Acoustic Pressure Characteristic Analysis in Cavity of 2-D Phononic Crystal

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Acoustic pressure characteristic analysis in cavity of 2-D phononic crystal

Jia-Yi Yeh1, Jiun-Yeu Chen2

1

Department of Information Management, Chung Hwa University of Medical Technology 2

Center for General Education, Hsing Kuo University of Management E-mail : yeh@mail.hwai.edu.tw

Abstract

This paper investigated the acoustic pressure characteristics analysis in cavity of the two-dimensional phononic crystal. Present phononic crystal is composed of PMMA cylindrical with square array embedded in air background. In order to obtain the band structures and acoustic pressure characteristics of the phononic crystal system, the plane wave expansion method and supercell are adopted and utilized. In addition, the effects of sizes and filling fractions are also investigated. Finally, the acoustic wave propagation and pressure characteristics in the PCs with point defect are simulated by the COMSOL Multiphysics software.

Keywords: phononic crystal, point defect, cavity, supercell, acoustic pressure.

1. Introduction

Elastic and acoustic wave propagation in periodic composite materials has been extensively studied in recent years. Phononic crystals (PCs) named acoustic/sonic band gap media are the elastic analogues of photonic crystals and have also received renewed attention recently [1, 2]. The PCs exhibit a unique characteristic called “phononic band gaps” (PBGs) within which sound and vibrations are forbidden, and those structures exhibit unique dynamic characteristics that make them act as elastic or acoustic filters for wave propagation.

James et al. [3] presented the propagation of an acoustic wave through one-dimensional PCs and calculated the transmission coefficients of various finite structures. Then, the PBGs were investigated and calculated theoretically by Kushwaha et al. [4] and experimentally by Montero et al. [5] in composite systems constituted by periodic inclusions of a given material in a host matrix. The contrasts in elastic properties and densities between the constituents are emerging as critical parameters in determining the existence of PBGs and their width about 2-D and 3-D composites. Sigalas et al. [6] obtained the defect modes in two-dimensional PCs. Thereafter, Torres et al. [7] studied the surface waves and localization phenomena in linear and point defect. Point defect characteristics of acoustic waves in square arrays of water rods in a mercury host were studied by Wu et al. [8].

Several theoretical methods have been used to study the dynamics for wave propagation in PCs, such as, the plane-wave expansion (PWE) method, the finite difference time domain (FDTD) method and the multiple-scattering theory (MST). Among these methods, the PWE method is most extensively used to calculate band structure because of its convenience. The PWE method was adopted to investigate the characteristics of the periodic system by Kushwaha et al. [9]. The finite difference time domain method was presented to study the wave propagation problems of the PCs by Tanaka, Tamura [10] and Garcia-Pablo [11]. Kafesaki, Economou [12] and Lai [13] studied the wave propagation problems by multiple scattering theory.

The properties of the PCs are studied and the defect bands, band structures are also determined in this study. The calculation is based on the plane wave expansion method and the defect mode is obtained by supercell

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calculation. This work investigates the acoustic pressure characteristics in cavity of two-dimensional PCs which are composed of PMMA cylinders with square arrays embedded in air background. A defect is introduced by removing one cylinder from the middle of the PCs and the point defect can act as the resonant cavity and by using commercial software, COMSOL Multiphysics [14], the pressure in the cavity was calculated in this study.

2. Analytical Model and Calculation Method

The band structure of the two-dimensional periodic PCs with point defect is presented in this paper. The justification is that this seems to be the only case in which the wave equation for inhomogeneous solids greatly simplifies. Then, the wave equation of the PCs is known to be:

) ( ) ( ₂ 1 2 1 11 P P         _ t C , (1)

in which,  is the mass density, C11cl2 is the longitudinal elastic constant,

c

l is the longitudinal speed of

acoustic wave, P is the pressure, and  is the two-dimensional nabla, respectively.

Making use of the periodicity of the PCs system, the quantities 1(r) and C₁₁1(r) can be expanded in the two-dimensional Fourier series as the following equations:



  _ G r iG e r) ( ) ( 1 G   _, ₍₂₎



  _ G r iG e r C₁₁1( ) (G) _, ₍₃₎

in which, G is the two-dimensional reciprocal-lattice vector.

The supercell with 5×5 circular cylinders (material A, PMMA) embedded in a background medium (material B, air) as shown in Fig. 1. And, introduces a defect by removing a central cylinder, to form two dimensional lattices with lattice spacing

a

0. The corresponding densities of the system are  and A  , respectively. B

Then, the Fourier coefficient can be written as follows [15]:

           _          _              



) ( ) ( ), for 0 ( ) ( ) ( 0 for , ] ) 1 ( 1 [ ) 1 ( ) ( ₁ ₁ _[( _, ₎ _] ₍₀_,₀₎ ₁ 1 1 2 1 2 1 2 2 1 _G _G G G G _G _G F e e F f N f N N N m i N N m a m m i B A B A        (4) where 2 0 2 2 0 a N r

f  is the filling fraction of one cylinder in the supercell and F(G) is the structure factor. An equation analogous to Eq.(4) can be written for (G)_{in terms of}C₁₁1_{. For the cylinder with radius}

r

0 in present system, the structure factor F(G) can be written as follows:

0 0 1( ) 2 ) ( r r J f F G G G  , (5)

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where J1 is the Bessel function of the first kind of order 1.

Fig.1. The 5×5 supercell with a cavity (point defect).

Then, the eigenvalue equation can be obtained as the following form by applying the Bloch theorem:



      _ _ _ _ _ _ _ _ _ __ _ _ _ _ G G k k G k G k G G G G G k ( ) ]P ( ) [ ( ) ( ) ( ) ] ( )P ( ) 0. [ 111 2 1 2 1 11 2 1 F C C     ₍₆₎

This is a set of linear, homogeneous equations for the eigenvectors P_k(G) and eigenfrequencies (k) if G is permitted to take all the possible values. By letting k scan the area of the irreducible region of the Brillouin zone as shown in Fig. 2, the Band structures of the PCs can be obtained:

Fig. 2. The first Brillouin zone of a square lattice crystal

Additionally, the COMSOL Multiphysics software is used to simulate the acoustic wave propagation in the PCs with a point defect and the equation utilized to analyze present problem and pressure fields in the cavity of the PCs is the following equation:

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P P 2 2 l c         ₍₇₎

3. Results and Discussions

The acoustic pressure characteristic analysis in cavity of 2-D PCs system consisting of PMMA cylinders in air background is presented in this study. The materials properties and relative parameters are _A 1190kg/m3,

3 / 25 . 1 kg m B 

 , c_A 2694m/s_{, and} c_B 343m/s_{. Besides, the filling fractions of the cylinder in the unit cell,} 2

0 2 0 a r

f , investigated and discussed in this paper are 0.4 and 0.6

The band structures with filling fraction f0.4_{and 0.63 for the PCs with a point defect are shown in Fig.3}

and 4, respectively.

Fig. 3. The band structure for 2-D PCs with cavity ( f0.4₎

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The numerical results are calculated by the plane wave expansion method with considering 625 plane waves. According to the results, the defect mode can be found about 4.02kHz for f0.4_{and 3.9kHz, 5.16kHz for}

63 . 0  

f . After obtaining the results, it can be utilized to design some novel acoustic devices.

Then, the acoustic pressure characteristics analysis in the cavity of the PCs can be analyzed after getting the frequency of the defect mode. In order to simulate the pressure field characteristics of the cavity, the finite element analysis software (COMSOL multiphysics 3.5a) is adopted. Fig. 5 shows the pressure field simulation results of the 5×5 PCs with a cavity for filling ratio f0.4. The maximum pressure field in the cavity can be observed in the resonant frequency about 4.01kHz and the simulated result is similar with the numerical results obtained by the plane wave expansion method.

Fig. 5. Pressure field simulation of the 5×5 PCs with a cavity ( f0.4)

In addition, the pressure field simulation results of the 5x5 PCs with a cavity for filling ratio f0.63 are present in Fig. 6. According to the figure, the maximum pressure field in the cavity can be observed in the resonant frequency about 5.15kHz and the simulated result is also similar with the numerical results obtained by the plane wave expansion method in Fig.4.

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In order to analyze and increase the pressure in cavity of the 2-D PCs, the 5×5, 5×7 and 7×7 supercell combinations for filling ratio f0.4_{and 0.63 are discussed and present in Fig. 7 and 8, respectively. It can}

seen that the central pressure will increase with different combination of the PCs system and we can use the characteristics to design some novel acoustic devices, such as energy harvesting devices.

3600 3800 4000 4200 4400 Hz -2 0 2 4 6 8 Pa 5x5 5x7 7x7

Fig. 7. Central pressure field characteristics in cavity of 2-D PCs (f0.4)

4600 4800 5000 5200 5400 Hz -5 0 5 10 15 20 25 Pa 5x5 5x7 7x7

Fig. 8. Central pressure field characteristics in cavity of 2-D PCs ( f0.63₎

4. Conclusions

The acoustic pressure characteristic analysis in cavity of the PCs is investigated in this study. The defect bands, band structures of the PCs system with point defect are also calculated. The numerical calculation is based on the plane wave expansion method and the defect mode is obtained by supercell calculation. The following conclusions can be obtained according to the numerical and simulated results:

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system.

(2) According to the simulated results, the pressure fields in cavity for the PCs can be utilized to design some novel acoustic devices, such as high-efficacy acoustic cavity and acoustic energy harvesting devices.

(3) With different combination of the PCs system, the central pressure in the cavity will be increasing and strengthening. Those analytical results can be utilized in the development of the novel acoustic applications for the PCs with cavity.

Acknowledgements

This research was partially supported by the National Science Council in Taiwan through Grant NSC 100-2221-E-273-003-.

Reference

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