# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## 5 Adverse Effect of a Finite Number of Resonators and Its Remediation
**(b)** The surface impedance of the metamaterial unit plotted as functions of frequency. The circles are deduced from the measured reflection coefficient while the curves are the theory predictions. The arrow denotes the frequency at which the simulations were carried out on the sound pressure field, shown in the right panel below. **(c)** Right panel: The simulated sound pressure field at 0.22 mm above the front surface of the metamaterial unit at 612 Hz, which is the anti-resonance frequency between the FP resonances of the 5 th and 6 th channels (blue and red squares), shown in the left panel. The color is indicative of the pressure amplitude normalized to that of the incident wave. At anti-resonance, there is no coupling to the propagating modes; only an evanescent mode exists as explained in the text. hence, the inverse of the impedance (which is just the Green function apart from the factor \(i/\omega\)) must be additive:  
\[Z=\frac{i}{\omega}\left(\frac{1}{16}\sum_{n=1}^{16}g_{n}^{(e)}\right)^{-1} \tag{15}\]  
Very good agreements with the experiment are seen in Figures 3b and 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 \(d_{\text{min}}=10.55\) 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 \(d_{\text{min}}=9\) cm. Figure 4: Absorption performance of the metamaterial unit with a thin layer of acoustic sponge placed on top. Red circles are the data points measured by the impedance tube, and the red solid line is the calculated absorption by using the impedance evaluated from Equation (15). Here, the value of \(\beta=1.4\). The dashed line indicates the designed cutoff frequency. Excellent agreement between theory and experiment is seen. The absorption below the cutoff frequency is largely due to the acoustic sponge. Figure 5: A designed absorption spectrum as indicated by the dashed line. The performance of the actual implementation with 16 FP resonators, with 3 mm of acoustic sponge on top, is shown by red circles. The red solid line is the theory prediction. It agrees very well with the experimental data as measured by the impedance tube. 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.