## 3 Validation and Analysis
For case 1, the complex frequency plane associated with the transmission coefficient is calculated as \(\log_{10}1\,\tau\,1^{2}\), as shown in Figure 5. It is worth noting that the data on the real axis are the \(-\)STL/20, and the distance between the zero and pole is positively correlated with the effective bandwidth. Considering the thermo-viscous losses, the imaginary axis coordinate of the zero point is not zero, as shown in Figure 5a, and the STL has a predictable maximum because of the thermo-viscous effect. When the effect of thermo-viscous losses is neglected, the complex plane is shown in Figure 5b. The zero point lies on the real axis, and the STL tends to be infinity. In this state, the system does not produce a sound absorption coefficient. The sound pressure distribution in the thermo-viscous zone at the frequency corresponding to the STL peak of case 1 is shown in Figure 6a. At the end of channel C1, the sound pressure is highest, and the sound waves resonate significantly. Thus, the length \(l_{1}\) of channel C1 has a greater effect on the peak frequency. The sound pressure level (SPL) distribution in the pressure acoustic zone is shown in Figure 6b, with the incoming port of the sound wave on the left side and the outgoing port on the right side, and the SPL shows significant attenuation after the sound wave passes through the entrance of the channel. The maximum sound pressure level occurs within the WU, and it is clear that there is a resonance. The radiated field at the entrance of the side branch suppresses the wave propagation and achieves noise blocking. As shown in Figure 2b, the four WUs are configured as an LC with different resonant frequencies achieved by adjusting the length \(l_{2}\) of channel C2, whose STL is shown in Figure 7. Without considering thermo-viscous losses, the maximum value of STL tends to infinity, which corresponds to Figure 5b when the minimum value of the transmission coefficient tends to 0. In practice, the thermo-viscous effect cannot be neglected, especially when the waveguide cell geometry is sufficiently small. Considering the thermo-viscous losses, the STL peak and resonance frequencies decrease, which is favorable for low-frequency noise suppression. At the same time, the valleys are increased. In this case, the  
Figure 5: Complex frequency plane. (**a**) With thermo-viscous losses and (**b**) without thermo-viscous losses. Figure 6: Acoustic performance at resonant frequency. (**a**) Sound pressure distribution and (**b**) SPL distribution. valleys are all greater than 10 dB. The comparison of the experimental results with the numerical calculation results provides an excellent agreement and verifies the accuracy of the numerical calculation model. To further investigate the mechanism of thermo-viscous loss action and the noise-reduction principle of the cell, the impedance at the entrance of the waveguide cell was extracted, as shown in Figure 8. In Figure 8a, the effect of thermo-viscous loss is neglected, and the resistance and reactance are close to 0 at a specific frequency, and the corresponding \(\ket{Z_{n}}\) tends to 0. A perfect impedance matching is realized in this state, and the wave is guided into the channel to resonate. When thermo-viscous loss is considered, the impedance is shown in Figure 8b, where the impedance is still close to 0 at specific frequencies. However, the resistance is significantly higher, caused by thermo-viscous waves. Correspondingly, the minimum value of \(\ket{Z_{n}}\) is greater than 0 while generating a sound absorption coefficient. As shown in Figure 9, the transmission coefficient \(\tau\), the absorption coefficient \(\alpha\), and the reflection coefficient r exhibit significant variations at specific frequencies. As a ventilated acoustic material, the transmission coefficient is a direct variable for evaluating the effectiveness of noise reduction, which in this case can be minimized to less than 0.1. In real noise environments, 0.5 is usually the expected value.