# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## Appendix B Inclusion of Higher Order FP Resonances in the Design Strategy
That is, since \(\sum_{m=1}^{\infty}\alpha(\Omega)D(\Omega)/(2m-1)|_{\Omega=\omega/(2m-1)}=2d/( \pi v_{0})\) (this can be easily deduced from Equations (113)-(115)), Equation (112) can be integrated to yield  
\[\frac{Z}{Z_{0}}=\frac{\pi}{\pi-2i\text{tanh}^{-1}(\Omega_{\text{c}}/\omega)} \tag{116}\]  
The relevant reflection coefficient \(R=(Z-Z_{0})/(Z+Z_{0})\) is given by  
\[R=\frac{\text{tanh}^{-1}(\Omega_{\text{c}}/\omega)}{\pi-i\text{tanh}^{-1}( \Omega_{\text{c}}/\omega)} \tag{117}\]  
That is, at high frequencies the reflection is zero, i.e., the absorption coefficient must approach 1. Therefore, in the broadband absorber design one can use a relatively small number of FP channels, designed for the low frequencies by following the proposed recipe above, and high absorption in the high frequencies regime is guaranteed. In particular, this would ensure high absorption above 5000 Hz for the broadband absorber presented in the main text, where there are no measured data. So far, the parameter \(\phi\) remains un-determined. Below, we show that its value should not be arbitrary. Instead, it serves as the critical link between the designed mode density, the sample thickness, and the causal constraint. In the broadband absorber, the channel length of the FP resonator is given by  
\[\ell_{n}=\frac{\pi v_{0}}{2\Omega_{n}}=\pi v_{0}\frac{\exp[2\phi(\overline{n}_{ i}-\overline{n})\widetilde{Z}_{i}/Z_{0}]}{2(2i-1)\Omega_{\mathrm{c}}} \tag{10}\]  
provided its 1 st-order resonance is located in the frequency range \(\Omega\in[(2i-1)\Omega_{\mathrm{c}},(2i+1)\Omega_{\mathrm{c}}]\). Since the channel length can vary, we wish to know the minimum thickness of the sample by optimally folding the FP channels, without changing the overall area exposed to the incident wave. This minimum thickness \(\overline{d}\) can be obtained through volume conservation of the FP channels. Here, we evaluate \(\overline{d}\) by focusing on only the air channels of the FP resonators. Since the FP channels' cross sections occupy a fraction \(\phi\) of the surface area, \(\overline{d}\) is given by  
\[\overline{d}=\lim_{N\to\infty}\sum_{n=1}^{N}\frac{\ell_{n}}{N}=\pi v_{0}\lim_ {N\to\infty}\int_{\Omega_{\mathrm{c}}}^{\Omega_{\mathrm{c}}(\phi)}\frac{1}{ \Omega}\frac{dn}{d\Omega}d\Omega/(2N) \tag{11}\]  
where the upper limit of the integral, \(\Omega_{\mathrm{e}}(\phi)=\lim_{N\to\infty}\Omega(n=N,\phi)\), is determined by the total number of 1 st order mode number \(N\), which is equal to the FP channel number. The numerical evaluation of Equation (10), with \(N=16\), gives \(\overline{d}=[0.6395-859.74\exp(-12.82\phi)]v_{0}/(\phi\Omega_{\mathrm{c}})\). By requiring \(\overline{d}=d_{\mathrm{min}}=2v_{0}/(\phi\Omega_{\mathrm{c}}\pi)\) given in the main text, we obtain the causally optimal value \(\phi_{optimal}=0.982\), with the upper limit \(\Omega_{\mathrm{e}}=28.4\Omega_{\mathrm{c}}\) (indicated in Figure 10 by the blue dotted line). Since in experimental implementation the value of \(\phi\) is determined by the wall thickness in our design, such a high value of \(\phi_{optimal}\) is not realizable in practice. However, a lower value of the actual \(\phi\) is seen to only degrade the absorption somewhat, as long as the mode distribution (and hence the length of the channels) is designed in accordance with the ideal value \(\phi_{optimal}\). As another example, other than the broadband absorber presented in the main text, we have also considered a target absorption spectrum which starts with near-perfect absorption from 345 Hz and has a notch in the frequency interval [562 Hz, 995 Hz] where the absorption is close to zero. The target impedance is given by \(Z(\omega)=Z_{0}[2-A(\omega)+2\sqrt{1-A(\omega)}]/A(\omega)\). Based on this target impedance the impedance \(\widetilde{Z}(\omega)\) can be obtained from Equation (11) through iterations. Substitution of \(\widetilde{Z}\) into Equation (10) gives the designed resonance frequencies \(\Omega_{n}(N,\phi)\) as a function of total channel number \(N\) and the parameter \(\phi\).