The Effect of Geometrical Parameters on Resonance Characteristics of Acoustic Metamaterials with Negative Effective Modulus

There has been an explosion of interest in acoustic metamaterial in the last ten years. The tunable negative acoustic metamaterial is an important issue for designing metamaterials. The acoustic metamaterial is restricted by the narrow bandgap of sound waves for the local resonance of metamaterial unit cells. By shifting the sizes of the unit cell, the acoustic metamaterial could potentially overcome the limit of the narrow resonance frequency. In this research, we focus on the resonant behavior of split hollow sphere (SHS) in the waveguide. Firstly, we analyze the resonance characteristics of SHS and get an analytical formula for the effect of geometrical parameters on the resonance frequency. Furthermore, the resonance frequency of SHS is verified with finite-element method analysis based on COMSOLMultiphysics simulation. The results are in good agreement with theory model. It is observed that there is a blue shift of the resonance frequency with the gradual increase of the neck radius of SHS, a V-type response curve with the increase of inner radius of SHS, and a red shift with the increase of outer radius of SHS. Using the method of estimate of resonance, we could get a precisely controllable unit structure with negative effective modulus and offer a way to optimize the realization of double negative acoustic metamaterial.


Introduction
Acoustic metamaterial is a kind of man-made material that exhibit superior properties, and it is applied to control the direction of sound wave.The idea of negative effective density of acoustic metamaterial using local resonance unit was first proposed by Liu et al. in 2000 [1].In this way, more researchers pay attention to local resonant acoustic metamaterial with negative effective density mass [2][3][4][5][6][7][8][9][10].At the year of 2006, Fang et al. [11] designed a one-dimensional array consisted of split hollow cavity with subwavelength which is a new class of acoustic metamaterial with negative effective modulus at the ultrasonic frequency.Later on, Lee et al. [12] made a kind of one-dimensional acoustic metamaterial with splitting hollow on the side of hollow tube and confirmed that the acoustic metamaterial has the property of negative effective modulus at the frequency range of 0∼450 Hz.A simpler version of two-dimensional negative effective modulus based on SHS was prepared by Ding et al. [13].By 2015, Jing et al. [14] proposed a new kind of monopole resonance characteristics of balloon soft resonator to realize negative effective modulus.However, the main disadvantage of the local resonant acoustic metamaterials is that the response frequencies of different units are different and the frequency bands are narrow.Therefore, similar to electromagnetic metamaterials, the tunable negative acoustic metamaterial is an important issue in the acoustic research.The tunable range of resonant frequency is one of the important parameters in the application of acoustic metamaterials.
At present, active and passive methods have been adopted to control the resonance frequency of metamaterials with negative modulus [15][16][17].Akl and Baz [15] controlled the Helmholtz resonator with negative modulus with by coupling with piezoelectric film and applying the electric field.There are many factors that influence the resonance characteristics of acoustic metamaterial unit, such as the shape, size, medium, and number of holes.Ding et al. [18] controlled the resonance frequency with negative modulus by shifting the hole size of SHS.Hao et al. [19] tuned the resonant frequency by changing the medium type of SHS.In addition, the resonant frequency can be controlled by multiple holes on the SHS [20].At present, little attention has been paid to the precise relationship between the size of structure units and resonance frequency.
In this work, based on the Helmholtz theory and transmission line theory, the resonance characteristic of SHS was discussed.The effective bulk modulus and the amendment formula of resonance frequency were established.Furthermore, COMSOL Multiphysics with the finite-element method was used to design and simulate 3D structure of the acoustic metamaterials unit cell and to verify the effects of geometrical parameters on resonance characteristics consisting of size parameters of the neck radius, inner radius, and outer radius.This provides the basis and guidance for the design and fabrication of double negative acoustic metamaterials in arbitrary frequency by coupling structure units with negative mass density in the next step.

Resonant Metamaterial Unit
Cell.The Helmholtz resonators in [11] are arranged orthogonal to the propagation direction of the sound in the waveguide and the structure is successfully achieved negative effective bulk modulus.From transmission line concept, the equivalent circuit of a Helmholtz resonator is given by an LC circuit with no dissipation.And there is another LC section of the waveguide which is connected in parallel with the LC circuit.Based on this idea, we place a SHS in the middle of the cylindrical waveguide in the proposed metamaterial unit cell, as shown in Figure 1(a).The SHS is a type of Helmholtz resonator, including the sphere cavity and the short-tube formed by split hole, also called the neck.Here, it will be shown that the structure can be also used to support negative effective bulk modulus of acoustic metamaterial.
In Figure 1(b) a cross section of the metamaterial unit cell is shown.The cylindrical waveguide is driven at left end by a sound source and is measured at right end by a receiver.The cross section area of the waveguide is , and the length is .The SHS (a spherical cavity with the inner radius   and the outer one   , and a narrow neck with the radius ) is positioned at the source and levitated in the middle of the waveguide.
The equivalent circuit of the unit cell with lossless is given by an analogy circuit as Figure 2(a).In this model, the impedance of the waveguide is represented by the acoustic inertance   given by [21] and the acoustic compliance   , which is given by where  is the density of air, V  is the sound velocity, and  is the volume of the air in the waveguide except the SHS.Additionally, the SHS can be modeled with acoustic inertance and acoustic compliance where   is the effective length of the neck,   is the crosssectional area of the one, and   is the effective volume of cavity.

The Effective Bulk Modulus of the Unit Cell.
It is well known that, according to the dynamical theory, the vibration system is characterized by its mechanical impedance.
Similarly, acoustic impedance   describes the relationship between the sound pressure and the particle velocity, which is given by where   is the acoustic resistance and   is the acoustic reactance.The acoustic resistance represents the energy losses and the acoustic reactance is a concentrated expression of the inertia and elasticity of the system.Since the fluid medium is air in the waveguide, the shear viscosity and thermal conductivity of the gas are too small enough and can be neglected.Therefore, the acoustic resistance is equal to zero in the analog circuit.
In the case of the acoustic metamaterial unit cell, if the waveguide acts as a simple low-pass filter (LPF), the SHS can be viewed as an "internal heat" as shown in Figure 2(a).The resonance frequency of the SHS falls within the passband of the LPF.In the cylindrical waveguide, there is just one propagating acoustic mode below some cut-off frequency.
In order to simplify the question, the equivalent parameters method is used to the analysis of acoustic reactance.We use the effective acoustic compliance Δ  representing the quantity of the inertia and elasticity of SHS which is connected in parallel with   as shown in Figure 2(b).Consequently, the parallel combination of the two capacitors has one all effective acoustic compliance  eff , Figure 2(c). eff could vary in a large number for local resonance and plays a major role in the circuit.
In the context of acoustics, the bulk modulus of unit cell   is an important parameter of sound propagation [11].Suppose that sound propagation is an isentropic process, for which V 2  = / =   /; bulk modulus   = V 2  is reciprocal to the acoustic compliance from (2) and can be expressed as follows: Because the gas acts like a spring in the waveguide, the acoustic compliance is written as where  is the mechanical stiffness of the spring.Since the waveguide is the host frame of unit cell, the effective bulk modulus  eff is defined as Under the above assumption, acoustic impedance of the unit cell becomes Let us suppose that the waveguide has a larger cross section area and therefore the effective acoustic compliance  eff is significantly greater than acoustic inertance   at near resonance frequency.Equation ( 9) is now where  2 0 =  2  +  2  ,   = 1/√    , and   = 1/√    .The effective acoustic compliance and effective bulk modulus are determined by the following expressions, respectively: From ( 12), it can be seen that the inclusion of SHS in the waveguide results in the variation of effect bulk modulus of unit cell with frequency, as shown in Figure 3.
It can be seen that effective bulk modulus between frequencies 0 <  <   is greater than zero but is less than   .With the increasing of frequency, the effective bulk modulus is decreasing, because the acoustic inertance of SHS begins to take more effect.The frequency   is the series resonance frequency of SHS unit.At this frequency, acoustic impedance is equal to zero which means the sound wave is by-passed and the effective bulk modulus is also zero.
In the frequencies range   <  <  0 , the inertia of SHS is so big that the effective bulk modulus is negative.The effective bulk modulus tends to minus infinity as frequency increases from   to  0 .It should be noted that the frequency  0 is the parallel resonance frequency of the Δ  and   and is also antiresonance frequency.At the frequency, the acoustic impedance is equal to minus infinity.The effective bulk modulus is greater than zero in the frequency range above  0 and is greater than   .With the increasing of frequency, the effective bulk modulus is decreasing to   .It is also because of the more influence of the acoustic inertance of SHS.

The Resonance Frequency of SHS.
From the above analysis, we can see that the resonance of SHS plays an important role in the negative effective bulk modulus of unit cell.Based on Helmholtz theory [21], the resonance frequency of SHS can be calculated by the similar LC circuit shown in Figure 1(a).The resonance frequency of SHS is The effective volume of cavity   , that is, the volume of inner sphere minus the volume of spherical crown as part of shorttube, is thus given by where   =  2 is the area of cross section of the short-tube.The geometric length of the short-tube is Since the effective length of air in short-tube is longer than geometric length, we define it as the effective length   .In this model, the short-tube has two end corrections, where the outside of short-tube is considered as nonflange and the correction length is 0.61; the inside is under the flange state and the correction length is (8/3) [22].Thus, the effective length of short-tube is From the results given by ( 14) and ( 16), the amendment formula of resonance frequency is therefore In order to analyze the influence of geometrical parameters of SHS on tunable frequency, the relation between the three parameters of SHS and the resonance frequency was shown in Figure 4 based on formula (17).It can be seen that there is a lower resonance frequency as the neck size is smaller, and there is a higher frequency as the neck size is larger.This provides us an important way to precisely control acoustic metamaterial with negative effective modulus.

Simulation Results
. The acoustic field is simulated by COMSOL Multiphysics 4.3 and the pressure acoustic and solid mechanics are coupled in the acoustic model for the continuity.The model consists of a SHS unit and a waveguide.The plane sound wave was used as a sound source and the other open end of outlet waveguide was modeled with PML boundary condition.The remaining boundaries of waveguide are specified as sound hard condition for the sound waves reflecting from waveguide wall completely.The material of wall of SHS is polyethylene and the fluid domain is full of air.At standard temperature and pressure conditions, the bulk modulus   of air is 1.42 × 10 5 Pa for an isentropic process.Suppose that  = 20.0 mm,  = 225 mm 2 ,   = 5.0 mm,   = 4.5 mm, and  = 0.5 mm to 2.5 mm (Figure 1(b)).
The results of the transmission spectrum are got and shown in Figure 5. Except that the curve of the radius of 0 mm has no transmission dip, there is a dip in every transmission curve.Furthermore, the blue shift of transmission dip occurs as the neck radius of SHS becomes bigger.According to (3),  the increased quality of the air in the neck enhances its acoustic inertance, while the volume value of cavity has decreased slightly.In terms of the serial inductor-capacitor circuit, if the radius of cavity is constant, the inductance of the SHS decreases with increasing neck radius.Although the inner radius of SHS has not been changed, some volume of them becomes a part of the short-tube.The ability of storing energy of SHS gets weak and makes the capacitance of the resonator smaller.Hence, resonance frequency goes up with the increase of the neck radius of SHS.
The values of resonance frequency are shown in Table 1.For example, the SHS with 0.5 mm neck radius has phase reverse near the frequency of 2190 Hz for the local resonance.As is shown in Figure 6, the real part of effective modulus is negative near the resonance frequency of SHS, and it represents the character of the negative dynamic response.Similarly, effective modulus with different neck radius was obtained, and negative effective modulus near the resonance frequency is achieved individually (see Figure 6).This result is consistent with that described in [11].
For verifying the validity of formula (17), we calculate the values of resonance frequency with COMSOL method and amendment formula (17), and they were listed in Table 1.The simulated value of resonance frequency is in good agreement with the analysis results of the equivalent circuit.Thus, using formula (17), we can obtain the precise value and then control resonance frequency of SHS accurately.

Influence of Geometrical Parameters of SHS on Resonance Frequency
Similarly, the effects of other parameters on the tunable frequency of SHS were analyzed with COMSOL Multiphysics and formula (17) in the section.Formula (17) suggests that the tunable parameters of SHS include the hole radius, outer radius, inner radius of sphere cavity, and so on.Because the inner radius and neck radius affect the cavity volume, the difference between outer and inner radius acts as the length of short-tube, and the neck radius affects the area of cross section of short-tube.Thus, three characteristic parameters of SHS have a simultaneous influence on the tunable frequency.In this section, the effects of the inner and outer radius on the resonance frequency were mainly discussed.

The Inner Radius of SHS.
Supposing that   = 9.0 mm,  = 0.5 mm, and   = 4.5 mm to 8.5 mm, the transmission spectrum are measured and shown in Figure 7.The resonance frequency decreases firstly and then increases with increasing the inner radius gradually.In addition, the negative modulus can be obtained near their respective resonance frequency.The results of formula calculation and the simulation agree well, which suggests that we can obtain the precise resonance frequency by formula (17) and accurately control the AM with SHS.
There is a V-type character of the resonant frequency and the minimum frequency value is at the inner radius with 7.5 mm.This phenomenon explained that the acoustic inductance of SHS decreases with increasing the inner radius; on the other hand, the acoustic capacitance increases with increasing the volume of cavity.Under the influence of two opposite aspect, the resonance frequency decreases firstly and then increases with the increase of the inner radius.

The Outer Radius of SHS.
Supposing that   = 4.5 mm,  = 0.5 mm, and   = 5.0 mm to 9.0 mm, Figure 8 is the relation curves between the outer radius and related parameters, and the resonance frequency has a red shift with increasing the outer radius of SHS.This simulated result is in agreement with Helmholtz theory as well.In addition, the negative modulus can also be achieved near their respective resonance frequency.
The red shift explained that when the inner radius of SHS unchanged and the capacitance is constant, the sound energies of SHS unit with different outer radius are the same.However, the acoustic inductance of SHS increases with the increase of length of short-tube acted as inductor.Hence, the resonant frequency has a red shift.
The results of formula calculation and the simulation agree well, which suggests that we can obtain the precise resonance frequency by formula (17) and accurately control the acoustic metamaterial of SHS with negative modulus.
The results of formula calculation and the simulation agree well, which suggests that we can obtain the precise resonance frequency by formula (17).

Conclusion
Based on SHS unit, the tunable mechanism of structure size on controlling the resonance frequency of acoustic metamaterial with negative effective modulus was proposed from aspects of Helmholtz theory and the finite-element method.The amendment formula of resonance frequency of SHS was derived, and it can precisely describe the relation between resonance frequency and neck radius, inner radius and outer radius.Furthermore, COMSOL simulation was used to verify effectiveness of amendment formula, and it is revealed that simulated results of resonance frequency are in good agreement with theoretical results.The results indicate that the blue shift of resonance frequency occurs when the neck radius of SHS increases; the resonance frequency has a "V" change with the increase of inner radius of SHS; the resonance frequency has a red shift with increasing the outer radius of SHS.The method can precisely control the resonance frequency of acoustic metamaterial with negative effective modulus.There is a potential value on ultrasonic imaging, underwater acoustic, architectural acoustic, and sound absorbing material, and so on.

Figure 1 :Figure 2 :
Figure 1: (a) The 3D scheme of the metamaterial unit cell.(b) The cross section of the cell.

Figure 3 :
Figure 3: Effective bulk modulus of the unit cell obtained from the analytical expression.

Figure 4 :
Figure 4: Relation between the parameters of SHS and the resonance frequency.

Figure 7 :
Figure 7: The relation curves present the relationship between the inner radius and related parameters (  = 9.0 mm,  = 0.5 mm).(a) The resonance frequency of SHS.(b) The transmission coefficient.(c) The effective modulus.

Figure 8 :
Figure 8: The relation curves present the relationship between the outer radius and related parameters (  = 4.5 mm,  = 0.5 mm).(a) The resonance frequency of SHS.(b) The transmission coefficient.(c) The effective modulus.