## 2 Theory Model  
### Analytical Model  
The VAMLP proposed in this paper is shown in Figure 1a, and consists of an array of LCs, as shown in Figure 1b; each LC contains a ventilated region with side length \(a\) and four WUs. In the WU, the labyrinth is divided into channels C1 and C2 with lengths \(l_{1}\) and \(l_{2}\), respectively. Noise attenuation at different frequencies is achieved by adjusting \(l_{1}\) and \(l_{2}\) of the four WUs while ensuring ventilation performance. It is worth noting that, in practice, the geometric parameters of the LC can be adjusted according to the specific noise conditions and spatial conditions, including the side length \(a\) of the ventilated area, the lengths of the channels \(l_{1}\) and \(l_{2}\), the width of the channels \(w\), the number of turns of the channels, and the thickness of the labyrinth unit \(d\). The acoustic analytical model for the WU can be established by considering the two channels as connected in series. A backtracking procedure was employed to calculate \(Z_{2}\) and \(Z_{1}\) sequentially, where \(Z_{2}\) is the impedance at the beginning of channel C2 and \(Z_{1}\) is the impedance at the entrance of the side branch. Considering the thermo-viscous effect of the wave and the radiative impedance resulting from the discontinuities on the waveguide surface, the surface acoustic impedance \(Z_{2}\) is derived as  
\[Z_{2}=Z_{\tau 2}\frac{-Z_{\epsilon 2}\cot(k_{2}l_{2})\cot(k_{\tau}t_{e})+Z_{ \tau 2}}{-iZ_{\epsilon 2}\cot(k_{2}l_{2})-iZ_{\tau 2}\cot(k_{\tau}t_{e})} \tag{1}\]  
where \(Z_{\tau 2}\) is the radiative impedance according to the Johnson-Champoux-Allard (JCA) theory [26; 27], \(k_{2}\) is the number of waves in the channel C2, \(k_{\tau}\) is the number of radiative waves according to the JCA theory, \(t_{e}\) is the effective thickness of the rigid wall, and \(Z_{\epsilon 2}\) is the effective impedance taking into account the thermo-viscous effect. In the JCA model, the near-field radiative impedance is calculated as \(Z_{\tau}=\rho_{r}c_{r}\), where \(\rho_{r}\) and \(c_{r}\) are the density and sound velocity associated with the radiation, and \(c_{r}\) is calculated as [28]  
\[c_{r}=\sqrt{\frac{\kappa_{r}}{\rho_{r}}} \tag{2}\]  
where \(\kappa_{r}\) is the bulk modulus associated with radiation and \(\rho_{r}\) and \(\kappa_{r}\) are calculated as  
\[\rho_{r}=\frac{\alpha_{\text{inf}}\rho_{0}}{\phi}+\frac{\sigma}{i\omega}\sqrt {1+\frac{4i\alpha_{\text{inf}}^{2}\mu\mu\rho_{0}\omega}{\sigma^{2}\chi^{2}\phi ^{2}}} \tag{3}\]  
\[\kappa_{r}=\frac{\gamma P_{0}/\phi}{\gamma-(\gamma-1)\Bigg{(}1+\frac{8\mu}{i \rho_{0}\omega P_{\chi}\chi^{\prime 2}}\sqrt{1+\frac{i\rho_{0}\omega P_{\chi} \chi^{\prime 2}}{16\mu}}\Bigg{)}^{-1}} \tag{4}\]  
where, the tortuosity \(\alpha_{\text{inf}}=1\), \(\rho_{0}\) is the air density, the porosity \(\phi=1\), \(\sigma\) is the airflow resistivity [29; 30; 31], \(\omega\) is the angular frequency, \(\mu\) is the dynamic viscosity, and \(\chi\) and \(\chi^{\prime 2}\) are the viscous characteristic length and thermal characteristic length, respectively. Pr is Prandtl's number. Thermo-viscous losses are extremely important in generating the sound absorption coefficient. When thermo-viscous losses are considered in the model, the effective impedance in the channel is expressed as \(Z_{c}=\rho_{\epsilon}c_{e}\)[32]. \(k_{c}=\omega/c_{e}\), is the effective complex wave number in the channel, and \(\rho_{e}\) and \(c_{e}\) are the effective density and the effective speed of sound, respectively, which are calculated as  
\[\rho_{e}=\frac{k_{e}Z_{e}}{\omega},c_{e}=\frac{\omega}{k_{e}} \tag{5}\]  
where \(k_{e}\) and \(Z_{e}\) are given as  
Figure 1: Geometric model. (**a**) VAMLP and (**b**) LC. \[k_{e}^{2}=k_{0}\bigg{(}\frac{\gamma-(\gamma-1)\psi_{h}}{\psi_{\
u}}\bigg{)} \tag{6}\]  
\[Z_{e}^{2}=\frac{Z_{0}^{2}}{\psi_{\
u}(\gamma-(\gamma-1)\psi_{h})} \tag{7}\]  
where \(k_{0}\) is the wavevector, \(Z_{0}=\rho_{0}c_{0}\) is the characteristic impedance of air, and \(\gamma\) is the specific heat ratio.