# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## 4 Integration Strategy for Achieving Absorption by Design
If we take the effective medium expression for \(B_{eff}=B_{0}/\phi\), then \(d_{\min}=2v_{0}/(\phi\Omega_{c}\tau)\). So far we have neglected the effects of the higher order resonances in the FP resonators, which would become more numerous as frequency increases. This is done for the clarity of presentation so that the main thesis of this work can be cleanly delineated without the complications of mathematical details, which are reproduced in Appendix B. To complete the description of the integration strategy, we now focus on the determination of the value for \(\phi\) through the condition of \(d=d_{\min}\) and Equation (1). For the total absorption example described above, \(\Omega_{\overline{n}}=\Omega_{c}\exp[2\phi\overline{n}]\) gives the optimal form for the distribution of the resonance frequencies. If we specialize to the case of a discrete number of resonators, e.g., by using 16 FP resonators as a single, 4 \(\times\) 4 acoustic metamaterial unit, then \(N=16\), and the length of the \(n\)th FP resonator is given by \(\ell_{n}=\pi v_{0}/(2\Omega_{n})\). If we can fold the longer FP resonators so as to make the metamaterial unit a compact cuboid, then the minimum thickness \(\overline{d}\) of the unit is determined by volume conservation, i.e.,  
\[\overline{d}=\frac{1}{16}\sum_{n=1}^{16}\ell_{n}=\frac{\pi v_{0}}{32\Omega_{c }}\sum_{n=1}^{16}\exp[-2\phi(n-1)/16]\cong\frac{\pi v_{0}}{4\phi\Omega_{c}}[1 -\exp(-2\phi)] \tag{12}\]  
In Equation (12), the final expression is noted to be exact when \(N\rightarrow\infty\). By setting \(\overline{d}=d_{\min}\), one obtains an explicit expression for \(\phi\):  
\[\phi=-\frac{1}{2}\log(1-8/\pi^{2})=0.832 \tag{13}\]  
It is noted that since \(d_{\min}\) is calculated by using the target absorption spectrum, hence self-consistency is achieved, simultaneous with attaining the minimum sample thickness. It is clear that \(\phi\) sets the channel length for each FP resonator. It also plays a role in the area fraction of the reflecting surface. One might ask what happens if \(\phi\) deviates from its optimal value. For example, if one makes \(\phi\) smaller, it is clear from Equation (12) that \(\overline{d}\) can be smaller. However, the price to pay is that the resulting absorption spectrum would deviate from the target absorption spectrum. The most important consequence of neglecting the higher order FP resonances is in the determination of the \(\phi\) value, as well as in the form for the distribution of the resonance frequencies. If we use as the target absorption spectrum the one described above, then the consideration of all the higher order FP resonances (shown in Appendix B) would lead to a value of \(\phi=0.982\). Also, whereas  
Figure 1: **(a)** A plot of the normalized impedance as a function of frequency, calculated from Equation (11). Here, the imaginary part of the impedance, denoted by the blue line, is seen to decay quickly, as expected from the oscillatory nature of the integrant in Equation (8). The real part of the impedance approaches the target value in an exponential fashion. **(b)** The calculated absorption coefficient is seen to approach 1 in an exponential fashion. At frequencies close to the cutoff frequency, the effect of the imaginary part can be significant, but its influence decreases quickly with increasing frequency. in the above we have derived an expression \(\log\Omega_{\overline{n}}\propto\overline{n}\), the inclusion of higher order modes makes \(\log\Omega_{\overline{n}}\) super-linear in \(\overline{n}\). Here \(\Omega_{\overline{n}}\) denotes the lowest-order FP resonance, and hence it is inversely proportional to the FP channel length. Since \(\phi=0.982\) implies very thin walls separating the different FP channels, in actual implementation it is necessary to deviate from this optimal value. However, there is nothing to prevent us from setting the FP channel lengths by using the correct optimal value of \(\phi\) in conjunction with the corrected \(\Omega_{\overline{n}}\) distribution, which would guarantee maximum consistency.