1. IntroductionAcoustic metamaterials are broadly defined as artificial composite materials that can freely manipulate their constituent parameters, such as mass density ρ and bulk modulus B. For instance, acoustic metamaterials with extreme positive, negative, or near-zero parameters can be achieved with different types of structures. In previous studies, a kind of tubular acoustic metamaterial with negative density was constructed using an array of membranes. [1, 2] The negative bulk modulus metamaterials were fabricated based on periodically distributed Helmholtz resonators (HRs) [3–5] or side holes. [6, 7] Double negative (DNG) metamaterials have been designed by mixing two structures that have independent negative bulk modulus and negative mass density. [8–10] These acoustic metamaterials have potential applications in insulating broadband frequency noise, [11] tunable acoustic radiation patterns, [12] and acoustic invisible cloaking. [13–15]
Recently, the class of metamaterials that has a refractive index of near zero (RINZ) has become an important topic because of their anomalous properties in wave propagation. In an electromagnetic field, a RINZ metamaterial indicates that the medium has the characteristics of an infinitely large phase velocity and zero phase delay. These remarkable properties can be used for applications such as electromagnetic energy squeezing, [16] phase coupling, [17, 18] and transparency/cloaking devices. [19, 20] In principle, electromagnetic metamaterial research methods are also suitable for acoustic metamaterials. Although research into acoustic metamaterials has focused on the properties of negative mass density and/or negative bulk modulus, recent research on RINZ acoustic metamaterials has drawn considerable attention. To obtain RINZ acoustic metamaterials, the effective bulk modulus
should be infinite or the effective mass density
should be equal to zero. To manipulate the value of B accordingly, methods using an array of HRs [21] or side holes [22] have been proposed for one-dimensional (1D) acoustic metamaterials. To make ρ equal to zero, methods using membranes have been discussed for 1D [23, 24] and two-dimensional (2D) [25] RINZ acoustic metamaterials. Depending on the working frequency, the methods mentioned above can make the refractive index n of the metamaterial near zero, imaginary, or positive.
In this paper, we present a 1D membrane-based RINZ acoustic metamaterial that has a near zero effective mass density around the resonant frequency of the system. The effective density, refractive index, and phase velocity of the acoustic metamaterial are derived by introducing the concept of hidden force, which has been discussed for analogue mechanical systems. [26] To investigate the energy squeezing, transmission enhancement, and super coupling properties of the RINZ metamaterial, a normal waveguide material with a bigger cross-sectional area is directly coupled at the right end of the metamaterial. Using acoustic transmission line theory and fluid impedance theory, we discuss the properties of acoustic energy transmission and acoustic pressure distribution. In addition, numerical finite element method (FEM) simulations are performed to test and demonstrate the energy squeezing, transmission enhancement, and super coupling phenomena through the acoustic metamaterial. The results show that, at the resonant frequency
, the phase is almost uniform (super coupling) along the metamaterial, and at the specific frequency f
1 (which is slightly larger than
, zero reflection, energy squeezing, and high transmission are achieved despite the large geometric mismatch at the interface of the two materials. Such anomalous acoustic characteristics are mainly inspired by the metamaterial with near zero mass density.
2. Membrane-based RINZ metamaterialsThe structure of a 1D RINZ acoustic metamaterial is schematically shown in Fig. 1(a). The metamaterial consists of an array of very thin elastic membranes (M
i
) located along a waveguide pipe. The length of each unit cell and the inner diameter of the waveguide are L = 70 mm and l = 30 mm, respectively, as shown in Fig. 1(b). The tension of the membrane is τ = 20 N·m
. A sound source and an absorber are located at both ends of the metamaterial to generate and absorb acoustic waves, thus preventing reflection from the end of the metamaterial. As the wavelength is much longer than the periodic distance between the two adjacent membranes (
), this acoustic metamaterial can be regarded as a homogenized medium. [4]
2.1. Refractive index and phase velocityA mechanics-based concept can be used to describe the propagation of acoustic waves in this acoustic metamaterial, as recently demonstrated by Lee et al. [26] In this study, when an acoustic wave is transmitted into a waveguide, the center of the membrane in a unit cell of the acoustic metamaterial is subject to two forces, the applied force and the hidden force. The applied force
originates from the two adjacent unit cells, where
and
are the pressure difference across the unit cell and the cross-sectional area of the waveguide (or the area of the membrane), respectively. The hidden force
stems from the membrane's spring-like properties, where
and ξ are the effective stiffness of the membrane and the displacement of the center of the membrane, respectively. In general, once the applied force impacts the membrane, the hidden force generates and acts on the membrane. These two forces are collinear, the equivalent mechanical prototype of a unit cell is shown in Fig. 1(c). According to Newton's law, the motion equation of a unit cell is
, where
is the mass of a unit cell. Here,
and
are the density of air and the mass of the membrane, respectively. Hence, the effective mass of a unit cell can be obtained as
. Using the displacement harmonic function
, the effective mass of a unit cell can be expressed as
 | (1) |
where
is the resonant angular frequency. It can be seen from Eq. (
1) that, as the frequency increases through
, the value of
makes a sharp transition from negative to positive. As the acoustic metamaterial can be regarded as a homogenized medium, the effective mass density can be defined by
, where
is the volume of the unit cell. Therefore,
 | (2) |
where the average density of membrane-loaded air
(
is a constant that is slightly greater than the density of air
. This equation shows that the metamaterial exhibits negative effective mass density from zero frequency to the resonant frequency
, which provides a wide low-frequency forbidden band (LFB). The other constituent parameter for acoustic metamaterials is the effective modulus
. According to the pressure–volume relation, the existence of membranes does not change the effective modulus of this acoustic metamaterial, so
,
[1, 26] where
B
0 (
Pa) is the bulk modulus of air. For this system, the refractive index can be defined as the ratio of the phase velocity in free space to that in the acoustic metamaterial, i.e.,
, where
and
. Therefore,
 | (3) |
The effective density, phase velocity, and refractive index of the acoustic metamaterial with respect to frequency are shown in Figs. 2(a)–2(c), respectively. In the frequency range
(
, the effective density
is negative. Thus, the refractive index and the phase velocity are imaginary. In this situation, the acoustic metamaterial works in a forbidden band and can be used for the absorption of noise. In the frequency range
, we observe that the effective density, phase velocity, and refractive index are all positive, and an increase in the frequency causes the phase velocity to decrease continuously until it is equal to v
0. Hence, in this frequency domain, the acoustic waves propagate well in the metamaterial. When the frequency is close to the resonant frequency
, the effective density and refractive index simultaneously converge to zero. Notably, the phase velocity is very large and tends to infinity near the resonant frequency
. These results indicate that the acoustic metamaterial has potential applications as an acoustic wave filter and wave front transformer.
To confirm the preceding analytical results, we simulated the propagation of acoustic waves in the 1D RINZ acoustic metamaterial by FEM. In theory, the length of the acoustic metamaterial does not change the space distribution characteristics of the pressure field. Thus, for convenience, we selected an acoustic metamaterial with 20 unit cells for this calculation. In the FEM simulation, the membrane was considered as an additional acoustic impedance
, where
. Along the x-direction, a plane acoustic wave with an amplitude of 1 Pa enters the left inlet of the waveguide. Figure 3 shows the distribution of the pressure field in the acoustic metamaterial at frequencies of 250, 453.7, and 900 Hz, which are below, equal to, and above the resonant frequency
, respectively. From Fig. 3(a), we can see that if the frequency of the incident wave is below
, the acoustic wave can only penetrate into the first few units. This is because, in the frequency range
, the effective mass density
is negative, and the phase velocity and refractive index are imaginary. In this case, the metamaterial is opaque, and most of the incident energy is reflected back to the source. When the frequency of the incident wave is equal to the resonant frequency
, the effective density and the refractive index are near zero, as shown in Fig. 1(b). In this case, the wave can transmit through all the metamaterial units and the distribution of the pressure field is nearly constant, which means that the phase is nearly uniform in the RINZ metamaterial. Therefore, this RINZ metamaterial can be used as a phase super coupler. Figure 3(c) shows the acoustic pressure distribution in the metamaterial at a frequency of 900 Hz. At this frequency, the effective density and refractive index are positive and the metamaterial is transparent. The FEM results are consistent with the preceding analysis obtained using the mechanics-based concept (see Fig. 2).
2.2. Acoustic reflection and energy transmission coefficientsIn this section, we theoretically analyze the transmission properties of an acoustic wave from the RINZ acoustic metamaterial to a normal waveguide material that has a large cross-sectional area and is directly connected to the metamaterial. The composite structure of the system is shown in Fig. 4(a), where the normal waveguide material of length 3 m and radius 45 mm is terminated by the absorber. The geometric structure of the metamaterial is the same as that shown in Fig. 1(a), with a length of 7 m. At the interface of the two materials, the cross-section suddenly changes and the impedances of each material are different. Figure 4(b) shows a schematic of the cross-section of the two materials, in which
is the incident wave and
is the reflected wave in the metamaterial,
is the transmitted wave in the normal material, where
,
, and
are the incident, reflected, and transmitted waves at the interface, respectively. Here, k and
are the wave vectors for the metamaterial and the normal material, respectively. According to acoustic transmission line theory and fluid impedance theory, [3] the impedance relation between the sides of each membrane can be given by
 | (4) |
where
and
are the impedances of the fluids on the left and right of the membrane, respectively, and
z is the distributed impedance of the waveguide. The impedance relation between the membrane and the fluid in the waveguide can be expressed as
 | (5) |
Applying Eqs. (4) and (5) recursively N times, the impedance
of the acoustic metamaterial with N unit cells can be obtained. Based on the acoustic boundary condition, the acoustic pressure and volume velocity are simultaneously continuous at the interface (x = 7 m) between the two materials. This gives the following equations:
 | (6) |
where
is the cross-sectional area of the normal material and
is the impedance of the medium in the normal material. By combining the two parts of Eq. (
6), the acoustic pressure reflection coefficient
and transmission coefficient
at the interface of the two materials can be written as
 | (7) |
 | (8) |
Equations (7) and (8) show that the parameters determining the acoustic pressure reflection and transmission coefficients at the interface of the two different media are the acoustic impedance
and
, rather than the characteristic impedances
and
. [21, 27] Furthermore, it can be seen from Eq. (7) that there exist two critical frequency points. The first corresponds to the case
, where the effective density
, the reflection coefficient is 1, and the acoustic wave is completely reflected. The second frequency point corresponds to the situation in which the acoustic impedances between the metamaterial and the normal material are matched, i.e.,
, which indicates that the reflection coefficient is zero and the acoustic wave realizes non-reflection transmission. Figure 5(a) shows the acoustic wave reflection from the normal material to the metamaterial with respect to the frequency given by Eq. (7). At the lower frequencies of
, the reflection coefficient is near 1 and the almost acoustic waves are reflected to the source; thus, the metamaterial is opaque and no waves can propagate into the normal material. When the frequency is 465 Hz
, the reflection coefficient becomes zero, as shown in the magnified part of Fig. 5(a). This means that the acoustic impedances of the two materials are matched and the acoustic waves are totally transmitted. The value of f
1 is slightly larger than
, as shown in Fig. 2(a). At frequency f
1, the effective density of the metamaterial is close to zero. We can predict that if
, the two frequencies f
1 and
will be exactly the same, because in this situation the large difference in cross-sectional area at the connection compensates for the mismatch between the acoustic impedance at a frequency of
, which leads to the total transmission associated with the uniform phase along the metamaterial. In the frequency range
, the composite system is transparent and the wave can propagate from the metamaterial to the normal material. As the cross-sectional area of the normal material is nine times that of the metamaterial, i.e.,
, the interface of the two materials can be regarded as a soft boundary, and in this case the calculated reflection coefficient is negative. The energy transmission coefficient can be defined as
. The evolution of
with respect to frequency is plotted in Fig. 5(b), where we can see that the energy transmission is sensitive to the frequency. At the specific frequency f
1, it is indeed possible to achieve total energy transmission, despite the strong geometrical mismatch between the metamaterial and the normal material. This is the result of acoustic impedance matching between the two materials in the RINZ condition. Consequently, the acoustic wave provides a unique capability of high transmission through the RINZ metamaterial.
2.3. Acoustic wave propagation characteristicsWe now investigate the acoustic wave propagation characteristics for various transmission distances from the RINZ acoustic metamaterial to the normal material. In our structure, for a monochromatic forward traveling acoustic wave, the acoustic pressure in the RINZ acoustic metamaterial can be expressed as
 | (9) |
where
and
are the amplitudes of the incident and reflected acoustic pressures at the left entrance to the acoustic metamaterial.
can be determined by the initial condition, and we assume
. According to Eq. (
7),
can be calculated using the relation
, where
k is the wave vector of the metamaterial given by
. In the case
,
k is a real number, and the acoustic wave propagates in the
x-direction without attenuation. The acoustic energy moves in the same direction away from the source, and the incident acoustic pressure at the interface is given by
. However, when
, the wave vector
k is an imaginary number, and the wave attenuates in the
x-direction rapidly and cannot propagate. The sound energy moves in the negative
x-direction toward the source, and the incident acoustic pressure at the interface is
. To explain this characteristic explicitly, when
, Eq. (
9) can be simplified as
 | (10) |
where the wave vector
k
1 is given by
. Equations (
9) and (
10) express the acoustic pressure in the acoustic metamaterial. According to the definition of the acoustic pressure transmission coefficient
, the amplitude of acoustic pressure in the normal material can be written as
. Therefore, the acoustic pressure in the normal material can be expressed as
, where
is the wave vector of the medium in the normal material. By combining the expressions for
with Eqs. (
9) and (
10), the forward traveling wave solutions can be obtained as
 | (11) |
 | (12) |
The theoretical curves of the acoustic pressure with respect to distance are plotted in Fig.
6. These show the propagation of acoustic waves in the acoustic metamaterial and normal material for excitation frequencies of 250, 350, 450, 453.7, 465, and 900 Hz. For excitation frequencies below
, the acoustic waves exponentially decay along the metamaterial, e.g., if
f = 250 Hz, the black curve in Fig.
6(a) indicates that the amplitude of acoustic pressure decreases about 75% in the first three unit cells. When the excitation frequency is 453.7 Hz
, as shown in Fig.
6(b), the acoustic pressures in the metamaterial and the normal material are close to 0 and 2 Pa, respectively. This is because the acoustic pressure reflection and transmission coefficients are 1 and 2 at this excitation frequency. Care must be taken at this critical frequency point, as the hard boundary condition at the interface of the two materials means that only the static pressure with an amplitude of 2 Pa exists, and in the normal material there is no particle vibration and no acoustic wave propagation. When the excitation frequency is 465 Hz
, the acoustic metamaterial has a near zero effective density and the acoustic impedances are matched between the two materials. Thus, as shown in Fig.
6(c), the acoustic pressures in the metamaterial and normal material have the same amplitude, and the phase change inside the metamaterial is apparently slower than inside the normal material. Moreover, transmission enhancement and energy squeezing are achieved, despite the large geometrical mismatch. These extraordinary capabilities may have very appealing applications in sound squeezing and acoustic interconnects. At a frequency of 900 Hz
, the acoustic pressure in the metamaterial is nine times that in the normal material, as shown in Fig.
6(d). This is in inverse proportion to the cross-sectional areas of the two materials.
Using the same method of FEM as in the preceding section, we simulated the propagation of acoustic waves in the composite system. In this FEM simulation, the normal material had the same length as the metamaterial, which consisted of 20 unit cells. The distributions of acoustic pressure in the acoustic metamaterial and the normal material along the propagation path at frequencies of 250
, 465
, and 900 Hz
are plotted in Fig. 7. From Fig. 7(a), we can see that if the frequency is below f
1, the acoustic wave can only propagate into the first few units and the composite system is opaque, similar to that shown in Fig. 6(a). If the frequency is equal to f
1, the acoustic metamaterial has near zero effective density, as plotted in Fig. 7(b). The acoustic pressures in the two materials have the same amplitudes, and the pressure distribution in the propagation path is the same as that shown in Fig. 6(c). Moreover, as expected, we can see from Fig. 7(b) that the phase changes slowly and the effective wavelength in the metamaterial is much longer than in the normal material. Good transmission and energy squeezing are achieved through the RINZ metamaterial. If the frequency is above f
1, e.g., 900 Hz, the acoustic wave propagates well in the composite system, and the propagation velocity in the two materials is the same, as shown in Fig. 7(c). These simulation results are completely consistent with those obtained in the preceding theoretical analysis.
