# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## 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 \(\phi=0.982\) 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. Figure 2: The integration scheme can be schematically represented by the “circle of consistency”. 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  
\[\left(g_{n}^{\left(\mathbf{e}\right)}\right)^{-1}=g_{n}^{-1}-\omega^{2}\rho_{ 0}(\tau+i\beta/\omega)\Lambda \tag{14}\]  
where the last term on the right-hand side is the self-energy term, with \(g_{n}=\phi\tan\left[\omega\ell_{n}\sqrt{\rho_{0}/B_{0}}\right]/\left(\omega Z _{0}\right)\) denoting the bare Green function and \(g_{n}^{\left(\mathbf{e}\right)}\) denoting the renormalized effective Green function of the \(n\)th 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 \(\Lambda\) 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;  
Figure 3: Metamaterial unit and its features. **(a)** Schematics of the metamaterial unit consisting of 16 Fabry-Perot (FP) channels, arrayed in a 4 \(\times\) 4 square lattice. A photo image of a sample comprising 4 units arranged in a mirror-symmetric pattern is shown in the lower left panel.