# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## Appendix B Inclusion of Higher Order FP Resonances in the Design Strategy  
In this section we give the derivation of the design algorithm that includes all the higher order FP resonances. For a FP channel with length \(\ell_{n}\), its surface impedance is defined at its mouth, \(z=0\), by \(Z=p(0)/v(0)\), with  
\[p(z)=\cos\biggl{[}\omega(z+\ell_{n})\sqrt{(1+i\beta/\omega)\rho_{0}/B_{0}} \biggr{]}\]  
\[v(z)=-i\sin\biggl{[}\omega(z+\ell_{n})\sqrt{(1+i\beta/\omega)\rho_{0}/B_{0}} \biggr{]}/Z_{0}\]  
For an array of \(N\) FP channels with various lengths facing the incident sound wave in parallel, their total impedance is given by  
\[ \begin{split} Z=iZ_{0}\biggl{\{}&\phi\sum\limits_{n =1}^{N}\tan\Bigl{[}\omega\ell_{n}\sqrt{(1+i\beta/\omega)\rho_{0}/B_{0}} \Bigr{]}\biggr{\}}^{-1}\\ &=i\frac{Z_{0}d}{\omega v_{0}}\biggl{[}\sum\limits_{n=1}^{N}\sum \limits_{m=1}^{\infty}\frac{\alpha_{n}}{(2m-1)^{2}\Omega_{n}^{2}-\omega^{2}-i \beta\omega}\biggr{]}^{-1}\end{split} \] (A11)  
where \(\phi\) is the structure's surface porosity (fraction of the total surface area occupied by the open mouths of the FP channels), \(\Omega_{n}=\pi v_{0}/(2\ell_{n})\) is the 1 st-order FP resonance of the \(n\)th FP resonator, the terms with \(m>1\) stand for higher order FP resonances, and oscillator strength \(\alpha_{n}=2d\phi/(\ell_{n}N)=4d\phi\Omega_{n}/(\pi v_{0}N)\). It is easy to see that Equation (A11) is equivalent to Equation (7) in the main text if we take only the terms with \(m=1\). In the ideal case, \(\ell_{n}\) is continuously distributed, i.e., \(\Omega_{n}\) is a continuous variable, Equation (A11) can be converted into an integral:  
\[ \begin{split} Z&\simeq i\underset{\beta\to 0}{\lim}\frac{Z_{ \text{ord}}}{\omega\omega_{0}}\bigg{[}\int_{0}^{\infty}\sum_{m=1}^{\infty} \frac{\alpha(\Omega)D(\Omega)}{(2m-1)^{2}\Omega^{2}-\omega^{2}-i\beta\omega}d \Omega\bigg{]}^{-1}\\ &=i\underset{\beta\to 0}{\lim}\frac{Z_{\text{ord}}}{\omega\omega_{0}} \bigg{[}\int_{0}^{\infty}\frac{\sum_{m=1}^{\infty}\alpha(\widetilde{\omega})D( \widetilde{\omega})/(2m-1)}{\widetilde{\omega}^{2}-\omega^{2}-i\omega\beta}d \widetilde{\omega}\bigg{]}^{-1}\end{split} \] (A12)  
where \(\widetilde{\omega}=(2m-1)\Omega\), and \(D(\Omega)\) is the modes density of the 1 st-order FP resonances. For \(\beta\to 0\), the real part of the integral in Equation (A12) contributes negligibly, owing to the oscillatory nature of the integrant. The imaginary part of \(\lim_{\beta\to 0}\left(\widetilde{\omega}^{2}-\omega^{2}-i\beta\omega\right)^{-1}\) can be accurately approximated by a delta function; hence, we have  
\[Z(\omega)\simeq\frac{Z_{\text{ord}}}{\pi\omega_{0}}\bigg{[}\int_{0}^{\infty} \sum_{m=1}^{\infty}\frac{\alpha(\widetilde{\omega})D(\widetilde{\omega})}{2m- 1}\delta(\omega^{2}-\widetilde{\omega}^{2})d\widetilde{\omega}\bigg{]}^{-1}= \frac{ZZ_{\text{ord}}}{\pi\pi_{0}}\bigg{[}\sum_{m=1}^{\infty}\frac{\alpha( \Omega)D(\Omega)}{2m-1}\bigg{|}_{\Omega=\omega/(2m-1)}\bigg{]}^{-1}\] (A13)  
If we omit the higher order FP resonances and consider only the term \(m=1\), then by recalling that \(D(\Omega)=\Delta n/\Delta\Omega\), we have \(\Delta\Omega/\Delta n=\pi v_{0}\alpha(\Omega)Z(\Omega)/(2Z_{0}d)\). Since \(\alpha(\Omega)=4d\phi\Omega/(N\pi v_{0})\), we have \(\Delta\Omega/\Delta n=(2\phi\Omega/N)[Z(\Omega)/Z_{0}]\). By letting \(\Delta n/N\to d\overline{n}\) in the limit of \(N\to\infty\), where \(\overline{n}=(n-1)/N\), we have thus derived Equation (10) in the main text. To include the higher order FP resonances, we recognize that the additional impedances that arise from the higher order resonances are in parallel to those arising from the 1 st order FP resonances. Since now we have to deal with multiple impedances even from a single FP resonator, we would like to denote that impedance related to the 1 st order FP resonance to be \(\widetilde{Z}(\Omega)\).