## 2 Electro-Acoustic Analogue Modeling
### Linear Limit  
We now consider the linear limit of Equation (6) in order to obtain the dispersion relation of the system. Assuming plane waves of the form \(P\propto\exp[i(k\chi-\omega\tau)]\), we obtain the following complex dispersion relation connecting the wavenumber \(k\) and frequency \(\omega\),  
\[ \begin{split}& D(\omega,k)=-i\omega(-\omega^{2}+k^{2}-\zeta k^{4}+m ^{2})-i\omega\gamma_{H}\gamma_{\omega}\\ &+\gamma_{H}(-\omega^{2}+k^{2}-\zeta k^{4})+\gamma_{\omega}(- \omega^{2}+m^{2})=0,\end{split}  \tag{8}\]  
where \(\gamma_{\omega}\) and \(\gamma_{H}\) account for the viscothermal losses in the waveguide and the losses (radiation losses and viscothermal losses) due to the side holes, respectively. In the absence of losses, Equation (8) is reduced to:  
\[D(\omega,k_{r})=-\omega^{2}+k_{r}^{2}-\zeta k_{r}^{4}+m^{2}=0, \tag{9}\]  
which is the familiar dispersion relation of the linear Klein-Gordon model [39; 40], with a higher-order spatial derivative term \(\zeta k^{4}\) accounting for the influence of the periodicity of the system to the dispersion relation. Although this term appears to lead to instabilities for large values of \(k\), in the long-wavelength limit where \(k\) is sufficiently small, such instabilities do not occur. For frequencies between \(0\leq\omega<m\), there is a band gap, and, for \(m<\omega<\omega_{B}\), there is a propagating band, with the dispersion curve \(\omega(k)\) having the form of hyperbola (see Figure 2). Considering that all quantities in the dispersion relation (Equations (8) and (9)) are dimensionless, it is also relevant to express them in physical units. The frequency \(\omega_{ph}\) and wavenumber \(k_{ph}\) in physical  
Figure 2: (color online) (**a**,**b**) show the real and imaginary parts of the complex dispersion relation respectively. The blue circles (green stars) show the results from the TL (TMM) approach from Equation (10) [Equation (11)]; the blue dashed (green continuous) lines show the lossless dispersion relation obtained from the TL (TMM) approach from the lossless limit of Equation (10) (Equation (11)); (**c**) the frequency dependent \(qk^{\prime\prime}\), the product of the nonlinearity and the dispersion coefficients of NLS equations. units are connected with their dimensionless counterparts through \(\omega=\omega_{ph}/\omega_{B}\) and \(k=k_{ph}c/\omega_{B}\). Then, we rewrite Equation (8) with physical units as follows:  
\[ \begin{split}(-i\frac{\omega_{ph}}{\omega_{B}}+\gamma_{H})(-\frac{ \omega_{ph}^{2}}{\omega_{B}^{2}}+\frac{k_{ph}^{2}c_{0}^{2}}{\omega_{B}^{2}}- \zeta\frac{k_{ph}^{4}c_{0}^{4}}{\omega_{B}^{4}}-i\frac{\omega_{ph}}{\omega_{B}} \gamma_{\omega})\\ -i\frac{\omega_{ph}}{\omega_{B}}m^{2}+m^{2}\gamma_{\omega}=0.\end{split}  \tag{10}\]  
The real and imaginary parts of Equation (10), shown in Figure 2, are found to be almost identical with the lossless case i.e., \(\gamma_{\omega}=0\) and \(\gamma_{H}=0\) (dashed blue lines). Thus, the medium is considered to be weakly lossy. In particular, below we neglect viscothermal losses in the waveguide (\(\gamma_{H}=0\)), which are small compared to radiation losses. Furthermore, we assume that the remaining losses are sufficiently small, such that \(\gamma_{H}=\epsilon^{2}\gamma_{H}^{\prime}\). On the other hand, the green stars in Figure 2 show the corresponding results for the lossy dispersion relation obtained by using the transfer matrix method (TMM) [20]:  
\[\cos(k_{ph}d)=\cos(kd)+i\frac{Z_{c}}{2Z_{H}}\sin(kd), \tag{11}\]  
where \(k\) and \(Z_{c}\) are given by Equations (1) and (2), respectively, and \(Z_{H}=i\omega_{ph}L_{H}+R_{H}\) is the input impedance of the side holes. For the lossless case, \(k\), \(Z_{c}\) and \(Z_{H}\) are respectively reduced to \(k=\omega/c_{0}\), \(Z_{c}=\rho_{0}c_{0}/S\) and \(Z_{H}=i\omega_{ph}L_{H}\).