Sound transmission, band structure and negative bulk modulus in a mirror-symmetry metamaterial

Abstract

The control of sound energy in reflection and transmission problems using acoustic metamaterials has become a relevant field, especially considering scientific, technological, and application aspects. However, the relationship between this control and the unusual effective parameters that metamaterials can exhibit is poorly explored, and therefore needs to be investigated. In this work, a scattering matrix and an experimental method based on the transfer matrix (one-load method) are used to recover the effective properties and evaluate the negative effective parameters in specific frequency bands of a low-dimensional with mirror-symmetric metamaterial ($\lambda /19$). The relationship between sound transmission behavior ($T\to 0$) and the negative effective bulk modulus ($K_{ef}<0$), which occurs between 620–1480 Hz and 1960–2750 Hz, is investigated and provides a deep understanding of the physical mechanisms of the structure. Finally, this study contributes to advancements in the field of sound energy control, as well as to the understanding of the unusual effective properties of mirror-symmetric structures.

Keywords:
Acoustic metamaterial; Sound transmission loss; Mirror-symmetric; Negative bulk modulus

1. Introduction

Acoustic metamaterials technology, which represents the association between wave physics and periodic structures with unique properties and behaviors, especially at low frequencies, is a research area that has gained notoriety in recent decades. This attention is due to the technological and innovative aspects, as well as the wide applicability that these materials provide [¹, ², ³]. Acoustic metamaterials are structures that replicate a primitive unit cell and exhibit: (i) unusual effective properties and (ii) behavior not found in common natural materials, especially for long wavelengths. In practice, the different types of metamaterials are based on different physical principles, the most prominent being: (i) Helmholtz resonance, (ii) Fabry-Pérot resonance, (iii) Fano resonance and (iv) hybrid-type resonances [⁴]. It is well-known that the most prominent application of metamaterials is in the control of sound energy. In general, this control is evaluated through two physical parameters: the reflection coefficient (sound absorption) and the transmission coefficient (sound transmission loss) [⁵]. Although several studies on the behavior of these structures in problems of reflection and sound transmission have been consolidated in the literature, research associating their behavior with their unusual effective parameters is rare and deservesattention.

Note that, from the perspective of physics education and according to the topics discussed here, few studies address this subject [⁶, ⁷]. On the other hand, evaluating the behavior of metamaterials taking into account their unusual effective parameters is important to gain a better understanding of the underlying physical mechanisms that govern the performance of these structures. In this regard, Lee and Wright [⁸] presented a fundamental review and a physical interpretation of the density and effective bulk modulus of different types of acoustic metamaterials. The authors introduced the terminology of hidden force and hidden volume source: the effective density or effective bulk modulus is negative when the hidden force or volume source is greater than and operates in antiphase with the force or volume change that would be obtained in its absence, respectively. Zhou et al.[⁹] presented the behavior of a membrane-type acoustic metamaterial with negative effective parameter characteristics. The authors experimentally and numerically evaluated the sound transmission loss and the behavior of the dynamic effective mass and dynamic effective stiffness with negative values. Jena et al. [¹⁰] used an analytical method and the theory of effective properties to estimate and evaluate the mass density and bulk modulus properties of eight configurations of a metamaterial based on Helmholtz resonators. Also noteworthy is the research on a fractal metamaterial proposed by Xiang et al. [¹¹] in which they evaluated the bandgap and sound attenuation properties using the theory of effective properties. In addition, they evaluated the mass density and equivalent bulk modulus behavior for first, second, and third order fractals. It is important to note that the theory of effective properties or effective parameters is a method that recreates the characteristics of the acoustic field through an effective homogeneous medium. In this regard, the theory of effective properties has been used to obtain the properties of various types of metamaterials, from electromagnetic [¹², ¹³], acoustic [⁵, ¹⁴, ¹⁵, ¹⁶] and even branched Helmholtz resonators [¹⁰].

This work presents the behavior of a mirror-symmetric metamaterial for a sound transmission problem, analyzing its effective bulk modulus and effective mass density. The low-dimensional physical structure is based on a waveguide loaded with symmetric quarter-wavelength ($\lambda /4$) scatterers, such that as sound waves propagate through a longer path within the scatterers, more energy can be dissipated. Consequently, these scatterers are often called Fabry-Pérot dispersers. Although this study was influenced by the aforementioned studies, the structure presented here differs from the models mentioned in References [⁹, ¹⁰, ¹¹, ¹⁴, ¹⁵, ¹⁶] by having a smaller thickness, which is therefore the originality of the work. Furthermore, its novelty and relevance lie in presenting a study that associates the behavior of an acoustic metamaterial with its unconventional effective parameters. Consequently, the proposed structure significantly expands the possibilities for controlling and manipulating sound waves and finds diverse practical applications in scientific and technological contexts, including, among others, acoustic insulation, acoustic filtering, sound camouflage, and much other. Theoretical discussions about the behavior of the structure are corroborated by experimental results. Hence, this work contributes to the advancement in the control of sound energy through mirror-symmetric structures and deepens the understanding of their behavior based on their respective negative effective properties. Furthermore, from the perspective of didactic contributions and extensions, it is expected that the themes addressed here will gain greater prominence in higher education disciplines in Brazil, as well as in interdisciplinary research projects.

2. Materials and Methods

2.1. Mirror-symmetric metamaterial geometry

The description of the geometry of the metamaterial with mirror-symmetry (symmetric with respect to the median plane at $x=0$) is presented in Figure 1(a). The physical system exhibits symmetry when the symmetry transformation performed on this system does not change, that is, it is invariant [¹⁷]. In physical acoustics, reflection invariance plays an important role because, due to mirror symmetry, the coefficients of forward and backward reflection become identical. Note that the system can be rewritten in two complementary subsystems [¹⁸]: 1) symmetric system, which is established when the Neumann boundary condition – NBC is imposed in the symmetric plane (see Fig. 1(b)); 2) anti-symmetric system, which is established when the Dirichlet boundary condition – DBC is imposed on the symmetric plane (see Fig. 1(c)). A symmetric, antisymmetric, and reciprocal system can be characterized from the scattering matrix as [¹⁹, ²⁰]

Figura 1
(a) Overview of the proposed problem, i. e., the geometry of the metamaterial with mirror-symmetry with respect to the median plane at $x=0$. (b) Symmetric subsystem and (c) antisymmetric subsystem of the acoustic metamaterial with mirror-symmetry. $I$ represents the amplitude of the incident wave and $P$ is the sound pressure.

(1) $$\mathbf{\text{S}}=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right]=\left[\begin{array}{cc}{T}_f& R_f\\ R_b& T_b\end{array}\right],$$

with $T_f=T_b=T$ representing the transmission coefficients and $R_b=R_f=R$ the forward and backward reflection coefficients of the system. The eigenvalues of the matrix S for a mirror-symmetric structure are obtained by the characteristic equation.

(2) $$\mathbf{\text{det}}\left(\left[\begin{array}{cc}T& R\\ R& T\end{array}\right]-\mathrm{\Delta }\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right)=0,$$

(3) $$T^2-2\mathrm{\Delta }T+{\mathrm{\Delta }}^2-R^2=0.$$

From equation 3, we have ${\mathrm{\Delta }}_{f,b}=T\pm R$, while the eigenvectors are given by

(4) $$\left(\begin{array}{cc}T-T\pm R& R\\ R& T-T\pm R\end{array}\right)\left(\begin{array}{c}{\overrightarrow{v}}_f\\ {\overrightarrow{v}}_b\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)⟹$$

(5) $$⟹\begin{array}{c}\pm \left(R\right){\overrightarrow{v}}_f+\left(R\right){\overrightarrow{v}}_b=0\\ \left(R\right){\overrightarrow{v}}_f\pm \left(R\right){\overrightarrow{v}}_b=0.\end{array}$$

From Eq. 5 we have ${\overrightarrow{v}}_{f,b}=(1,\pm 1)$. For the symmetric subsystem in Figure 1(b) and considering plane waves propagating forward ($e^{-j\omega t}$), the reflection coefficient is obtained from $R_{\text{sym}}=T+R_f$. In this case, it selects only those modes that are symmetric in the symmetry plane to which the NBC condition is applied. Otherwise, the antisymmetric subsystem (see Figure 1(c)) acts as a mirror whose image has a phase shift of ${180}^{\circ }$ and only the antisymmetric modes are selected in the DBC condition; therefore, the reflection coefficient is obtained from $R_{\text{asym}}=T-R_b$. Thus, the reflection and transmission coefficients of the general problem in Figure 1(a) are expressed as $R=\left(R_{\text{sym}}-R_{\text{asym}}\right)/2$ and $T=\left(R_{\text{sym}}+R_{\text{asym}}\right)/2$. Consequently, the eigenvalues of the scattering matrix can be expressed as ${\mathrm{\Delta }}_f=R_{\text{sym}}$ and ${\mathrm{\Delta }}_b=R_{\text{asym}}$. The components of equation 1 for the analytical and experimental methods are presented below.

2.2. Mathematical model

The basic configuration of the mirror-symmetric structure and the geometric parameters are shown in Figure 2. For a time harmonic excitation ($e^{-j\omega t}$) and considering analyses below the cutoff frequency, the transmission and reflection coefficients can be obtained by the transfer matrix method ($T_g$), which relates the state vector $T=(P,U)$ on both sides of the structure.

Figura 2
The mirror-symmetric metamaterial consists of a main waveguide (slit) loaded with $N$ identical Fabry-Pérot dispersers, from the top to the symmetrical median plane. All walls of the system are considered rigid and with thickness $b$. The dispersers have a fixed periodicity $h$. Each cell has height $l$, width $m$, and total depth $2L=2\times \left(Nh\right)$. The main waveguide has height $d_f$ and width $L_g=m-b$. The Fabry-Pérot dispersers have height $W=\left(h-b-\left(n-1\right)\times b\right)/n$, width $a=L_g$ and total length $L_{ef}\approx n\times \left(l-2b\right)+\left(n-1\right)\times W$ where $n$ is the number of coiled.

(6) $$\begin{array}{l}{\left[\begin{array}{c}P\\ \mathbf{\text{-n}}\cdot U\end{array}\right]}_{x=-L}={\text{T}}_g{\left[\begin{array}{c}P\\ \mathbf{\text{-n}}\cdot U\end{array}\right]}_{x=L}\\ =\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]{\left[\begin{array}{c}P\\ \mathbf{\text{-n}}\cdot U\end{array}\right]}_{x=L},\end{array}$$

with n represents the normal vector of the wave propagation direction, $P$ the acoustic pressure, and $U$ the volume velocity. The global transfer matrix ${\text{T}}_g$ of the structure is expressed as

(7) $$\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]={\left(M_S{M}_T\right)}^N{M}_i{\left(M_T{M}_S\right)}^N,$$

where $M_S$, $Mi$ and $M_T$ are, respectively, the matrices of the main waveguide and the dispersers. According to the fundamental principle of continuity of pressure and volume velocity, these matrices are given by

(8) $$M_S=\left[\begin{array}{cc}\cos \left[k_{S}t\right]& j{Z}_S\sin \left[k_{S}t\right]\\ j\sin \left[k_{S}t\right]/Z_S& \cos \left[k_{S}t\right]\end{array}\right],$$

(9) $$M_i=\left[\begin{array}{cc}\cos \left[k_S\left(2b\right)\right]& j{Z}_S\sin \left[k_S\left(2b\right)\right]\\ j\sin \left[k_S\left(2b\right)\right]/Z_S& \cos \left[k_S\left(2b\right)\right]\end{array}\right],$$

(10) $$M_T=\left[\begin{array}{cc}1& 0\\ 1/Z_{{\text{T}}^{{}^{\prime }}}& 1\end{array}\right],$$

where the effective wavenumber $k_{\text{S}}$ and the characteristic acoustic impedance in the main waveguide $Z_{\text{S}}$ are [⁵, ²¹]:

(11) $$k_S=\frac{\omega }{c_0}\left[1+\frac{\left(1-j\right)\sqrt{\eta }}{r_{\text{S}}\sqrt{{\rho }_0\omega }}\cdot \left(1+\frac{\left(\gamma -1\right)}{\sqrt{\text{Pr}}}\right)\right],$$

(12) $$Z_S=\frac{{\rho }_0{c}_0}{A_S}\left[1+\frac{\left(1-j\right)\sqrt{\eta }}{r_{\text{slit}}\sqrt{{\rho }_0\omega }}\cdot \left(1-\frac{\left(\gamma -1\right)}{\sqrt{\text{Pr}}}\right)\right],$$

with $\eta =1.8134\times {10}^{-5}$ Pa$\cdot$s, $\gamma =1.41$ and $\text{Pr}=0.71$ representing, respectively, the viscosity of air, the specific heat ratio and the Prandtl number. ${\rho }_0=1.21$ kg$\cdot$m^-3 and $c_0=343$ m$\cdot$s^-1 are the density and sound speed of air, respectively. In equation 8 $t=h-W$, and in equations 12 $A_S=L_g\times d_f$ and $r_{\text{S}}$, are the cross-sectional area and the radius of the main pore are, respectively. Details regarding the derivation of each matrix in Eq. 7 can be found in Reference [²²].

In equation 10, the term $Z_{{\text{T}}^{{}^{\prime }}}$ represents the total impedance of the disperser, which is obtained as $Z_{{\text{T}}^{{}^{\prime }}}=-j{Z}_{\text{T}}\cot \left(k_{\text{T}}{L}_{ef}\right)$, with $k_T=\omega \sqrt{{\rho }_T/K_T}$ being the effective wavenumber. $Z_T=\left(\sqrt{{\rho }_T{K}_T}\right)/A_T$ and $A_{\text{T}}=W\times a$ are the characteristic acoustic impedance and the disperser area, respectively. ${\rho }_T$ and $K_T$ are the complex functions of the density and the bulk modulus, correspondingly. The first function describes viscous losses, while the second represents thermal losses. For dispersers with rectangular cross-section, both expressions are derived from viscothermal acoustic theory [²³, ⁶]. Finally, from the global transfer matrix (equation 7) and respecting the reciprocal and symmetrical nature of the physical system ($T_{11}=T_{22}$), the reflection and transmission coefficients for symmetrical and reciprocal structures become:

(13) $$R=R_f=R_b=\frac{T_{12}/Z_0-T_{21}{Z}_0}{T_{12}/Z_0+T_{21}{Z}_0+2T_{22}},$$

(14) $$T=T_f=T_b=\frac{2e^{jkNh}}{T_{12}/Z_0+T_{21}{Z}_0+2T_{22}}.$$

Consequently, the sound transmission loss can be obtained as

(15) $$STL=20{\text{log}}_{10}\left(\frac{1}{T}\right).$$

2.3. Acoustic band structure

The acoustic band structure is characterized by a passband in which waves propagate freely within the structure and a stopband in which waves are exponentially attenuated. These waves are called Bloch waves (or the Bloch-Floquet theorem) and exhibit the same symmetry and periodicity as the structure. Thus, the sound pressure and velocity fields of these waves can be expressed as a Bloch function which has the following form:

(16) $$P\left(x\right)=P\left(x+h\right)e^{-jqh},$$

(17) $$U\left(x\right)=U\left(x+h\right)e^{-jqh}.$$

where $P\left(x+h\right)$ and $U\left(x+h\right)$ are the pressure and velocity amplitudes that vary periodically with period $h=L$ and $q$ is the Bloch wavenumber, which is a complex number. Since only plane waves propagate in the structure periodically, the Bloch-Floquet theorem can be used to derived the acoustic band structure. Therefore, the global transfer matrix of a single unit cell can be described as follows [²⁴]

(18) $$\begin{array}{l}{\left[\begin{array}{c}P\\ U\end{array}\right]}_{x=-L}=\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]{\left[\begin{array}{c}P\\ U\end{array}\right]}_{x=L}\\ =\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]{\left[\begin{array}{c}P\cdot \left(e^{-jqL}\right)\\ U\cdot \left(e^{-jqL}\right)\end{array}\right]}_{x=-L}.\end{array}$$

From equation 18 we obtain the following characteristic equation

(19) $$\mathbf{\text{det}}\left(\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]-\left[\begin{array}{cc}{e}^{-jqL}& 0\\ 0& e^{-jqL}\end{array}\right]\right)=0.$$

Consequently, the following eigenvalue problem can be obtained

(20) $$\begin{array}{l}\left|\begin{array}{cc}{T}_{11}-e^{-jqL}& T_{12}\\ T_{21}& T_{22}-e^{-jqL}\end{array}\right|\\ ={\left(e^{-jqL}\right)}^2-e^{-jqL}\left(T_{11}+T_{22}\right)+\left|T\right|=0.\end{array}$$

From the reciprocity principle, the determinant of the equation above $\left|T\right|=\left[T_{11}{T}_{22}-T_{12}{T}_{21}\right]=1$, therefore, the Bloch waves propagating forward and backward exhibit the same acoustic band structure. Thus, the usual form of the dispersion relation can be found by rewriting equation 20 as

(21) $$e^{-jqL}+\frac{1}{e^{-jqL}}=T_{11}+T_{22}⟹q=\frac{{\cos }^{-1}}{L}\left(\frac{T_{11}+T_{22}}{2}\right).$$

Since the terms $T_{11}$ e $T_{22}$ vary with frequency note that the acoustic band structure becomes a functional relationship between the real temporal frequency ($\omega$) and the Bloch spatial frequency ($q$) [²⁴].

2.4. Experimental model

The reflection and transmission coefficients of the mirror-symmetric structure were obtained in a cylindrical impedance tube. The experimental setup is based on the transfer matrix method, since the structure is symmetric, the one-load method was used, that is, anechoic termination [²⁵]. The impedance tube has a diameter of $\Phi =46$ mm, corresponding to a cutoff frequency $f_c=0.585c_0/\Phi =4367$ Hz; however, the results presented were carried out in the maximum frequency range of $100$ to $3100$ Hz. The random signal and four $1/2$ inch condenser microphones (Bruel Kjaer 4971-H 041) were used to measure the sound pressure. A data acquisition system (Bruel Kjaer LAN-XI type 3677-A-041 with BK Connect software) allowed the recording and processing of the signals. The experimental setup used is illustrated in Figure 3. The distances between the downstream face of the sample and the microphones 1 and 2 are $x_1$ and $x_2$, respectively. Similarly, $x_3$ and $x_4$ are the distance between the upstream face of the sample and microphones 3 and 4. Using the one-load method, the objective is to determine a matrix T${}_{\mathbf{\text{t}}}$ that relates two state vectors $T=(P,u)$ as a function of the acoustic pressure ($P$) and the velocity of particles ($u$) on both sides of the samples [²⁵],

Figura 3
Experimental setup based on the transfer matrix method to evaluate the behavior of mirror-symmetric metamaterial. (a) Measuring equipments. (b) Schematic diagram of the experimental setup. The distance considered are: $x_1=0.136$ m, $x_2=0.105$ m, $x_3=2L+0.105$ m and $x_4=2L+0.136$ m.

(22) $${\left[\begin{array}{c}P\\ u\end{array}\right]}_0={\mathbf{\text{T}}}_{\mathbf{\text{t}}}{\left[\begin{array}{c}P\\ u\end{array}\right]}_{2L}=\left[\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right]{\left[\begin{array}{c}P\\ u\end{array}\right]}_{2L}.$$

The acoustic waves field inside the tube is decomposed as:

(23) $$\begin{array}{cc}\hfill A& =j\frac{H_{11}{e}^{-j{k}_o\left(x_2\right)}-H_{21}{e}^{-j{k}_0\left(x_1\right)}}{2\sin \left(k_0\left(x_1-x_2\right)\right)};\hfill \\ \hfill B& =j\frac{H_{21}{e}^{j{k}_o\left(x_1\right)}-H_{11}{e}^{j{k}_0\left(x_2\right)}}{2\sin \left(k_0\left(x_1-x_2\right)\right)},\hfill \end{array}$$

(24) $$\begin{array}{cc}\hfill C& =j\frac{H_{31}{e}^{j{k}_o\left(x_4\right)}-H_{41}{e}^{j{k}_0\left(x_3\right)}}{2\sin \left(k_0\left(x_4-x_3\right)\right)};\hfill \\ \hfill D& =j\frac{H_{41}{e}^{-j{k}_o\left(x_3\right)}-H_{31}{e}^{-j{k}_0\left(x_4\right)}}{2\sin \left(k_0\left(x_4-x_3\right)\right)}.\hfill \end{array}$$

In equations 23 and 24, we have four frequency response functions (FRF), the first being reference FRF $H_{11}$, a vector with unit values of the same dimension as the frequency vector adopted in the analyses; the other FRFs are obtained directly from the tests. Hence, the sound pressures and velocities exactly on the downstream and upstream faces of the structure are given by:

(25) $$P_0=A+B;\mathrm{}{u}_0=\frac{\left(A-B\right)}{{\rho }_0{c}_0},$$

(26) $$\begin{array}{cc}\hfill P_{2L}& =C{e}^{-j{k}_{0}2L}+D{e}^{j{k}_{0}2L};\hfill \\ \hfill u_{2L}& =\frac{\left(C{e}^{-j{k}_{0}2L}-D{e}^{j{k}_{0}2L}\right)}{{\rho }_0{c}_0}.\hfill \end{array}$$

Thus, the matrix T${}_{\mathbf{\text{t}}}$ can be written as

(27) $$\begin{array}{l}\mathbf{\text{T}}{}_{\mathbf{\text{t}}}=\left[\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right]\\ =\frac{1}{P_0{u}_{2L}+P_{2L}{u}_0}\times \left[\begin{array}{cc}{P}_{2L}{u}_{2L}+P_0{u}_0& {\left(P_0\right)}^2-{\left(P_{2L}\right)}^2\\ & \\ {\left(u_0\right)}^2-{\left(u_{2L}\right)}^2& P_{2L}{u}_{2L}+P_0{u}_0\end{array}\right],\end{array}$$

this allows us to obtain the reflection (equation 13) and transmission (equation 14) coefficients for symmetrical ($m_{11}=m_{22}$) and reciprocal ($m_{11}{m}_{22}-m_{12}{m}_{21}=1$) structures. It can be seen that the acoustic band structure (equation 21) in the experimental method is obtained directly from the coefficients of equation 27, that is, replacing $T_{ij}$ by $m_{ij}$. In addition, the relationship between the amplitudes of the forward and backward waves in the mirror-symmetric and reciprocal structure can be characterized theoretically and experimentally by the symmetrical scattering matrix given by equation 1.

2.5. Effective parameters model

Theory of the effective medium is used to better understand the physical mechanisms underlying the proposed mirror-symmetric metamaterial. Essentially, the method aims to recreate the characteristics of the acoustic field through a homogeneous medium, applying experimental data to obtain the effective parameters. A schematic representation of the parameter recovery process is shown in Figure 4. The structure in Figure 4(a) is considered a homogeneous fluid material in Figure 4(b), with the same amplitude and phase of the reflection ($R$) and transmission ($T$) coefficients. The effective parameters are obtained using the inverse method, as shown in Figure 4(c), which will be presented below.

Figura 4
Procedure for acquiring effective properties. (a) Representation of the structure. (b) Structure of an effective material equivalent to the proposed structure. (c) Illustration of the method.

Considering plane waves incident on the plane of the metamaterial, with the structure surrounded by air on both sides, and assuming an ideal anechoic termination in the experimental apparatus, the effective acoustic impedance is expressed as follows. Interested readers can find a detailed description of the method in Reference [²⁶]:

(28) $$Z_{ef}=\frac{\chi }{1-2R+R^2-T^2},$$

where $R$ and $T$, respectively, are the experimental sound reflection and transmission coefficients. The parameter $\chi$ is defined as follows:

(29) $$\chi =\pm \sqrt{{\left(R^2-T^2-1\right)}^2-4T^2}.$$

In equation 29, we choose the root that provides a positive solution in equation 28[⁵, ²⁶]. Furthermore, the acoustic refractive index is expressed as

(30) $$n_{ef}=\frac{-j\ln \zeta +2\pi F}{k_{0}2L},$$

where $F$ is the number of dispersers and a minimum metamaterial thickness corresponding to $F=0$ is used during the recovery of the effective parameters. $\zeta$ is defined as:

(31) $$\zeta =\frac{1-R^2-T^2+\chi }{2T}.$$

From the effective acoustic impedance (equation 28) and the acoustic refractive index (equation 30), the effective mass density and the effective bulk modulus can be estimated using the following mathematical relationships:

(32) $${\rho }_{ef}=Z_{ef}\cdot n_{ef},$$

(33) $$K_{ef}=\frac{Z_{ef}}{n_{ef}}.$$

3. Results and Discussions

Figure 5 illutrates the wide control of sound energy for both forward and backward sound waves, as shown in the figure above. A sample structure was manufactured using 3D printing vat photopolymerization technology and ABS resin as the fabrication material. The geometric parameters used are described in the figure caption. Furthermore, to compose the symmetrical structure, a Fabry-Pérot disperser type was chosen with its first and second resonance frequencies at $f_{RH}=c/4L_{ef}=718$ Hz and 2154 Hz (see vertical dashed black line). The scattering coefficients are presented in Figures 5(a) and (b), while the sound transmission loss is presented in Figures 5(c), and a good agreement can be observed between the theory and the experimental results. Note that the resonances for the reflection problem in the structure occur before the resonance frequency of the disperser ($f_{HR}$), this behavior is due to strong dispersion in the structure which occurs due to the coupling of the Fabry-Pérot tubes in the main waveguide. As expected, due to mirror symmetry, the metamaterial exhibits reflection invariance, that is, the structure is indifferent to the direction of the incident waves ($R=R_f=R_b$ and $T=T_f=T_b$). Some discrepancies are observed between methods; this behavior can be attributed to imperfect sample manufacturing, primarily due to inaccuracies in the dimensions of the main waveguide. Note that above $f_{HR}$, a bandgap is introduced into the system; consequently, the transmission coefficient is significantly reduced; thus the structure acts as a rigid-backed wall. Further on, in Figure 6, we will explore the the acoustic band structure. Focusing now on STL behavior, good agreement can be observed and two sound reduction peaks exceeding 57 dB at 716 Hz and 53 dB at 2140 Hz in the bandgap regions, indicating that the behavior of $T$ is strongly dependent on the acoustic band structure, and these STL peaks occur close to the $f_{HR}$ of the Fabry-Pérot-type disperser. Furthermore, the structure exhibits a sound attenuation greater than 20 dB between 656 Hz and 1484 Hz and greater than 30 dB between 1994 Hz and 2659 Hz (see red horizontal lines). In addition, the sample thickness is $2L=2\times \left(Nh\right)=41$ mm, which corresponds to $\approx \lambda /19$ for the lowest experimental peak (445 Hz) in the reflection problem. This result reveals that the structure achieves a subwavelength scale.

Figura 5
Theoretical and experimental results of the metamaterial with mirror-symmetrical. (a) and (b) Reflection and transmission coefficients. (c) Sound transmission loss. Geometric parameters (in mm) used: $m=15.0$, $l=62.0$, $h=20.5$, $2L=41.0$, $d_f=2.0$, $b=1.50$, $n=2$ curled and $N=1$. The vertical dashed black line represents the resonance frequency of a Fabry-Pérot disperser.

Figura 6
Theoretical and experimental behavior of the acoustic band structure of the metamaterial. (a) Real part of the wavenumber and experimental sound transmission coefficient. (b) Imaginary part of the wavenumber.

The acoustic band structure of the symmetrical metamaterial and its experimental sound transmission coefficient are showed in Figure 6. The band structure diagram is presented in the form of an extended zone with a non-absolute normalized value. In the bandpass regions (see the blue horizontal lines in Figure 6(a)), Bloch waves propagate freely in the structure, since the wavenumber is real and a nonlinear function of frequency; therefore, the waves in these bands are dispersive. On the other hand, in the bandforbidden regions (see the brown horizontal lines), Bloch waves are strongly attenuated and are of the evanescent type, since the imaginary part of the wavenumber is non-zero (see Figure 6(b)). Thus, this component is also a non-linear function of frequency; consequently, the sound attenuation is exponential and the waves are dispersive. Bandgaps are therefore bands associated with a single Bloch wavelength [²⁴]. The first bandgap begins before the first resonance frequency $f_{RH}=718$ Hz and has its physical origin in the Fabry-Pérot resonances of dispersers, while the second bandgap begins before the second resonance frequency $f_{HR}=2154$ Hz and has its origin in the periodicity of the structure due to the destructive interference phenomenon of Bloch waves. Finally, by deduction, we conclude that the behavior of the acoustic band structure corroborates the results of the transmission coefficient and the sound transmission loss in Figure 5.

We can now better understand the physical mechanisms underlying the sound transmission behavior, the acoustic band structure, and the unusual effective parameters of the mirror-symmetric structure. To realistically determine these effective parameters, we can verify that the experimental procedure performed provides reliable characteristics. This can be done by observing whether the experimental apparatus has an ideal anechoic termination across the analyzed frequency range and whether the real component of the effective acoustic impedance is positive [⁵]. Figure 7(a) shows that the experimental setup guarantees a condition of absence of reflected sound waves, since sound absorption greater than 96.12% is observed across the frequency spectrum. On the other hand, Figure 7(b) shows that the real part of the effective impedance is positive.

Figura 7
(a) Behavior of sound reflection and absorption coefficients with the impedance tube using the anechoic load method. (b) Experimental behavior of the normalized effective acoustic impedance.

The theoretical and experimental behavior of the real part of the effective mass density and the effective bulk modulus are shown in Figure 8. These parameters provide information about the characteristics of the structure, i. e., its band structure and, consequently, the behavior of the sound transmission coefficient. A negative effective mass density in the structure indicates that the velocity of the particles is out of phase with the dynamic pressure gradient of the medium. This negative mass density becomes more pronounced when the resonance of the structure is strong enough to disperse the field to a level higher than the background sound field. Figure 8(a) reveals that in the adopted frequency range, the effective mass density of the mirror-symmetric metamaterial is positive, meaning that the velocity of the particles is in phase with the pressure gradient of the medium. Therefore, ${\rho }_{ef}>0$ implies that a sound wave within the structure exhibits the same direction as the acoustic excitation. However, a negative effective volume modulus indicates that in certain regions of the frequency spectrum, the metamaterial medium expands when subjected to external compression and contracts when stretched. Figure 8(b) shows a comprehensive overview of how this parameter varies in different acoustic bands, elucidating the complex behavior of the mirror-symmetric metamaterial in response to sound waves. The results reveal that, in the first (620–1480 Hz) and second (1960–2750 Hz) bandgaps, a negative effective bulk modulus ($K_{ef}<0$) is obtained. These negative regions arise from the resonances of the Fabry-Pérot tubes, since these tubes act as a local scatterer and induce the sound reflection and the periodicity of the structure.

Figura 8
Theoretical and experimental behavior of the real parts of the parameters. (a) Effective mass density and (b) effective bulk modulus. Where $K_0=\gamma P_0$ is the bulk modulus of air with $P_0=101325$ Pa.

In the general, we can observe that the discrepancies between experimental and theoretical results occur mainly in the amplitudes of the analyzed parameters (see Figures 5, 6 and 8). As mentioned previously, although the printer used has a tolerance of $\pm 0.15$ mm, which introduces a maximum dimensional deviation of up 0.3 mm, these discrepancies are due to imperfections in 3D printing, such as the roughness of the internal walls of the structure and, especially, in the main waveguide. The roughness of the internal surfaces of the sample causes the damping effect that necessarily exists and, in practice, produces a decrease in the amplitude of the parameters. Note that similar results were observed by [²⁷] and [²⁸]. Additionally, another factor contributing to the discrepancies is the unwanted leakage of sound waves at the coupling boundaries between the impedance tube and the sample.

Next, we will present an alternative interpretation for the underlying physics discussed above. Consider the classical one-dimensional wave equation

(34) $${\nabla }^{2}p(x,t)=\frac{1}{c_{ef}^2}\frac{{\partial }^{2}p(x,t)}{\partial t^2},$$

which can be solved by solving for the pressure field $p(x,t)=p_a{e}^{j\left(-kx+\omega t\right)}$ of a harmonic plane wave, with $k=\omega /c_{ef}$ representing the wave vector and $c_{ef}$ the speed of sound. When $K_{ef}<0$, this implies that the speed of sound becomes complex, that is, $c_{ef}=j\sqrt{K_{ef}/{\rho }_{ef}}$, consequently the wave vector also becomes complex ($k=\omega /c_{ef}=-j{k}_{real}$). Substituting the complex $k$ in the pressure field solution, we have $p(x,t)=p_a{e}^{-k_{real}x}{e}^{j\omega t}$. Thus, note that when $K_{ef}<0$ we have a wave that propagates in time ($e^{j\omega t}$), but exhibits evanescent behavior, that is, it decays exponentially in space ($e^{-k_{real}x}$). Therefore, due to the evanescent nature of the wave, there will be no spatial propagation, and consequently the bandgap regions are identified, and these bands allow $T\to 0$. Note that a similar behavior would also be observed if ${\rho }_{ef}$ were negative. These observations contribute to the understanding of the mechanisms behind the sound transmission properties in the structure. Hence, the unconventional parameter exhibited by the mirror-symmetric metamaterial significantly expands the possibility of controlling the sound energy in this class of material.

4. Conclusions

A new type of metamaterial with a negative effective bulk modulus, based on the Fabry-Pérot disperser, was proposed, being very relevant for the control and manipulation of sound waves in various practical applications, such as acoustic insulation and filtering. Invariance to reflection was demonstrated due to the mirror-image symmetry of the structure. Using the scattering matrix and an experimental method based on the transfer matrix (one-load method) to obtain the effective properties, the behavior of the unusual effective bulk modulus in specific bands at the structure was established. In this regard, the relationship between the sound transmission behavior ($T\to 0$) and the negative effective bulk modulus ($K_{ef}<0$) was investigated. In addition, an alternative interpretation of the relationship between the behavior of $T$ and $K_{ef}<0$ was presented, providing a deeper understanding of the underlying physical mechanisms of the metamaterial. In summary, metamaterial with mirror-image symmetry properties has the ability to manipulate sound transmission, mainly due to its wide bandgaps. This wide bandgap refers to the evanescent nature of the wave, since there is no spatial propagation. Therefore, our findings may find applications in sound wave attenuation, such as sound barriers, noise mitigation, acoustic insulation, and in the control and manipulation of sound waves.

5. Didactic Extensions

By combining mathematical and experimental models, this study offers a perspective for analyzing topics such as transmission coefficient, reflection coefficient, sound transmission loss, and acoustic impedance – all addressed in undergraduate and graduate courses related to the behavior of mechanical waves (sound). To illustrate the application of these topics by teachers for physics education, for example, the use of the Vibra¹ computational simulation software as a teaching resource is recommended. Note that the Vibra is an open-source software developed in Python for modeling acoustic and structural physics problems using the Finite Element Method (FEM). In this context, Vibra can be used to illustrate to illustrate, through animations of the acoustic pressure field, the behavior of sound waves propagation (transmission coefficient) between two media with different acoustic impedance’s. Therefore, this approach can encourage greater student engagement with topics in mechanical waves and support the development of critical thinking in analyzing the behavior of sound wave behavior in different media or different types of structures.

Acknowledgements

This research was supported by the Graduate Program in Mechanical Engineering (POSMEC) of the Federal University of Santa Catarina, Brazil. And Fundação de Ensino e Engenharia de Santa Catarina (FEESC) granted through Programa Conexões para Inovar (count number 9075). The authors Gildean do N. Almeida and Erasmo F. Vergara are grateful for funding from the National Council for Scientific and Technological Development (CNPq).

Data Availability

Data supporting the findings of this study are available in the article.

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• 1
  Vibra software is the result of a project developed at the Laboratory of Vibrations and Acoustics by the Multidisciplinary Modeling and Optimization (MOPT) group and sponsored by Petrobras. For more details, please visit: https://github.com/MOPT-UFSC/VIBRA/tree/main.

Edited by

• Editor-in-Chief:
  Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  05 June 2026
• Date of issue
  2026

History

• Received
  15 Oct 2025
• Reviewed
  01 May 2026
• Accepted
  02 May 2026