# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## Appendix B Inclusion of Higher Order FP Resonances in the Design Strategy
In that case  
\[\frac{d\Omega}{dn}=\frac{\pi v_{0}}{2d}\bigg{[}\frac{\widetilde{Z}(\Omega)}{Z_ {0}}\bigg{]}\alpha(\Omega)\] (A14)  
Substitution of Equation (A14) into Equation (A13) and separating out the term \(m=1\) from the \(m\)-summation, yields an equation for \(\widetilde{Z}(\Omega)\),  
\[\widetilde{Z}(\omega)^{-1}=Z(\omega)^{-1}-\sum_{m=2}^{\infty}\frac{\widetilde{ Z}(\Omega)^{-1}}{2m-1}\bigg{|}_{\Omega=\omega/(2m-1)}\] (A15)  
The value of \(\widetilde{Z}\) can be obtained from Equation (A15) through iterations, based on a given target impedance \(Z\). Simultaneously, Equation (A15) also expresses the fact that the target impedance at frequency \(\omega\) is now the consequence of impedance from the order FP resonance, plus the impedance from all the higher order FP resonances, added in parallel. For example, if the target \(Z=Z_{0}\) for \(\omega>\Omega_{\text{c}}\) and divergent for \(\omega<\Omega_{\text{c}}\), then the value of \(\widetilde{Z}\) can be determined in a piecewise fashion as follows. The piecewise fashion of the result is a natural consequence (upon iteration) of the step-function nature of the target impedance. The iteration results show that \(Z_{0}/\widetilde{Z}_{1}=1\) in the first frequency range \(\Omega\in[\Omega_{\text{c}},3\Omega_{\text{c}}]\), \(Z_{0}/\widetilde{Z}_{2}=2/3\) in the second frequency range \(\Omega\in[3\Omega_{\text{c}},5\Omega_{\text{c}}]\), \(Z_{0}/\widetilde{Z}_{3}=7/15\) in the third frequency range \(\Omega\in[5\Omega_{\text{c}},7\Omega_{\text{c}}]\), \(Z_{0}/\widetilde{Z}_{4}=34/105\) in the fourth frequency range \(\Omega\in[7\Omega_{\text{c}},9\Omega_{\text{c}}]\), \(Z_{0}/\widetilde{Z}_{5}=269/1155\) in the fifth frequency range \(\Omega\in[9\Omega_{\text{c}},11\Omega_{\text{c}}]\), etc. In each frequency interval \(i\), i.e., for \(\Omega\in[(2i-1)\Omega_{\text{c}},(2i+1)\Omega_{\text{c}}]\), the 1 st-order FP resonance frequency distribution can be determined by Equation (14). That is, with the initial condition \(\Omega=(2i-1)\Omega_{\text{c}}\) when the continuous variable \(\overline{n}=(n-1)/N=\overline{n}_{i}=N_{i}/N\), where \(N_{i}\) denotes the total number of 1 st-order FP resonances below \((2i-1)\Omega_{\text{c}}\), Equation (A14) gives \(\Omega_{n}=(2i-1)\Omega_{\text{c}}\exp[2\phi(\overline{n}-\overline{n}_{i}) \widetilde{Z}_{i}/Z_{0}]\). From such 1 st order FP resonance frequencies one can easily determine the required lengths of the FP resonators in the design. In Figure 11a we plot the natural logarithm of \(\Omega_{n}\) as a function of \((n-1)/N\). Here, the value of \(\phi\), needed for the evaluation of \(\Omega_{n}\), is taken to be the causally optimal value determined below. The function \(\ln\Omega_{n}\) versus \((n-1)/N\) is seen to be piecewise hyper-linear. By using this result, discretization of the resonators in the actual design can be easily determined by locating the frequencies on the vertical axis with the associated (equally-spaced) values of \((n-1)/N\) with \(N\) being the total number of FP channels one wants to use. For the broadband absorber presented in the main text with \(N=16\), these frequencies are explicitly indicated by the red dotted lines in Figure 11a. One important feature for the sequence \(Z_{0}/\bar{Z}_{i}\) is that it decays to zero very quickly (Figure 11b), i.e., the required 1 st-order FP modes density in the high frequency regime is very low. This fact is relevant to the high frequency absorption behavior for the broadband absorber presented in the main text.