Resonant Modes of One-Dimensional Metamaterial Containing Helmholtz Resonators with Point Defect

The metamaterial constructed by Helmholtz resonators (HR) has low-frequency acoustic forbidden bands and possesses negative mass density and effective bulk modulus at particular frequencies. The resonant modes in one-dimensional HR structure with point defect were studied using finite element method (FEM). The results show that the acoustic energy is localized between the resonant HR and the opening in the local-resonant-type gap. There is a high pressure area around the defect resonator at the frequency of defect mode. In the Bragg type gap, the energy mainly distributes in the waveguide with harmonic attenuation due to the multi-scattering. Phase opposition demonstrates the existence of negative dynamic mass density. Local negative parameter is observed in the pass band due to the defect mode. Based on further investigation of the acoustic intensity and phase distributions in the resonators corresponding to two different forbidden bands, only one local resonant mode is verified, which is different from the three-component local resonant phononics. This work will be useful for understanding the mechanisms of acoustic forbidden bands and negative parameters in the HR metamaterial, and of help for designing new functional acoustic devices.


Introduction
Helmholtz resonator (HR) is normally constructed by a large cavity with a short neck [1].Due to its resonance, the resonator possesses capability of low-frequency sound absorption and elimination [2].Recently, with the increasing research on phononic crystals and acoustic metamaterials, the structure based on HRs has been reconsidered for its property of sound forbidden [3]- [8].Furthermore, it is found that the structure possesses negative effective bulk modulus [4] and negative dynamic mass density [5] in its band gap, and therefore it is considered as a possible material to realize new functional devices of transformation acoustics [9].
Based on the different mechanisms, there are two kinds of acoustic forbidden bands in the HR metamaterial.One is called Bragg type gap (BG), which is appeared due to the Bragg scattering in the material with periodically arrayed cells [10].The BG can only forbid the sound waves with wavelength comparable or shorter than the lattice constant.It is unpractical to control low frequency sound using this kind of metamaterial for its huge sizes.On the other hand, the second type acoustic forbidden band is brought by local resonance of HR [11], which can be called local-resonant-type gap (LRG).The LRG exists around the eigenfrequency of the resonator.As the sound wavelength corresponding to the eigenfrequency is usually some times of magnitude larger than the geometric parameters of the resonator, low frequency sound waves can be well controlled.
The band structure is much richer when defect exists [7] comparing that of perfect periodical case.Localized mode can be observed due to the coupling of the defect units and perfect units [12] [13] as well as several new gaps of BG and/or LRG.A localized mode is that, at a particular frequency, the linear free oscillations are trapped around the defect resonators and decay exponentially away from them [7].In this case, the acoustic energy can be captured by the point defect or limited directionally transmitting along the line defect and area defect.With this character, wave-control devices can be designed [14] [15].Recently, Fey et al. [8] indicated that a wide bandgap material could be get with a subwavelength collection of detuned HRs which are considered as a series of defects.However, the problem turns complicated with the increase of the number of defects.
Comparing with two-and three-dimensional metamaterials, one-dimensional (1D) systems can be calculated with higher accuracy [1] [4].It is also understood that the results of 1D system are helpful for understanding the property of more complex cases.In previous researches, theoretical studies on the 1D HR structures were based on the theory of Bloch wave and scattering [1] [2].However, due to its strict periodicity assumption, this method is infeasible to deal with more complicated composites with quasiperiodicity or disperiodicity.Recently, some reduced methods were developed to analyze the acoustic transmission property of the HR structure.Cheng et al. [5] analyzed the acoustic transmission properties of 1D HR metamterial by means of acoustic transmission line method (ATLM).Based on the interface response theory (IRT), Wang et al. [7] studied 1D phononic crystals containing HRs systematically, especially on the acoustic transmission properties of structure with point defect.So far, these studies gave more attention to the transmission property of the HR metamaterial with simplified parameters than the details inside the structure.We believe that, with full view of distribution of the oscillation modes in the structure, a clear understanding on the mechanisms about the acoustic band gaps and negative parameters can be obtained, which is useful for designing new acoustic energy concentrator and creating high pressure environment for acoustic experiments.
In practice, since the complex geometry is simplified in former theoretical methods which are unable to investigate the detailed field distribution in the structure, an accurate approach must be introduced to analyze the resonant modes property of the HR metamaterial.The Finite Element Method (FEM) is an appropriate approach to minutely study the characteristics of the acoustic field for complex structures.On the basis of FEM, the distributions of acoustic intensity and phase for different oscillation modes in the 1D metamaterial with HRs were studied in this paper.Local resonant modes were also investigated for different forbidden gaps.We studied the 3-dimensional model using COMSOL Multiphysics software (Version 4.2) which is based on the Finite Element Method (FEM).We set up the boundary conditions as shown in Figure 2, in which a perfect matched layer (PML) was used at the end of the waveguide to simulate the absorbing boundary condition.All the other boundaries were set to be hard walls, except that a radiation boundary condition with a harmonic wave was used as the incident wave.

Model and Verification of the Method
The host medium in the waveguide and the resonators is water.
To the computational mesh, in our simulation, at least 8 elements per wavelength were used, which guaranteed the accuracy of the method, and also satisfied the general six-element-per-wavelength rule in acoustic mesh [16].All elements are hexahedral.
To validate the feasibility of the software using FEM, we first made a comparison between the results of FEM and ATLM [5] [17] for the acoustic transmission property of the metamaterial.
In ATLM, based on the transformation relationship between acoustic impedances of the inlet and outlet, the transmission coefficient can be obtained by applying this formula recursively.
The impedance transfer formula [16] of ATLM can be written as where, Z l (Z r ) is the effective impedance of the inlet (outlet) of the unit cell.Z 0 = ρ 0 c 0 /S g is the distributed impedance of the duct.S g is the cross-section area of the waveguide.k is the wave vector of the host medium.L is the distance between two adjacent HRs.
With the assumption of long-wavelength, the transfer impedance of the waveguide parallels to the HR impedance Z h [16].The parallel impedance is which can be considered as the terminal-end impedance of its left neighbor.
By repeating this process over the N units, the effective acoustic impedance (Z effect ) of 1D metamaterial with N unit cells can be obtained.Then, the sound pressure reflection coefficient can be calculated as The sound intensity reflection coefficient and intensity transmission coefficient are 2 , 1 Figure 2. Finite element model of the metamterial.
In Equation ( 4), the band gap exists only if T = 0, which means 1 p r = , and therefore Z effect = 0 or ∞.It indicates that Z l or/and Z h vanishes in Equation (2).Z l = 0 corresponds to that the real and imaginary parts of Equation (1) equal zero simultaneously, which is mathematically impossible.In fact, based on Equation (1), we get that, when KL = nπ, viz.f = nc/2L, the value of Z l reaches its minimum (equals to Z r ).This frequency corresponds to the central frequency of BG.
On the other hand, when Z h = 0, the incident wave frequency equals to the resonant frequency of the HR, which means the appearance of LRG in this case.

Simulation Based on FEM
Now we take a detail observation on the acoustic intensity distributions of the structure with point defect for several specified frequencies using the results with full-wave simulation based on FEM.The choice of the frequencies was based on FEM results in Figure 3(b).The acoustic intensity distributions are displayed in Point (e) (4.08 kHz) corresponds to a defect mode, which is a narrow transmission band is the forbidden band.It is obvious that, in Figure 4(e), the acoustic energy is localized around the defect HR and its neighbors.The intensity reaches the largest value at the defect resonator, and then attenuates sharply to both sides.This is a typical property of the defect mode [12].Since the defect mode is created by the coupling of the defect HR and perfect HRs, the defect mode frequency is not the same with both resonant frequencies.The defect mode is useful for realizing new filter, energy harvester and acoustic cloaking.
Point (f) (5.1 kHz) is in the pass band outside the LRG.As shown in Figure 4(f), with the increment of the pressure in the waveguide, the pressure in the HRs decreases.Now, the energy is not localized in HR, but released to the waveguide.In this case, the HRs are like obstacles to short-wavelength sound.With this conclusion, it is imaginable that the harmonicity would be more obvious and the intensity would be higher in the waveguide for Figure 4(g) (7.5 kHz) and (h) (8.8 kHz).Point (h) just locates in the BG.In Figure 4(h), due to multiscattering, the intensity in the waveguide attenuates gradually, which tends to zero at the terminal end.On this condition, the BG appears.
To summarize, as shown in Figure 4, there are plenty resonant modes in the metamterial containing HRs with point defect.When frequency is lower than the resonant frequency, the acoustic energy distributes in the waveguide and resonators symmetrically.As frequency turns to the resonant frequency, local resonant mode can localize the energy between the first resonant HR and the incident port.In the defect mode, a high pressure zone exists around the defect resonator.Finally, the energy in the resonant HR is released to the waveguide and transmits in the waveguide only when frequency is higher than the resonant frequency.
Figure 5 shows the corresponding phase distributions of Figure 4.In Figure 5(a), the phase in the waveguide is the same as that in the shunted HR, which indicates that the resonator oscillates in-phase with the wave in the waveguide.
In this case, the dynamic mass density must be positive [5] [18].As frequency

Figure 1
Figure1shows the schematic diagram of a Helmholtz resonator which is connected with a section of waveguide forming a unit cell of the metamaterial.As a numerical example, here we consider a model with 11 HR unit cells, and the 6 th one is abnormal which can be considered as a defect.The overall geometric parameters are L = 0.09 m, and d 1 = 0.025 m.For the cells, the geometrical parameters of the ten perfect units are a 2 = 0.02 m, d 2 = 0.02 m, V = a 3 × l 3 × d 3 = 0.03 × 0.04 × 0.05 m 3 , while the only difference for the defect unit is that d 2 = 0.04 m.The background media is water (ρ 0 = 998 kg/m 3 , c 0 = 1483 m/s).Here, we analyzed the acoustic band gap structure of the metamaterial in the region of 1 -10 kHz.

Figure 1 .
Figure 1.Schematic diagram of a Helmholtz resonator and a section of waveguide.

Figure 3 Figure 3 .
Figure3shows the acoustic transmission coefficient curves for both perfect metamaterial and structure with point defect basing on FEM and ATLM, respectively.Despite small differences, the results obtained based on FEM can also show all the properties of the HR metamaterial with point defect, such as transmission bands, forbidden bands and defect mode.Figure3demonstrates the feasibility of FEM, which can be a further approach to analyze the resonant

Figure 4 .Figure 4 .
Figure 4.In Figure4, point (a) locates at 1.5 kHz in the low-frequency pass band, where the acoustic intensity distributes periodically in the waveguide.Since the frequency does not reach the resonant frequency of HRs, the resonators are in the state of "pre-resonance", and the acoustic energy is being localized by the resonator.In Figure4(b) (2.38 kHz), energy is localized between the defect HR and the incident opening with small amount of acoustic energy penetrating.The acoustic intensity in the waveguide is obviously weaker than that in the resonators.As we know, point (b) corresponds to the resonant frequency of the defect HR, which indicates that, in the LRG, the resonant HR can localize almost all the energy passing across it.Point (c) (2.52 kHz) is another dip between the two LRGs.Comparing with Figure 4(b), the energy in the defect HR has been already released in Figure4(c).This is because that with the increase of frequency, the resonant mode of the defect HR vanishes.In this case, the sound is no

Figure 5 .
Figure 5. Phase distribution for the metamaterial containing HRs with point defect.Figures (a) to (h) correspond to the cases showed in Figure 4.

Figure 6 .
Figure 6.Acoustic intensity ((a) and (b)) and phase ((c) and (d)) distributions for Helmhotlz resonator in different forbidden bands.The frequency of (a) and (c) is 3.2 kHz corresponding to the LRG; and (b) and (d) is 8.8 kHz corresponding to the BG.