An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design

Abstract

As much of metamaterials’ properties originate from resonances, the novel characteristics displayed by acoustic metamaterials are a narrow bandwidth and high dispersive in nature. However, for practical applications, broadband is often a necessity. Furthermore, it would even be better if acoustic metamaterials can display tunable bandwidth characteristics, e.g., with an absorption spectrum that is tailored to fit the noise spectrum. In this article we present a designed integration strategy for acoustic metamaterials that not only overcomes the narrow-band Achilles’ heel for acoustic absorption but also achieves such effect with the minimum sample thickness as dictated by the law of nature. The three elements of the design strategy comprise: (a) the causality constraint, (b) the determination of resonant mode density in accordance with the input target impedance, and (c) the accounting of absorption by evanescent waves. Here, the causality constraint relates the absorption spectrum to a minimum sample thickness, derived from the causal nature of the acoustic response. We have successfully implemented the design strategy by realizing three structures of which one acoustic metamaterial structure, comprising 16 Fabry-Perot resonators, is shown to exhibit near-perfect flat absorption spectrum starting at 400 Hz. The sample has a thickness of 10.86 cm, whereas the minimum thickness as dictated by the causality constraint is 10.55 cm in this particular case. A second structure demonstrates the flexible tunability of the design strategy by opening a reflection notch in the absorption spectrum, extending from 600 to 1000 Hz, with a sample thickness that is only 3 mm above the causality minimum. We compare the designed absorption structure with conventional absorption materials/structures, such as the acoustic sponge and micro-perforated plate, with equal thicknesses as the metamaterial structure. In both cases the designed metamaterial structure displays superior absorption performance in the target frequency range.

1. Introduction

Even in the 21st century, acoustic noise can still be a pernicious problem. An obvious question facing the scientific and engineering community is: How can we do better to remediate this problem? While doing by trial and error and learning by experience clearly constitute a path forward, occasionally natural laws can offer us the ultimate limit on how well one can achieve. It is the purpose of this article to show that acoustic absorption is one such case, and knowing the limit represents a tremendous advantage since one can work backwards in designing strategies to achieve such limit. The relevant limit in the case of acoustic absorption is that imposed by the causality principle, which, combined with the resonant mode density determination based on the target absorption spectrum for a structure backed by a reflecting substrate, is shown to yield a self-consistent strategy for achieving absorption by design. By considering the absorption through evanescent waves, we also address the practical issue of using a finite number of resonances to best achieve the target absorption spectrum.

For absorption at least, our design approach presented in this article has overcome the Achilles’ heel of narrow frequency bandwidth and dispersive character [1,2] for acoustic metamaterials.

Acoustic response of materials and/or structures must obey the causality principle, which states that the response at any moment of time can only depend on what occurred prior to that moment. In other words, the future cannot affect what happens now. Translated into mathematical language, this simple and intuitive statement can have deep consequences. In 1926–1927, Hans Kramers and Ralph Kronig independently derived the well-known Kramers-Kronig relationship between the real and imaginary parts of the dielectric function for electromagnetic waves, based on the causality principle [3]. Another, much less-known consequence of the causality principle is that for any electromagnetic absorption spectrum, there is a minimum sample thickness [4,5]. Adapted to acoustics, we have derived the following relationship for a flat structure sitting on a reflecting substrate [6]:

$$d\ge \frac{1}{4{\pi }^2}\frac{B_{\mathrm{eff}}}{B_0}\left|{\displaystyle {\int }_0^{\infty }\ln }[1-A(\lambda )]d\lambda \right|=d_{\min }.$$

(1)

Here, $\lambda \left(=2\pi v_0/\omega \right)$ denotes the sound wavelength in air, with $v_0=340\mathrm{m}/\mathrm{s}$ being the speed of air-borne sound, $\omega$ the angular frequency, $A(\lambda )$ the absorption spectrum, $B_{\mathrm{eff}}$ the effective bulk modulus of the sound absorbing structure in the static limit, and $B_0$ the bulk modulus of air. For the sake of making the present article self-contained, we give a detailed derivation of Equation (1) in Appendix A that first appeared in the Supplementary Information section of ref. [6]. Equation (1) tells us that the acoustic absorption spectrum cannot be independent from the sample thickness. A somewhat interesting aspect of Equation (1) is that only the bulk modulus appears in the causality constraint (in the electromagnetic case only the magnetic permeability appears in the causality constraint), whereas it is well known in acoustics that there are two material parameters: bulk modulus and mass density. This is attributed to the fact that in the derivation, it is necessary to take the zero-frequency limit. Hence, no inertial effect can be present. Some immediate implications of Equation (1) are that there cannot be 100% absorption over a finite frequency range by a finite-thickness structure, and that closer the absorption gets to 100%, the thicker the sample must be. Also, for the same frequency bandwidth, a lower central frequency implies a thicker minimum sample thickness, and vice versa.

It should be mentioned that there have been numerous studies to broaden the frequency range of absorption for acoustic metamaterials, some of which were quite successful [7,8,9,10]. The present approach differs by delineating the ultimate limit as dictated by natural law, and using such constraint as part of the design strategy to achieve tunable absorption by design.

2. Turning the Causality Constraint into a Design Tool

By setting $d=d_{\min }$ in Equation (1), we can interpret the resulting equation as a design tool. That is, one can use the equation to find the solution to the following questions. For a given sample thickness d and a fixed frequency (wavelength) range, what is the best absorption achievable? From the causality constraint, it is clear that for a given sample thickness, there is only a limited amount of absorption resources. Therefore, in this context the real question is: How to allocate these limited resources, i.e., how to define “best”? If the purpose is noise remediation, then perhaps the best target absorption spectrum is that which matches the noise spectrum. Alternatively, for a fixed d and a desired absorption level, what is the broadest possible frequency range possible? But perhaps the most-often encountered scenario would be: For a target absorption spectrum $A(\lambda )$, what would be the minimum sample thickness achievable? Below we show examples for what can be achieved from this perspective, but first it is necessary to consider the requirement of impedance matching for a sample, which is the complementary consideration to the causality constraint in our strategy to design a structure with a tunable absorption spectrum.

We define any sound absorption sample to be “optimal” if it attains $d=d_{\min }$.

3. Impedance, Absorption and Green Function

At normal incidence, reflection coefficient R from a flat sample is given by

$$R=\frac{Z-Z_0}{Z+Z_0}$$

(2)

where $Z=\rho v$ denotes the sample impedance, $\rho$ is the mass density, and v is the sound speed. Here, $Z_0$ is the impedance of air. If the sample sits on a reflecting hard surface, then there is no transmission, and absorption A can be directly calculated as $A=1-|R{|}^2$. In other words, there is a one-to-one correspondence between absorption and the impedance ratio $Z/Z_0$. In what follows, we consider how this impedance ratio can be tuned via acoustic resonances, i.e., through the use of metamaterials.

Impedance and Green Function

When the lateral sample dimension is much smaller than the wavelength so that the diffraction effect can be neglected, impedance may be alternatively defined as

$$Z=p/\dot{\delta }$$

(3)

where p denotes pressure modulation, and $\dot{\delta }$ is the displacement velocity. The Green function is defined as

$$G=\delta /p$$

(4)

Hence, for harmonic waves we have a simple relationship between the impedance and the Green function:

$$Z=\frac{i}{\omega G}$$

(5)

i.e., the two are essentially the inverse of each other, with the real and imaginary parts switched. The advantage of expressing the impedance in this manner is that the Green function can be explicitly written down through the eigenfunction expansion approach. For the acoustic metamaterials, the eigenfunctions are the resonance modes, which must have the Lorentzian form. To illustrate how the resonance can tune the impedance, let us consider the case of a system having two discrete resonances with resonance frequencies ${\mathsf{\Omega }}_1<{\mathsf{\Omega }}_2$:

$$Z(\omega )=\frac{i{\rho }_{0}d}{\omega }{\left(\frac{{\alpha }_1}{{\mathsf{\Omega }}_1^2-{\omega }^2}+\frac{{\alpha }_2}{{\mathsf{\Omega }}_2^2-{\omega }^2}\right)}^{-1}$$

(6)

where ${\rho }_0$ denotes the density of air, and ${\alpha }_{1(2)}$ is the oscillator strength, to be made explicit later. Here, we treat the dissipation to be negligibly small. It is clear that at $\omega ={\mathsf{\Omega }}_{1(2)}$ the impedance is zero, and there is always a frequency ${\mathsf{\Omega }}_1<\tilde{\omega }<{\mathsf{\Omega }}_2$ where the sum of the two terms vanishes in the parenthesis, so that the magnitude of the impedance diverges. This is denoted the antiresonace point [11]. Hence, with resonances one can tune the impedance of a resonant metamaterial structure at will by varying the frequency. Since there is a one-to-one correspondence between the impedance and absorption, it follows that one might be able to obtain absorption by design, by properly integrating together multiple resonances. Below we show precisely how this proper integration can be carried out in accordance with the desired target absorption spectrum, and how the causality equality can be used to fix a crucial design parameter so as to achieve consistency with the goal of attaining absorption by design with the minimum sample thickness.

4. Integration Strategy for Achieving Absorption by Design

In order to be concrete, in the following we will consider the use of Fabry-Perot (FP) resonators to realize our integration strategy.

A simple-minded approach to putting together multiple resonances for achieving, e.g., broad-band absorption, is by integrating within a single metamaterial unit multiple resonators with equally-spaced resonance frequencies. However, we show below that there is a much better strategy than using equally-spaced resonances. To see how such a strategy can be derived, let us consider an idealized structure with a continuum of resonances so that its impedance can be written as [12]

$$Z(\omega )=i\underset{\beta \to 0}{\lim }\frac{Z_{0}d}{\omega v_0}{\left[{\displaystyle {\int }_0^{\infty }\frac{\alpha (\mathsf{\Omega })D(\mathsf{\Omega })}{{\mathsf{\Omega }}^2-{\omega }^2-i\beta \omega }}d\mathsf{\Omega }\right]}^{-1}$$

(7)

where $D(\mathsf{\Omega })$ is the density of the resonant modes, and $\beta$ is the dissipative constant that controls the width of the absorption peak. It should be noted that letting $\beta \to 0$ does not imply the system to have zero dissipation when integrated over frequency, since a zero $\beta$ means that the imaginary part of the quantity in the square bracket becomes a delta function with zero width, but which has a non-zero integrated effect. It follows that from Equation (7) we have

$$Z(\omega )\simeq \frac{Z_{0}d}{\omega v_0}{\left[-i\mathrm{P}{\displaystyle {\int }_0^{\infty }\frac{\alpha (\mathsf{\Omega })D(\mathsf{\Omega })}{{\mathsf{\Omega }}^2-{\omega }^2}d\mathsf{\Omega }+}{\displaystyle {\int }_0^{\infty }\pi \alpha (\mathsf{\Omega })D(\mathsf{\Omega })\delta ({\omega }^2-{\mathsf{\Omega }}^2)d\mathsf{\Omega }}\right]}^{-1}$$

(8)

Here, the symbol $\mathrm{P}$ denotes taking the principal value. In Equation (8), the integrant in the imaginary part is oscillatory in nature. Hence, it can be small. If for the moment we ignore the imaginary part in the square bracket and focus only on the real part, then

$$\frac{Z(\omega )}{Z_0}=\frac{2d}{\pi v_0}{\left[\alpha (\omega )D(\omega )\right]}^{-1}$$

(9)

Here, the mode density can be written as $D(\mathsf{\Omega })={(d\mathsf{\Omega }/d\overline{n})}^{-1}$, where $\overline{n}$ is a continuous variable that indexes the resonance modes, defined as $\overline{n}=\underset{N\to \infty }{\lim }(n-1)/N$, with n being an integer ranging from 1 to N. By recognizing that ${\left[\alpha (\mathsf{\Omega })\right]}^{-1}=\pi v_0/4\varphi \mathsf{\Omega }d$ for the lowest-order FP resonances, from Equation (9) we obtain the following simple differential equation for determining the mode density from the target impedance:

$$\frac{d\mathsf{\Omega }}{d\overline{n}}=2\varphi \left[\frac{Z(\mathsf{\Omega })}{Z_0}\right]\mathsf{\Omega }$$

(10)

Here, $\varphi$ is the area fraction of the FP resonators’ openings in the total surface area of the metamaterial unit. It will be seen that the determination of $\varphi$ requires the causal relationship to achieve both minimum sample thickness as well as consistency with the target absorption spectrum. Two comments are in order. First, since we want to target a certain absorption spectrum, then $Z(\mathsf{\Omega })/Z_0$ can be directly obtained from Equation (2) and the fact that $A=1-|R{|}^2$. Second, the mode density obtained by solving Equation (10) cannot exactly reproduce the target absorption spectrum because in the derivation of Equation (10) we have ignored the imaginary part of the impedance. In order to illustrate this point, let us choose for the target absorption spectrum a flat, perfect absorption spectrum starting at a lower cutoff frequency ${\mathsf{\Omega }}_c$. In that case $Z(\mathsf{\Omega })/Z_0=1$ above ${\mathsf{\Omega }}_c$ and we have ${\mathsf{\Omega }}_{\overline{n}}={\mathsf{\Omega }}_{\mathrm{c}}\exp [2\varphi \overline{n}]$. Then, $\alpha (\mathsf{\Omega })D(\mathsf{\Omega })=2d/(\pi v_0)$ is a constant, and Equation (7) can be integrated to obtain:

$$\frac{Z(\omega )}{Z_0}=\frac{\pi }{\pi -2i{\tanh }^{-1}({\mathsf{\Omega }}_{\mathrm{c}}/\omega )}$$

(11)

This expression is graphically illustrated in Figure 1 below.

From Equation (11) one can easily evaluate $d_{\min }$ as defined by Equation (1). If we take the effective medium expression for $B_{eff}=B_0/\varphi$, then $d_{\min }=2v_0/(\varphi {\mathsf{\Omega }}_{\mathrm{c}}\pi )$.

So far we have neglected the effects of the higher order resonances in the FP resonators, which would become more numerous as frequency increases. This is done for the clarity of presentation so that the main thesis of this work can be cleanly delineated without the complications of mathematical details, which are reproduced in Appendix B.

To complete the description of the integration strategy, we now focus on the determination of the value for $\varphi$ through the condition of $d=d_{\min }$ and Equation (1). For the total absorption example described above, ${\mathsf{\Omega }}_{\overline{n}}={\mathsf{\Omega }}_{\mathrm{c}}\exp [2\varphi \overline{n}]$ gives the optimal form for the distribution of the resonance frequencies. If we specialize to the case of a discrete number of resonators, e.g., by using 16 FP resonators as a single, 4 × 4 acoustic metamaterial unit, then N = 16, and the length of the nth FP resonator is given by ${\ell }_n=\pi v_0/(2{\mathsf{\Omega }}_n)$. If we can fold the longer FP resonators so as to make the metamaterial unit a compact cuboid, then the minimum thickness $\overline{d}$ of the unit is determined by volume conservation, i.e.,

$$\overline{d}=\frac{1}{16}{\displaystyle \sum _{n=1}^{16}{\ell }_n}=\frac{\pi v_0}{32{\mathsf{\Omega }}_{\mathrm{c}}}{\displaystyle \sum _{n=1}^{16}\exp [-}2\varphi (n-1)/16]\cong \frac{\pi v_0}{4\varphi {\mathsf{\Omega }}_{\mathrm{c}}}[1-\exp (-2\varphi )]$$

(12)

In Equation (12), the final expression is noted to be exact when N $\to \infty$. By setting $\overline{d}=d_{\min }$, one obtains an explicit expression for $\varphi$:

$$\varphi =-\frac{1}{2}\log (1-8/{\pi }^2)=0.832$$

(13)

It is noted that since $d_{\min }$ is calculated by using the target absorption spectrum, hence self-consistency is achieved, simultaneous with attaining the minimum sample thickness. It is clear that $\varphi$ sets the channel length for each FP resonator. It also plays a role in the area fraction of the reflecting surface. One might ask what happens if $\varphi$ deviates from its optimal value. For example, if one makes $\varphi$ smaller, it is clear from Equation (12) that $\overline{d}$ can be smaller. However, the price to pay is that the resulting absorption spectrum would deviate from the target absorption spectrum.

The most important consequence of neglecting the higher order FP resonances is in the determination of the $\varphi$ value, as well as in the form for the distribution of the resonance frequencies. If we use as the target absorption spectrum the one described above, then the consideration of all the higher order FP resonances (shown in Appendix B) would lead to a value of $\varphi$ = 0.982. Also, whereas in the above we have derived an expression $\log {\mathsf{\Omega }}_{\overline{n}}\propto \overline{n}$, the inclusion of higher order modes makes $\log {\mathsf{\Omega }}_{\overline{n}}$ super-linear in $\overline{n}$. Here ${\mathsf{\Omega }}_{\overline{n}}$ denotes the lowest-order FP resonance, and hence it is inversely proportional to the FP channel length. Since $\varphi$ = 0.982 implies very thin walls separating the different FP channels, in actual implementation it is necessary to deviate from this optimal value. However, there is nothing to prevent us from setting the FP channel lengths by using the correct optimal value of $\varphi$ in conjunction with the corrected ${\mathsf{\Omega }}_{\overline{n}}$ distribution, which would guarantee maximum consistency. The area fraction of the reflecting surface, however, would be larger in such case, which implies some degradation in the absorption performance from the optimal target.

The above scheme can be schematically illustrated below by the “circle of consistency” shown in Figure 2.

It should be noted that while we have used a particular target absorption spectrum as an example (flat, total absorption for frequencies greater than ${\mathsf{\Omega }}_c$), the design strategy presented above is applicable to any desired target absorption spectrum.

5. Adverse Effect of a Finite Number of Resonators and Its Remediation

In any actual implementation of the above design strategy, it is always the case that there can only be a discrete, finite number of resonators. In Figure 3a we show an illustration and a photo image of an acoustic metamaterial unit comprising 16 FP channels, designed in accordance with the integration strategy presented in the previous section and fabricated by 3D-printed polyactide. As we cannot fabricate the sample with the wall thickness as thin as that required by the $\varphi$ = 0.982 requirement, the final measured results are affected by the increased reflection surface due to the thicker walls. However, the effect is minimal. In Figure 3b, we plot the measured impedance as a function of frequency by blue circles. The solid line represents the analytical theory given below. Oscillations around $Z/Z_0$ are seen. This is the consequence of having only 16 resonators, which are not enough to insure a smooth, flat absorption spectrum. Since deviation from impedance matching implies imperfect absorption, we expect oscillations in the absorption spectrum as well.

In order to remedy the oscillations, we look into the underlying cause of a peak, indicated by the red arrow in Figure 3b. By using COMSOL to carry out full-waveform simulations, we obtained the picture shown in Figure 3c. It turns out that at the indicated peak frequency, the pressure modulation at a little distance away from the surface of the metamaterial unit is as shown in the right panel in Figure 3c. The most important feature is that there is lateral pressure modulation which averages to zero over the surface. Since the lateral size of the metamaterial unit is much less than the relevant wavelength range, such lateral pressure modulation feature implies lateral wavevector $k_{\parallel }$ values much larger than the sound wavevector in air, $k_0$. Since $k_{\parallel }^2+k_{\perp }^2=k_0^2$, at the indicated frequency the wave must decay exponentially away from the surface of the metamaterial unit, i.e., $k_{\perp }$ is purely imaginary. This is precisely the characteristic of the anti-resonance point. Moreover, as the right-hand side panel shows, lateral pressure modulations imply lateral oscillating pressure gradient. This intuitively tells us that there are interactions between the FP resonators. Hence, theoretically the individual Green function of the resonator should be renormalized by a self-energy term arising from the interactions, so that

$${\left(g_n^{\left(\mathrm{e}\right)}\right)}^{-1}=g_n^{-1}-{\omega }^2{\rho }_0(\tau +i\beta /\omega )\mathsf{\Lambda }$$

(14)

where the last term on the right-hand side is the self-energy term, with $g_n=\varphi \tan \left[\omega {\ell }_n\sqrt{{\rho }_0/B_0}\right]/(\omega Z_0)$ denoting the bare Green function and $g_n^{\left(\mathrm{e}\right)}$ denoting the renormalized effective Green function of the nth resonator. Here, we have used the expression for the effective mass density of a porous medium, with ${\rho }_0$ being the density of air, $\tau$ being a dimensionless parameter on the order of 1, to account for the tortuosity of the pores, and $\beta$ being the dissipative coefficient particular to the porous medium. The value of $\beta$ is fairly small for air, so we have neglected it in the expression for the bare Green function. The expression for $\mathsf{\Lambda }$ is derived in Appendix C. The blue theoretical curve in Figure 3b is calculated by using the following formula since the 16 FP resonators are integrated in parallel; hence, the inverse of the impedance (which is just the Green function apart from the factor $i/\omega$) must be additive:

$$Z=\frac{i}{\omega }{\left(\frac{1}{16}{\displaystyle \sum _{n=1}^{16}{g}_n^{(e)}}\right)}^{-1}$$

(15)

Very good agreements with the experiment are seen in Figure 3b and Figure 4, both shown as solid lines.

There is a reason why we use the effective mass density for a porous medium in Equation (14). Since there are lateral pressure gradient oscillations, in practical terms it can imply lateral displacement of air on the surface of the metamaterial unit at the anti-resonance point. Hence, a thin layer of acoustic sponge placed on top of the acoustic metamaterial unit can serve as a very effective absorber at the anti-resonance point. This expectation is indeed borne out, as shown by the absorption performance of the metamaterial unit in Figure 4, after a 3 mm thick layer of acoustic sponge is placed on top of the metamaterial unit. For this particular example, the sample thickness d = 10.86 cm, whereas $d_{\min }=10.55$ cm. Hence, the sample is only 3 mm over the minimum thickness as dictated by the causal constraint.

For the particular sample shown in Figure 4, one may ask how far the near-perfect absorption can be extended to the higher frequencies, beyond those shown in the figure, which is 3000 Hz as limited by the impedance tube measurement. In Appendix B we show analytically that owing to the plurality of the higher-order resonances of the FP resonators and their equally-spaced frequency characteristics, the absorption actually becomes better and better with increasing frequency and approaches 1 in an exponential fashion (see Equation (A17)). Such behavior is actually substantiated by numerical simulations up to 8000 Hz.

As an illustration of our integration strategy to attain the goal of absorption-by-design, in Figure 5 we show the performance of another sample with a reflection “notch” in its absorption spectrum. Again, theory and experimental results show excellent agreement. Here, the sample thickness d = 9.33 cm, whereas $d_{\min }=9$ cm.

It should be mentioned that owing to the subwavelength lateral dimension of our metamaterial units, the angular dependence of absorption is nearly flat up to 60 degrees. After that the absorption smoothly drops as the angle of incidence approaches 90 degrees. Such behavior is shown by simulation results presented in Appendix D.

6. Comparison with Conventional Acoustic Absorbers

It has to be emphasized that causal optimality can be attained in conventional acoustic absorbers such as micro-perforated panel (MPP) and acoustic sponge [13]. However, they lack freedom for adjusting the absorption spectrum to a dictated target. Therefore, within the same thickness, although the causal integral Equation (1) can give the same value, the proposed absorption-by-design strategy can concentrate the absorption resources into the frequency region of interest, e.g., the peak region in the noise spectrum, thereby achieving better noise remediation.

MPP is a well-known conventional resonance absorber, utilizing the acoustic resonance of the back cavity to enhance the dissipation provided by the micro-pores in the front panel. It can achieve relatively high absorption at low frequencies. In the original paper of Maa [14,15], an MPP comprising a 0.2 mm-thick aluminum plate perforated by 0.2 mm diameter circular holes with an area fraction of 2%, backed by a 6 cm cavity, can display an absorption spectrum as shown by the blue symbols (experiment) and blue line (theory) in Figure 6. The absorption peaks at 640 Hz and rapidly drop away at higher frequencies. In contrast, fiberglass is a commonly used porous material that dissipates sound by the frictions in its porous structures. Its absorption is thereby naturally broadband. A layer of 6-cm thick fiberglass (denoted by green symbols) is noted to have a flat absorption around 90% (10 dB) for frequencies higher than 800 Hz. Compared with the designed metamaterial absorber with the same thickness (5.63-cm metamaterial with 3 mm sponge in front), shown by the red circles (experiment) and line (theory), it is clear that within the target absorption frequency range of above 800 Hz, our designed metamaterial structure is 5 to 10 dB better than the fiberglass. This is achieved precisely by concentrating the absorption resources in the target frequency regime.

The above comparisons emphasize the point that under the constraint of causality, the absorption-by-design approach can provide an efficient way to utilize the limited absorption resources to achieve the optimal sound attenuation in frequencies of interest. This feature is particularly essential for solving noise problems in mechanical systems like large machines, transformers, and aircraft engines, etc., in which the noise is broadband but with featured spectra.

7. Conclusions

In this work we have presented a new approach, comprising the use of causality constraint and impedance matching to the target absorption spectrum, for the fabrication of acoustic metamaterial structures that can display for the first time the desired absorption by design tunability, without a theoretical upper frequency limit. The whole design concept can be summarized by a circle of consistency (Figure 2), and the implementation by using 3D-printed FP resonator structures has yielded measured results in excellent agreement with the theory prediction. A theoretical formalism for treating the evanescent waves is also presented, which not only can accurately predict the shifts in the resonance frequencies of the FP resonators when they are placed in close proximity to each other, but also reveals an effective way to remediate the negative absorption effects resulting from using a small, finite number of resonators. The whole approach amounts to a self-consistent algorithm, amenable to computerized sample fabrication starting with a desired target absorption spectrum.