows the experimental transmission spectra which are measured along the C–X direction. Fig. 10a is the experimental transmission spectrum of the perfect sonic crystal. Two measured
Fig. 10. Experimental transmission spectra along the C–X direction. (a) shows the case of the perfect woodpile sonic crystal; (b), (c) and (d) show the cases of the woodpile sonic crystal with a rod defect. The values of l for (b), (c) and (d) are 0 cm, 1.2 cm and 3.2 cm, respectively.
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band gaps are observed from 2.5 to 5 kHz and from 5.2 to 6 kHz, respectively. Fig. 10b–d shows the experimental transmission spectra of the woodpile structure with a rod defect, where the values of l are 0 cm, 1.2 cm and 3.2 cm, respectively. From Fig. 10b, it can be seen that two defect modes which are indicated by arrows can be observed in the first band gap. The frequencies of both defect modes are 3.17 and 4.93 kHz respectively. We can read the frequencies of both defect modes are 3.08 kHz and 4.8 kHz from the band structure calculation respectively. The measured frequencies of both defect modes are close to the numerical results. Comparing Fig. 10c with Fig. 10a, both experimental transmission spectra are similar. The defect mode in the second band gap cannot be mea-
sured and observed in Fig. 10c. From Fig. 10d, we can see that the middle pass band disappear and the width of the band gap become wider. This measured result is a good agreement with the numerical transmission spectrum shown in Fig. 3c. However, the defect modes in the first and second band gaps cannot be measured. It can be observed that a peak always exists at 2.83 kHz in Fig. 10. This peak is not the defect mode. This peak is resulted from the positions of the speaker and microphone since it exists in many experimental transmission spectra. Fig. 11 shows the experimental transmission spectra of the sonic crystal with a shift defect. No defect can be observed in the experimental transmission spectra. The middle pass band disappears as the shift distance d increases. Note
Fig. 11. Experimental transmission spectra of the woodpile sonic crystal with a shift defect. The transmission spectra are measured along the C–X direction. The values of d for (a)–(i) are 0.2 cm, 0.4 cm, 0.6 cm, 0.8 cm, 1 cm, 1.2 cm, 1.4 cm, 1.6 cm, and 1.8 cm, respectively.
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that the middle pass band disappears as the shift distance d is larger than 0.6 cm. A peak at 2.83 kHz still exists in the measured spectra of the case with a shift defect. 4.2. C–X0 direction The experimental transmission spectra of the C–X0 direction experimental model are shown in Fig. 12. Fig. 12a shows the transmission spectra of the sonic crystals without and with a rod defect. l is 0 cm, 1.2 cm and 3.2 cm for the cases with a rod defect, respectively. A measured band gap of the perfect sonic crystal is observed from 5.45 to 6.05 kHz. Fig. 12b shows the detail of the lower edge of the band gap which is indicated by a rectangular frame in
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Fig. 12a. Bold dashed and solid arrows indicate the defect modes in the experimental transmission spectra of l = 0 cm and 1.2 cm, respectively, and the frequencies of both defect modes are 5.61 and 5.57 kHz respectively. From the band structure calculation, the frequencies of the defect modes are 5.78 kHz and 5.55 kHz for l = 0 cm and 1.2 cm, respectively. The measured frequencies of both defect modes are near to the numerical results. Note that only one defect mode is observable in the experimental transmission spectrum of l = 1.2 cm. We know that there exists a deaf defect mode at 6.95 kHz in the numerical result. This defect mode cannot be observed in the experimental transmission spectrum, since the frequency of this defect mode is located in the pass band of the experimental transmission spectrum. Moreover, the frequency of
Fig. 12. Experimental transmission spectra along the C–X0 direction. (a) Experimental transmission spectra of the perfect woodpile sonic crystal case and the cases with a rod defect, where l is 0 cm, 1.2 cm and 3.2 cm, respectively. (b) Detail of the lower edge of band gap which is indicated by a rectangular frame in (a). Bold dashed and solid arrows indicate the defect mode of the case of l = 0 cm and l = 1.2 cm, respectively. (c) Experimental transmission spectra of the perfect woodpile sonic crystal case and the cases with a shift defect, where d is 1 cm and 2 cm, respectively. (d) Detail of the lower edge of band gap which is indicated by a rectangular frame in (c). A bold arrow indicates the defect mode of the case of d = 2 cm.
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the lower band gap edge for the case of l = 3.2 cm is slightly lower than that of the perfect periodic structure and no defect mode is found in the band gap. Fig. 12c shows the experimental transmission spectra of the sonic crystals without and with a shift defect. d is 1 cm and 2 cm for the cases with a shift defect, respectively. Fig. 12d shows the detail of the lower edge of the band gap which is indicated by a rectangular frame in Fig. 12c. We can see that the frequency of the lower band gap edge for the case of d = 1 cm is slightly higher than that for perfect periodic structure, and no defect mode can be observed in transmission spectrum. A defect mode indicated by a bold arrow exists at 5.48 kHz in the experimental transmission spectrum of d = 2 cm. This defect mode is very close to lower band gap edge, and can be regarded as the lower edge of the band gap. It reveals that a defect mode emerges from the lower band gap edge as the shift distance increases.
5. Conclusions We have studied the acoustic wave propagation in the woodpile sonic crystal with a defect. Wave propagation along the C–X and C–X0 directions are investigated separately. Defect bands and transmission spectra are obtained by using the FEM with periodic boundary condition and Bloch–Floquet theorem. The width and position of the middle rod in the periodic structure are varied to create rod and shift defects, respectively. The defect band frequencies are strongly dependent on the width and positions of the middle rods. The frequencies of defect modes in calculated transmission spectra have a good agreement with the results obtained from the band structures. The experimental transmission spectra of the woodpile sonic crystals with rod and shift defects are also presented. From the measured spectra of the C–X direction, some defect modes for the woodpile sonic crystal with a rod defect can be observed in the first band gap. However, defect modes in the second band gap cannot be measured. Moreover, for the woodpile sonic crystal with a shift defect, defect modes cannot be measured and observed in the first and second band gaps. From the measured spectra of the C–X0 direction, defect modes in the band gap can be measured and observed clearly. These defect modes are close to the lower band gap edge and the deaf defect mode cannot be observed. A good agreement is obtained between the measured results and the numerical results for the C–X0 direction. From the above, we know that the woodpile structure can be considered as a 1D periodic structure as acoustic wave propagate along a single direction. Here, the woodpile sonic crystal is compared with the layered sonic crystal. It is well known that a 1D layered sonic crystal is airtight, which means that one side of periodic structure is blind to another side. Moreover, the constituting materials of the layered sonic crystal are constrained. For example, the layered structure is not composed of gas, since the gas cannot be stacked with fluid or other gas. The layered structure can be composed of gas and solid. However, the total acoustic waves will be almost reflected by a solid layer, since the acoustic impedance of gas is very smaller than that of solid materials. It results in the bad filtering performance. The woodpile structure can solve these problems. The woodpile structure is not airtight and can be composed of air and solid. The filtering performance also can be enhanced. Such woodpile sonic crystal could be used to substitute for the 1D layered sonic crystal. The woodpile structure also has more geometric parameters to control the locations of band gaps and defect modes. The defect mode properties of the woodpile sonic crystal with a defect can be applied to design novel acoustic devices for filtering sound and trapping sound in defects.
Acknowledgments The authors sincerely acknowledge the financial assistance of National Science Council of Taiwan under the Grant No. NSC 972221-E-006-056-MY3. References [1] Kushwaha MS, Halevi P, Dobrzynski L, Djafari-Rouhani B. Acoustic band structure of periodic elastic composites. Phys Rev Lett 1993;71:2022–5. [2] Kushwaha MS, Halevi P, Martínez G, Dobrzynski L, Djafari-Rouhani B. Theory of acoustic band structure of periodic elastic composites. Phys Rev B 1994;49:2313–22. [3] Kushwaha MS, Halevi P. Band-gap engineering in periodic elastic composites. Appl Phys Lett 1994;64:1085–7. [4] Kuang W, Hou Z, Liu Y. The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals. Phys Lett A 2004;332:481–90. [5] Zhong L, Wu F, Zhang X, Zhong H, Zhong S. Effects of orientation and symmetry of rods on the complete acoustic band gap in two-dimensional periodic solid/ gas systems. Phys Lett A 2005;339:164–70. 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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Propagation of acoustic waves in the woodpile sonic crystal with a defect Liang-Yu Wu, Lien-Wen Chen ⇑ Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
a r t i c l e
i n f o
Article history: Received 10 May 2011 Received in revised form 1 October 2011 Accepted 3 October 2011 Available online 25 October 2011 Keywords: Sonic crystal Woodpile structure Defect band Defect mode
a b s t r a c t Acoustic wave propagation in a woodpile sonic crystal with a defect is studied theoretically and experimentally. The woodpile sonic crystal is composed of polymethyl methacrylate square rods which orthogonally stacked together, and it is embedded in air background. Defects are created by varying the width and positions of the middle rods in the periodic structure. Defect bands and transmission spectra are calculated by using the finite element method with the periodic boundary condition and the Bloch–Floquet theorem. Frequencies of defect bands are strongly dependent on the width and positions of the middle rods in the periodic structure. The experimental transmission spectra of the woodpile sonic crystals with a defect are also presented and compared with the numerical results. The defect mode properties of the woodpile sonic crystal with a defect can be applied to design novel acoustic devices for filtering sound and trapping sound in defects. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The acoustic and elastic wave propagation in periodic composite media (called sonic crystal or phononic crystal) has received much experimental and theoretical attention in recent years. Phononic crystals can exhibit acoustic or elastic band gaps in which acoustic wave and vibration are forbidden in any direction. The band gap phenomena can be used as filters, transducers, and for creation of vibration-free environments. The width of band gaps is determined by the contrast of elastic constants of constructed elements, the filling ratio and the lattice. Most of the studying works have been focused on the two-dimensional (2D) sonic crystals. The band structures and band gap width of sonic crystals consisting of lattices with various symmetry and scatterers of different shapes, orientations, and sizes have been investigated [1–5]. The acoustic band gaps have been experimentally presented for the rigid rods in air [6,7]. Martínez-Sala et al. [6] have measured the attenuation of sound in a minimalist sculpture consisting of steel cylinders in air; Sánchez-Pérez et al. [7] have demonstrated an experimental analysis of the acoustic transmission of a 2D periodic array of rigid cylinders in air with the square and triangular lattices. Moreover, one interesting property of these periodic structures is the possibility of generating defects that confine elastic or acoustic waves in localized modes [8–14]. As the periodicity of the sonic crystals is broken locally, the defect modes can be found within the acoustic band gaps, which are strongly localized around the ⇑ Corresponding author. Tel.: +886 6 2757575x62143; fax: +886 6 2352973. E-mail address: [email protected] (L.-W. Chen). 0003-682X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2011.10.002
defects. Point defects and line defects can act as resonant cavities and waveguides, respectively. An acoustic wave in range of gap frequencies is trapped in the point defect or propagates along the line defect. The defect modes in a 2D sonic crystal have been reported and discussed [8–10]. The point defect band can sweep inside the band gap as the defect radius changes. Khelif et al. [11] theoretically and experimentally have observed a full band gap and localized defect modes in an acoustic band gap in a sonic crystal that comprised steel cylinders and water from the transmission spectra. Moreover, Wu et al. [12,13] experimentally have measured the spectra and pressures in the defect of the sonic crystal, which is composed of polymethyl methacrylate (PMMA) cylinders and air, to observe the localization of the acoustic waves in the defect at the resonant frequency. Wu and Chen [14] have also studied the wave propagation of the 2D sonic crystal with a local resonant defect theoretically and experimentally. Three-dimensional (3D) sonic crystals have also been investigated by several authors. Kafesaki et al. [15] and Kushwaha et al. [16,17] have investigated band structures and acoustic band gaps of 3D sonic crystals with the face-centered-cubic, body-centeredcubic and simple cubic lattice for solid–solid, gas–liquid and liquid–liquid systems. Sainidou et al. [18] have studied the absolute band gaps in a 3D sonic crystal consisting of steel spheres in polyester, and reported how the width of band gaps depends on the lattice geometry. Kuang et al. [19] have reported the band structures of 3D solid sonic crystals consisting of four different scatterers (spherical, cubic, rhombic dodecahedral and truncated octahedral). Moreover, acoustic point defect states in 3D simplecubic periodic structure composed of water sphere in mercury background have also been investigated [20].
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The woodpile structure is one of the most popular 3D periodic structures in photonic crystals [21–26]. The woodpile structure, which is also called a layer-by-layer structure, differs from the other geometries by its simpler design. A 3D woodpile sonic crystal structure is built by blocks or rods that are orthogonally stacked together. It means that the adjacent layers are perpendicular to each other. The rods may touch each other, may overlap or be detached to a certain extent. The shape of the rods can be spherical, cuboidal, or ellipsoidal, etc. Between every other layer, the rods can be either shifted relative to each other by one-half of a period, or aligned without shift. These structures possess face-centered-tetragonal and body-centered-tetragonal lattice symmetries correspondingly. The woodpile sonic crystal has been proposed to demonstrate the self-collimation of the ultrasonic wave [27]. Jiang et al. [28] have also studied an acoustic woodpile structure fabricated by locally resonant materials, and the result shows that this structure has a strong sound absorbing capability in a wide frequency range. And then, in our previous researches, the acoustic band gaps of a 3D woodpile sonic crystal have been investigated theoretically and experimentally [29]. The purpose of this paper is to study the acoustic wave propagation in woodpile sonic crystals with a defect. The woodpile structure can be considered as a one-dimensional (1D) periodic structure as acoustic wave propagate along a single direction. A traditional 1D sonic crystal is composed of the layered structure [30,31]. The defect can be introduced into a 1D layered sonic crystal by varying the thickness or material of the middle layer in the periodic structure. The woodpile structure is more complex than the layered structure, so it has more geometric parameters to control the locations of band gaps and defect modes. Two kinds of defects are proposed in the woodpile structure. The first one is the rod defect, which is created by varying width of the middle rods in the periodic structure; the second one is the shift defect,
Fig. 1. (a) Schematic of a 3D woodpile structure with the body-centered-tetragonal lattice. The structure is composed of rectangular rods that are orthogonally stacked together. l1 and l2 are the width of rods, and a0 is the lattice constant. (b) 7 1 1 supercell of the C–X direction. (c) 1 1 7 supercell of the C–X0 direction.
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which is created by varying positions of the middle rods in the periodic structure. Defect bands and transmission spectra are obtained by using the finite element method (FEM) with the periodic boundary condition and the Bloch–Floquet theorem. Moreover, the transmission spectra of woodpile sonic crystals with the rod and shift defects are measured and compared with the numerical results. 2. Model and numerical method A woodpile sonic crystal composed of solid rod layers with a structural period of two layers in the z direction is considered to study, which is shown in Fig. 1a. This woodpile structure is of body-centered-tetragonal lattice. This structure also can be stacked by simple cubic unit cells, so it can be considered as the simple cubic lattice [29]. a0 is the lattice constant. l1 and l2 are the width of the rectangular rod. The z direction is called the C–X0 direction, and the x and y directions are called the C–X direction. We vary the width or shift the position of one of rods in the infinite woodpile structure to destroy its periodicity. This imperfect region is called as a defect. The C–X and C–X0 directions of the woodpile structure with the simple cubic lattice is proposed to study wave propagation in the woodpile sonic crystal with a defect. The woodpile structure can be regarded as a 1D periodic structure as wave prop-
Fig. 2. (a) Woodpile sonic crystal with a rod defect along the C–X direction. The width of the middle rod is varied to create a rod defect, where l is the width of the middle rod. (b) Defect mode frequencies in the first and second band gaps of the C– X direction as a function of the width of the middle rod. The inset is the zoom of the highest defect band in the second band gap in the region 0 5 l/a0 5 0.2, where is indicated by a rectangular frame. Solid lines represent edges of the first and second band gaps of the C–X direction for the perfect woodpile structure.
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agation along a single direction (C–X or C–X0 direction) is considered. In order to analyze the non-ideal periodical structure of the sonic crystal, the supercell method is adopted. The supercell structures of the C–X and C–X0 directions are shown in Fig. 1b and c, respectively. The supercell structure consists of many unit cells. Both supercell structures have seven periods, and a defect can be introduced into the middle of the supercell. Both of them are acted as unit cells in our calculations. The size of the supercells should be large enough since we must guarantee that the coupling effect between neighboring supercells can be ignored. The structure is surrounded by fluid background. The woodpile sonic crystals can be analyzed by using a commercial program, COMSOL MultiphysicsÒ, based on the FEM [32]. As the acoustic impedance of fluid background is much smaller than that of solid rods, the total longitudinal waves propagating in fluid will be almost reflected by solid rods. The wave propagation in such sonic crystal is predominant in the fluid. So, the transverse waves in solid rods can be ignored. In order to achieve this approximation, we can consider the solid rods as fluid with very high stiffness and specific mass. The structure is theoretically assumed to be infinite and periodic in the x, y, and z directions. Considering the periodic boundary conditions above allows us to reduce the model to a single unit cell. According to the Bloch–Floquet theorem, the relation between the
pressure distribution p for nodes lying on the boundary of the unit cell can be expressed as
pðx þ a1 ; y; zÞ ¼ pðx; y; zÞ expðiK 1 a1 Þ; pðx; y þ a2 ; zÞ ¼ pðx; y; zÞ expðiK 2 a2 Þ;
ð1Þ
pðx; y; z þ a3 Þ ¼ pðx; y; zÞ expðiK 3 a3 Þ; where a1, a2, and a3 are the basis vectors of the periodic structure, and K = (K1, K2, K3) is the Bloch wave vector. A phase relation is applied on the lateral faces of the unit cell, defining boundary conditions between adjacent units. This phase relation is related to the wave number of the incident wave in the periodic structure. The analysis of eigenfrequencies and the corresponding eigenvectors can be performed by giving Bloch wave vectors. The eigenvectors are related to the pressure distribution of the modes. And then, eigenfrequencies are assembled to build the band structures along the C–X and C–X0 directions of the first Brillouin zone. Defect bands of supercell structures can be found in the band structure. Furthermore, the woodpile sonic crystals can be studied by calculating the transmission spectra. We stack several unit cells to construct the periodic structure with and without a defect along a specific direction (ex. C–X or C–X0 direction). And then, the periodic boundary conditions are set on other directions. The acoustic wave at the pass band or band gap frequencies is incident into the
Fig. 3. Calculated transmission spectra of the woodpile sonic crystal with a rod defect along the C–X direction. The values of l/a0 for (a), (b) and (c) are 0, 0.2 and 0.8, respectively. Arrows indicate the defect modes in the band gaps.
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periodic structure and propagates in the structure along the specific direction. The power ratio of the transmitted wave to incident wave represents the transmission coefficient. wo and wi denote the transmitted power at the outlet of the periodic structure and the incident power at the inlet of the periodic structure, respectively. wo and wi are defined as [32]
wo ¼
Z X
jpo j2 dA and wi ¼ 2qair cair
Z X
jpi j2 dA; 2qair cair
ð2Þ
where po and pi are the pressure at the outlet and the inlet, respectively, and X is the area of the outlet and the inlet. The transmission coefficient can be expressed as
T ¼ wo =wi
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rod is square (l1 = l2) and the ratio of l1/a0 is 0.5. We have previously studied that the first and second band gaps of the C–X direction extend from 0.284 to 0.575 xa0/2pcair and from 0.628 to 0.831 xa0/2pcair, respectively, and the band gap of the C–X0 direction extends from 0.628 to 1 xa0/2pcair [29]. The supercells shown in Fig. 1b and c are adopted to calculate the band structures and defect bands. Band structure calculated by the FEM calculation with a periodic boundary condition is used to analyze the infinite structure. The FEM calculation is also adopted to calculate the transmission spectra of the finite structure. In this paper, the defect is located at the middle of periodic structure for all transmission calculation works, and both periodic structures along the C–X and C– X0 directions have seven periods.
ð3Þ
The transmission coefficients are functions of the frequency. The transmission spectrum can be built by assembling transmission coefficients of the different frequencies. 3. Numerical results A woodpile sonic crystal composed of PMMA rods in air background is proposed to study. The material parameters are density qPMMA = 1180 kg/m3 and qair = 1.2 kg/m3, and sound of speed cPMMA = 2700 kg/m3 and cair = 343 kg/m3. The cross-section of the
Fig. 4. (a) Woodpile sonic crystal with a shift defect along the C–X direction. The position of the middle rod is shifted to create a shift defect, where d is the shift distance of the middle rod from the original position. (b) Defect mode frequencies in the first and second band gaps of the C–X direction as a function of shift distance of the middle rod. Solid lines represent edges of the first and second band gaps of the C–X direction for the perfect woodpile structure.
3.1. C–X direction 3.1.1. Rod defect We vary the width of the middle rod to create a rod defect in the C–X direction as shown in Fig. 2a, where l is the width of the middle rod. The dependences of the defect band frequencies in the first two band gaps of the C–X direction on the value of l/a0 are presented in Fig. 2b. Solid lines represent the first and second band gap edges of the C–X direction of the perfect woodpile structure. The inset in Fig. 2b is the zoom of the highest defect band in the second band gap in the region 0 5 l/a0 5 0.2. From Fig. 2b, it reveals that there are two defect bands in the first band gap and three defect bands in the second band gap as the middle rod does not exist (l/a0 = 0). The defect bands in the first band gap and the lowest defect band in the second band gap disappear in the region 0.1 5 l/ a0 5 0.4. Note that the highest defect band in the second band gap becomes two defect bands as the width of a rod defect exists. Both defect bands move toward and immerge into the upper edge of the second band gap as l/a0 increases from 0.1 to 0.2. The middle defect band in the second band gap moves toward and immerges into the lower edge of the second band gap as l/a0 increases from 0 to 0.5. There is no defect mode at l/a0 = 0.5 since it is a perfect periodic structure. For l/a0 > 0.6, the defect bands emerge from the upper edge of the first and second band gaps moving toward the lower band gap edge as l/a0 increases. On the other hand, an additional defect band emerges from the lower edge of the first band gap at l/a0 = 0.8 and moves toward the upper band gap edge as l/a0 increases. The transmission spectra of the periodic structure with a rod defect are shown in Fig. 3, where the values of l/a0 are 0, 0.2 and 0.8, respectively. Since their upper edge frequency of the second band gap is higher than that of the perfect periodic structure, the defect mode which is close to the upper edge of the second band gap can be clearly observed in Fig. 3a and b. The frequencies of defect modes are in good agreement with the band structures. From Fig. 3c, only one defect mode can be found in the first band gap since the lower defect mode is very close to the lower edge of the first band gap. We can see that the middle pass band disappears but the defect modes still correspond to the band structure. 3.1.2. Shift defect We shift the middle rod along the C–X direction to create a shift defect as shown in Fig. 4a, where d is the shift distance of the middle rod from the original position. Fig. 4b shows that the defect band frequencies in the first two band gaps of the C–X direction are dependent on the value of d/a0. The structure is a perfect periodic structure as d/a0 is 0. Variation tendency of the defect band frequencies for the positive and negative shift distance is symmetric. In the first and second band gaps, defect modes emerge from the lower and the upper edges of both band gaps as the absolute value of d/a0 increases. An additional defect band emerges from the upper edge of the second band gap at |d/a0| = 0.3 and moves to-
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Fig. 5. Calculated transmission spectra of the woodpile sonic crystal with a shift defect along the C–X direction. The values of d/a0 for (a), (b) and (c) are 0.2, 0.3 and 0.4, respectively. Arrows indicate the defect modes in the band gaps.
Fig. 6. (a) Woodpile sonic crystal with a rod defect along the C–X0 direction. The width of the middle rod is varied to create a rod defect, where l is the width of the middle rod. (b) Defect mode frequencies of in the band gap of the C–X0 direction as a function of the width of the middle rod. Solid lines represent edges of the band gap of the C–X0 direction for the perfect woodpile structure.
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Fig. 7. Calculated transmission spectra of the woodpile sonic crystal with a rod defect along the C–X0 direction. The values of l/a0 for (a), (b) and (c) are 0, 0.3 and 0.8, respectively. Arrows indicate the defect modes in the band gaps.
Fig. 8. Woodpile sonic crystal with a shift defect along the C–X0 direction. The position of the middle rod is shifted to create a shift defect, where d is the shift distance of the middle rod from the original position.
ward the lower band gap edge as |l/a0| increases. Note that this defect band is very close to the upper edge of the second band gap.
The transmission spectra of the periodic structure with a shift defect are shown in Fig. 5, where the values of d/a0 are 0.2, 0.3
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and 0.4, respectively. Similarly, their upper edge frequency of the second band gap is also higher than that of perfect periodic structure, so we can obviously observe that there are three defect modes in the second band gap. The transmission spectrum of d/a0 = 0.2 is different from the result of band structure of d/a0 = 0.2. Because the highest defect band shown in Fig. 5a is out of the second band gap of the perfect periodic structure, it cannot be observed in Fig. 4b. Frequencies of the highest defect modes are almost independent of the value of d/a0. Variation tendency of defect modes in the transmission spectra also has a good agreement with the results obtained from the band structures.
3.2. C–X0 direction 3.2.1. Rod defect Fig. 6a shows the rod defect in the C–X0 direction, where l is the width of the middle rod. Fig. 6b shows that the defect band frequencies in the band gap are dependent of the value of l/a0. It reveals that there is one defect band in the band gap as l/a0 = 0, and two defect bands exists in the band gap as l/a0 = 0.05. It is because another defect mode would occur as the width of rod defect exists. Both of defect bands immerge into the upper and lower edges of band gap respectively. For l/a0 = 0.5, the defect band cannot be found in the band gap.
Fig. 7 shows the transmission spectra of the periodic structure with a rod defect, where the values of l/a0 are 0, 0.3 and 0.8, respectively. The transmission spectra of l/a0 = 0 and 0.8 have a good agreement with the band structures. However, from Fig. 7b, it reveals that only one defect mode is found in the band gap. The higher defect band cannot found in the transmission spectra. The higher defect mode can be seen as the deaf defect mode [7,14,29]. The incident plane wave along the C–X0 direction does not excite this defect mode. So, the higher defect mode cannot be observed in the transmission spectrum of the C–X0 direction.
3.2.2. Shift defect The shift defect in the C–X0 direction is shown in Fig. 8, where d is the shift distance of the middle rod from the original position. A defect band emerges from the lower edge of the band gap as the middle rod is shifted. But, it is noticed that the defect mode is very close to the lower edge of the band gap and can be acted as the edge of the band gap. The transmission spectra of the periodic structure with a shift defect are shown in Fig. 9, where the values of d/a0 are 0.2, 0.35 and 0.5, respectively. From Fig. 9a and b, there exists no defect mode in the band gap. However, it is noticed that there is a defect mode in Fig. 9c. This defect mode is very close to the lower edge of band gap.
Fig. 9. Calculated transmission spectra of the woodpile sonic crystal with a shift defect along the C–X0 direction. The values of d/a0 for (a), (b) and (c) are 0.2, 0.35 and 0.5, respectively. An arrow indicates the defect mode in the band gaps.
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4. Experimental results and discussion The experimental modal is a woodpile structure consisting of PMMA square rods in air background. PMMA square rods with a width l1 = l2 = 2 cm are orthogonally stacked together and the lattice constant a0 is 4 cm. The experimental models of the C–X and C–X0 directions both have seven periods, and the defect is located at the middle of periodic structure. The speaker (Fostex: FF85K) and the microphone (Brüel & Kjær: 4190) are used as a sound source and a receiver, respectively. The speaker connected with the function generator (NF: WF-1945B) can either set to specific frequencies, or sweep through a variable range of the frequencies. In this experiment, the range of sweeping frequencies is from 200 Hz to 15 kHz. Sound signals can be detected by the microphone and are displayed in the digital sampling oscilloscope (Tek-
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tronix: TDS5032B). Spectra can also be obtained by using the spectrum analysis. In order to reduce reflected waves from the environment, sound-absorption sponges are used to enclose the sonic crystal structure. The sonic crystal is placed between the speaker and the microphone. The distance between the speaker and sonic crystal is about 22 cm; the distance between the microphone and the sonic crystal is about 10 cm. Both of the microphone and speaker are located at the center of woodpile structure, and the microphone is collinear to the speaker. 4.1. C–X direction Fig. 10 shows the experimental transmission spectra which are measured along the C–X direction. Fig. 10a is the experimental transmission spectrum of the perfect sonic crystal. Two measured
Fig. 10. Experimental transmission spectra along the C–X direction. (a) shows the case of the perfect woodpile sonic crystal; (b), (c) and (d) show the cases of the woodpile sonic crystal with a rod defect. The values of l for (b), (c) and (d) are 0 cm, 1.2 cm and 3.2 cm, respectively.
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band gaps are observed from 2.5 to 5 kHz and from 5.2 to 6 kHz, respectively. Fig. 10b–d shows the experimental transmission spectra of the woodpile structure with a rod defect, where the values of l are 0 cm, 1.2 cm and 3.2 cm, respectively. From Fig. 10b, it can be seen that two defect modes which are indicated by arrows can be observed in the first band gap. The frequencies of both defect modes are 3.17 and 4.93 kHz respectively. We can read the frequencies of both defect modes are 3.08 kHz and 4.8 kHz from the band structure calculation respectively. The measured frequencies of both defect modes are close to the numerical results. Comparing Fig. 10c with Fig. 10a, both experimental transmission spectra are similar. The defect mode in the second band gap cannot be mea-
sured and observed in Fig. 10c. From Fig. 10d, we can see that the middle pass band disappear and the width of the band gap become wider. This measured result is a good agreement with the numerical transmission spectrum shown in Fig. 3c. However, the defect modes in the first and second band gaps cannot be measured. It can be observed that a peak always exists at 2.83 kHz in Fig. 10. This peak is not the defect mode. This peak is resulted from the positions of the speaker and microphone since it exists in many experimental transmission spectra. Fig. 11 shows the experimental transmission spectra of the sonic crystal with a shift defect. No defect can be observed in the experimental transmission spectra. The middle pass band disappears as the shift distance d increases. Note
Fig. 11. Experimental transmission spectra of the woodpile sonic crystal with a shift defect. The transmission spectra are measured along the C–X direction. The values of d for (a)–(i) are 0.2 cm, 0.4 cm, 0.6 cm, 0.8 cm, 1 cm, 1.2 cm, 1.4 cm, 1.6 cm, and 1.8 cm, respectively.
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that the middle pass band disappears as the shift distance d is larger than 0.6 cm. A peak at 2.83 kHz still exists in the measured spectra of the case with a shift defect. 4.2. C–X0 direction The experimental transmission spectra of the C–X0 direction experimental model are shown in Fig. 12. Fig. 12a shows the transmission spectra of the sonic crystals without and with a rod defect. l is 0 cm, 1.2 cm and 3.2 cm for the cases with a rod defect, respectively. A measured band gap of the perfect sonic crystal is observed from 5.45 to 6.05 kHz. Fig. 12b shows the detail of the lower edge of the band gap which is indicated by a rectangular frame in
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Fig. 12a. Bold dashed and solid arrows indicate the defect modes in the experimental transmission spectra of l = 0 cm and 1.2 cm, respectively, and the frequencies of both defect modes are 5.61 and 5.57 kHz respectively. From the band structure calculation, the frequencies of the defect modes are 5.78 kHz and 5.55 kHz for l = 0 cm and 1.2 cm, respectively. The measured frequencies of both defect modes are near to the numerical results. Note that only one defect mode is observable in the experimental transmission spectrum of l = 1.2 cm. We know that there exists a deaf defect mode at 6.95 kHz in the numerical result. This defect mode cannot be observed in the experimental transmission spectrum, since the frequency of this defect mode is located in the pass band of the experimental transmission spectrum. Moreover, the frequency of
Fig. 12. Experimental transmission spectra along the C–X0 direction. (a) Experimental transmission spectra of the perfect woodpile sonic crystal case and the cases with a rod defect, where l is 0 cm, 1.2 cm and 3.2 cm, respectively. (b) Detail of the lower edge of band gap which is indicated by a rectangular frame in (a). Bold dashed and solid arrows indicate the defect mode of the case of l = 0 cm and l = 1.2 cm, respectively. (c) Experimental transmission spectra of the perfect woodpile sonic crystal case and the cases with a shift defect, where d is 1 cm and 2 cm, respectively. (d) Detail of the lower edge of band gap which is indicated by a rectangular frame in (c). A bold arrow indicates the defect mode of the case of d = 2 cm.
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the lower band gap edge for the case of l = 3.2 cm is slightly lower than that of the perfect periodic structure and no defect mode is found in the band gap. Fig. 12c shows the experimental transmission spectra of the sonic crystals without and with a shift defect. d is 1 cm and 2 cm for the cases with a shift defect, respectively. Fig. 12d shows the detail of the lower edge of the band gap which is indicated by a rectangular frame in Fig. 12c. We can see that the frequency of the lower band gap edge for the case of d = 1 cm is slightly higher than that for perfect periodic structure, and no defect mode can be observed in transmission spectrum. A defect mode indicated by a bold arrow exists at 5.48 kHz in the experimental transmission spectrum of d = 2 cm. This defect mode is very close to lower band gap edge, and can be regarded as the lower edge of the band gap. It reveals that a defect mode emerges from the lower band gap edge as the shift distance increases.
5. Conclusions We have studied the acoustic wave propagation in the woodpile sonic crystal with a defect. Wave propagation along the C–X and C–X0 directions are investigated separately. Defect bands and transmission spectra are obtained by using the FEM with periodic boundary condition and Bloch–Floquet theorem. The width and position of the middle rod in the periodic structure are varied to create rod and shift defects, respectively. The defect band frequencies are strongly dependent on the width and positions of the middle rods. The frequencies of defect modes in calculated transmission spectra have a good agreement with the results obtained from the band structures. The experimental transmission spectra of the woodpile sonic crystals with rod and shift defects are also presented. From the measured spectra of the C–X direction, some defect modes for the woodpile sonic crystal with a rod defect can be observed in the first band gap. However, defect modes in the second band gap cannot be measured. Moreover, for the woodpile sonic crystal with a shift defect, defect modes cannot be measured and observed in the first and second band gaps. From the measured spectra of the C–X0 direction, defect modes in the band gap can be measured and observed clearly. These defect modes are close to the lower band gap edge and the deaf defect mode cannot be observed. A good agreement is obtained between the measured results and the numerical results for the C–X0 direction. From the above, we know that the woodpile structure can be considered as a 1D periodic structure as acoustic wave propagate along a single direction. Here, the woodpile sonic crystal is compared with the layered sonic crystal. It is well known that a 1D layered sonic crystal is airtight, which means that one side of periodic structure is blind to another side. Moreover, the constituting materials of the layered sonic crystal are constrained. For example, the layered structure is not composed of gas, since the gas cannot be stacked with fluid or other gas. The layered structure can be composed of gas and solid. However, the total acoustic waves will be almost reflected by a solid layer, since the acoustic impedance of gas is very smaller than that of solid materials. It results in the bad filtering performance. The woodpile structure can solve these problems. The woodpile structure is not airtight and can be composed of air and solid. The filtering performance also can be enhanced. Such woodpile sonic crystal could be used to substitute for the 1D layered sonic crystal. The woodpile structure also has more geometric parameters to control the locations of band gaps and defect modes. The defect mode properties of the woodpile sonic crystal with a defect can be applied to design novel acoustic devices for filtering sound and trapping sound in defects.
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