## 2 Electro-Acoustic Analogue Modeling
### Continuum Limit  
We now focus on the continuum limit of Equation (3), corresponding to \(n\to\infty\) and \(d\to 0\) (but with \(nd\) being finite); in such a case, the pressure becomes \(p_{n}(t)\to p(x,t)\), where \(x=nd\) is a continuous variable, and  
\[p_{n\pm 1}=p\pm d\frac{\partial p}{\partial x}+\frac{d^{2}}{2}\frac{\partial^{2} p}{\partial x^{2}}\pm\frac{d^{3}}{3!}\frac{\partial^{3}p}{\partial x^{3}}+ \frac{d^{4}}{4!}\frac{\partial^{4}p}{\partial x^{4}}+O(d^{5}), \tag{4}\]  
i.e., \(p_{n+1}-2p_{n}+p_{n-1}\approx d^{2}p_{xx}+d^{4}p_{xxxx}/12\), where subscripts denote partial derivatives. In order to understand more accurately the dynamics of the system, we keep terms up to order \(O(d^{4})\) coming from the periodicity of the side holes [14]. This way, we obtain the corresponding partial differential equation (PDE),  
\[\begin{array}{c}\left(L_{H}\frac{\partial}{\partial t}+R_{H} \right)\left(d^{2}\frac{\partial^{2}p}{\partial x^{2}}+\frac{d^{4}}{12}\frac{ \partial^{4}p}{\partial x^{4}}-L_{\omega}C_{\omega 0}\frac{\partial^{2}p}{ \partial t^{2}}\right.\\ \left.-R_{\omega}C_{\omega 0}\frac{\partial p}{\partial t}+\frac{1}{2}L_{ \omega}C_{\omega}^{\prime}\frac{\partial^{2}p^{2}}{\partial t^{2}}+\frac{1} {2}R_{\omega}C_{\omega}^{\prime}\frac{\partial p^{2}}{\partial t}\right)\\ \left.-\left(L_{\omega}\frac{\partial}{\partial t}+R_{\omega} \right)p=0.\end{array} \tag{5}\]  
It is also convenient to express our model in dimensionless form by introducing the normalized variables \(\tau\) and \(\chi\) and normalized pressure \(P\), which are defined as follows: \(\tau\) is time in units of \(\omega_{B}^{-1}\), where \(\omega_{B}=\pi c_{0}/d\) is the Bragg frequency; \(\chi\) is space in units of \(c_{0}/\omega_{B}\) and \(p=\epsilon P_{0}P\), where \(P_{0}=\rho_{0}c_{0}^{2}\) and \(0<\epsilon\ll 1\) is a formal dimensionless small parameter. Then, Equation (5) is reduced to the following dimensionless form:  
\[\begin{array}{c}\left(\partial_{\tau}+\gamma_{H}\right)\left[P_{\tau\tau}- P_{\chi\chi}-\zeta P_{\chi\chi\chi\chi}+\gamma_{\omega}P_{\tau}-\epsilon\beta_{0}(P ^{2})_{\tau\tau}\right.\\ \left.-\epsilon\beta_{0}\gamma_{\omega}(P^{2})_{\tau}\right]+m^{2}P_{\tau}+m^ {2}\gamma_{\omega}P=0,\end{array} \tag{6}\]  
where,  
\[\begin{array}{c}m^{2}=\frac{S_{H}d}{\pi^{2}l_{H}S},\quad\gamma_{H}=\frac{R_ {H}}{\omega_{B}L_{H}},\\ \gamma_{\omega}=\frac{R_{\omega}S}{\pi\rho_{0}c_{0}},\quad\quad\quad\zeta= \frac{\pi^{2}}{12}.\end{array} \tag{7}\]  
It is interesting to identify various limiting cases of Equation (6). First, in the linear limit (\(\beta_{0}=0\), or \(p^{2}\ll 1\)), in the absence of side holes (\(m^{2}\to 0\), \(\gamma_{H}\to 0\)) and without considering viscothermal losses (\(\gamma_{\omega}\to 0\)) and higher order spatial derivatives, Equation (6) is reduced to the linear wave equation, \(P_{\tau\tau}-P_{\chi\chi}=0\). In the linear limit, in the presence of side holes, in the long wavelength approximation and without considering viscothermal losses (\(\gamma_{\omega}\to 0\)), radiation losses (\(\gamma_{H}\to 0\)), and higher order spatial derivatives (\(\zeta\to 0\)), Equation (6) takes the form of the linear Klein-Gordon equation [39; 40], \(P_{\tau\tau}-P_{\chi\chi}+m^{2}P=0\). Finally, in the nonlinear lossless regime, and in the absence of sides holes, without considering higher order spatial derivatives, Equation (6) is reduced to the well-known Westervelt equation, \(P_{\tau\tau}-P_{\chi\chi}-\epsilon\beta_{0}\left(P^{2}\right)_{\tau\tau}=0\), which is a common nonlinear model describing 1D acoustic wave propagation [10].