## 2 Electro-Acoustic Analogue Modeling  
### Setup and Model  
We consider plane wave propagation in a structure composed of a waveguide of radius \(r\) periodically loaded with an array of side holes of radius \(r_{H}\) and length \(l_{H}\), and the distance between two consecutive cells is \(d\), as shown in Figure 1a. We choose to work in a frequency range well below the first cut-off frequency of the higher propagating modes in the waveguide, and therefore the problem is considered as 1D. In order to theoretically analyze the problem, we employ the electro-acoustic analogue modeling based on the transmission line (TL) approach. This allows us to derive a nonlinear, dispersive and dissipative discrete wave equation, which--in the continuum limit--can be treated analytically via the multiple scales perturbation method. The corresponding unit-cell circuit of the TL model is shown in Figure 1b, where the voltage \(v\) and the current \(i\) correspond to the acoustic pressure \(p\) and to the volume velocity \(u\) through the waveguide's cross-sectional area, respectively [12; 28]. The first part of the unit-cell circuit, corresponding to the waveguide, is modeled by the inductance \(L_{\omega}\), the resistance \(R_{\omega}\) and shunt capacitance \(C_{\omega}\). The linear part of the inductance and the capacitance are \(L_{\omega 0}=\rho_{0}d/S\) and \(C_{\omega 0}=Sd/(\rho_{0}c_{0}^{2})\), respectively, where \(S=\pi r^{2}\), \(\rho_{0}\) and \(c_{0}\) are the cross section of the waveguide, the density and the sound velocity of the fluid in the system, respectively. We consider weakly nonlinear wave propagation, where the celerity \(c_{NL}\approx c_{0}\left(1+\beta_{0}p/\rho_{0}c_{0}^{2}\right)\), with \(\beta_{0}\) being the nonlinear parameter, depends on the amplitude. Thus, we consider the capacitance \(C_{\omega}\) to be nonlinear, depending on the pressure \(p\), while the inductance \(L_{\omega}\) linear [12]. The pressure-dependentcapacitance [14] can be expressed as \(C_{\omega}=C_{\omega 0}-C_{\omega}^{{}^{\prime}}p_{n}\), where \(C_{\omega}^{{}^{\prime}}=2\beta_{0}C_{\omega 0}/(\rho_{0}c_{0}^{2})\). The resistance \(R_{\omega}=\text{Im}(kZ_{c})d\), corresponds to the viscothermal losses at the boundaries of the waveguide wall:  
\[k=\frac{\omega}{c_{0}}\left(1+\frac{1-j}{s}(1+(\gamma-1)/\sqrt{Pr})\right) \tag{1}\]  
is the wavenumber and  
\[Z_{c}=\frac{\rho_{0}c_{0}}{S}\left(1+\frac{1-j}{s}(1-(\gamma-1)/\sqrt{Pr})\right) \tag{2}\]  
is the characteristic impedance of the waveguide, with \(\gamma\) being specific heat ratio, \(Pr\) the Prandtl number, \(s=\sqrt{\omega\rho_{0}r^{2}/\eta}\), and \(\eta\) the shear viscosity. The second part of the unit-cell circuit describes the effects of the holes that are considered to be linear (assuming sufficiently smoothed edges of the holes) [34; 35]. The side holes could be modeled by a shunt \(LR\) circuit composed of the series combination of an inductance \(L_{H}\) and a resistance \(R_{H}\). We study the regime where the wavelength \(\lambda\) of the sound wave is much bigger than the geometric characteristic of the side holes, i.e., \(kr_{H}\ll 1\). The corresponding inductance could be given by \(L_{H}=\rho_{0}(l_{H}+\Delta l_{Hi}+\Delta l_{Ho})/S_{H}\), where \(\Delta l_{Hi}\) and \(\Delta l_{Ho}\) are length corrections due to the radiation inside the waveguide and to the outer environment, respectively [36; 37; 38] (see more details in Appendix A) and \(S_{H}\) is the area of the side holes. The resistance \(R_{H}=\rho_{0}c_{0}(kr_{H})^{2}/(2S_{H})=\rho_{0}\omega^{2}/(2\pi c_{0})\) corresponds to the radiation losses to the outer environment and the propagation losses due to viscous and thermal effects in the side holes [36]. Here, we should mention that we approximate the frequency-dependent viscothermal losses of the waveguide and the radiation losses due to the side holes by resistance, \(R_{\omega}\), and \(R_{H}\), respectively, with the corresponding constant value fixed at the characteristic frequency of the dark solitons (see below).