# An Ultra-Thin Composite Metasurface with Hybrid-Damping Modes for Broadband Sound Absorption
## 3 Compound HR Structure with Hybrid-Damping Modes  
### Structure of the Broadband Unit and Cell  
As depicted in Figure 1a, the broadband HR structure incorporates eight compound HR cells to expand the absorption band, enabling the acquisition of high-order peaks with the external dimensions unchanged [20]. For an absorption cell, its relative specific impedance can be expressed as \(z_{s}=Z_{s}/p_{0}c_{0}=x_{s}+iy_{s}\), where \(Z_{s}\) is the surface specific impedance, \(\rho_{0}c_{0}\) is the characteristic impedance of air, and \(x_{s}\), \(y_{s}\) are the relative specific resistance and relative specific reactance, respectively. With \(y_{s}=0\), the cell is in the resonance state and gains an absorption peak, and on this basis, the perfect absorption can be achieved with \(x_{s}=1\). This impedance is called matched-damping impedance, and it will turn into over-damping and under-damping impedance when \(x_{s}>1\) and \(x_{s}<1\), respectively, both of which can result in a reduction in the peaks. In the case of over-damping cells, the peaks are all equipped with over-damping impedance and can obtain imperfect absorption. Furthermore, in a hybrid-damping cell, the peaks simultaneously exhibit over-damping and matched-damping impedance. Figure 2a illustrates the sound absorption performances of the compound cells with various damping modes. The cell cavity lengths are \(l_{1}=26\) mm and \(l_{2}=28\) mm, and the cross-section area is \(a\times b=10\) mm \(\times\) 10 mm. For the matched-damping, over-damping and hybrid-damping cells, the hole diameters are \(d_{1}=4.8\) mm, \(3.3\) mm, \(4.5\) mm, and \(d_{2}=2.3\) mm, \(1.7\) mm, \(1.7\) mm, respectively. It can be observed that the results of the matched-damping cell achieve two nearly perfect absorption peaks at \(f_{1}=900\) Hz and \(f_{2}=2480\) Hz. According to the impedance-matched conditions, both peaks obtain impedances that are matched with that of air, i.e., the relative specific resistances \(x_{s}\approx 1\), as shown Figure 2b,c. Owing to the decrease in the hole diameters, the acoustic resistances of the two peaks are increased to  
Figure 1: (**a**) Basic unit of the metasurface composed of an HR structure, two porous cells and a porous layer. The unit has an incident area \(S_{0}=L\times W=68\) mm \(\times\) 44 mm, and the thickness of the porous layer and the HR structure are \(H_{1}=10\) mm and \(H_{2}=21\) mm, respectively. The numbers from 1 to 16 denote the cells with first-order peak frequency from lower to higher. The cells 17 and 18 are the porous cells. (**b**) The sound absorption coefficients of the metasurface and the melamine foam with the same thickness. \(x_{s}\approx 2\), resulting in a 10% decrement in the absorption coefficients. Furthermore, the peaks have shifted to \(f_{1}=745\) Hz and \(f_{2}=1990\) Hz attributed to the increased acoustic masses. In the hybrid-damping cell, the acoustic resistances for the two peaks are \(x_{s1}=2.02\) and \(x_{s2}=1.25\), so the first peak still maintains excellent low-frequency performance despite a minor deviation towards higher frequencies, whereas the second peak achieves almost 100% absorption. Figure 2d presents the complex frequency plane representation of the hybrid-damping cell. The reflection coefficient can be plotted on the complex frequency plane by replacing the frequency as \(f^{\prime}=f_{r}+j_{l}\), where \(f_{r}\) is the intrinsic frequency and \(f_{i}\) is the imaginary frequency. The reflection coefficient, therefore, exhibits a complex conjugate zero and pole in a lossless system. As the loss is incorporated into the system, the zero and pole will be shifted down along the imaginary axis. When the zero is located on the real frequency axis, the intrinsic losses perfectly balance the energy leakage, resulting in perfect absorption,  
Figure 2: (**a**) Sound absorption coefficients, (**b**) relative specific resistance and (**c**) relative specific reactance of the compound cell with different damping modes.