## 2 Method
### Cells That Combine the Concepts of \"mp\" and \"re\"
Similarly, as depicted in Figure 7b, with an increasing \"re\" factor, there is also a gradual rise in the relative equivalent density for each individual cell (from slightly below 3 to exactly 3) accompanied by a corresponding decrease in the relative equivalent sound velocity (from approximately 0.36 down to about 0.33) when compared to an impedance-matched slow cell operating at one-third speed. The band curves and structural forms of re2 and re3 are depicted in Figure 8b,d, and the illustrations are in the lower right corner. These cells exhibit distinct dual characteristics of \"mp\" and \"re\". The red dashed line representing the compression wave band reveals that both cells possess a broadband slow speed effect. From Figure 8b, it can be seen that re2 has a stable slow sound speed frequency band ranging from approximately 0 to 11 kHz, except for a \"bulge\" near 6000 Hz. From Figure 8d, it can be observed that re3 has a stable slow sound speed frequency band ranging from approximately 0 to 7.5 kHz,  
Figure 7: Two kinds of impedance-matching slow cells (re2 and re3) were obtained. **(a)** re2 acquisition process. Before the “re” factor is introduced, the relative equivalent density is slightly less than 2, and the relative equivalent sound velocity is 0.61. With the increase in the “re” factor, the equivalent density gradually increases, and the equivalent sound velocity gradually decreases. When it is 0.077 (red arrow), the equivalent density of the unit cell is 2, the relative equivalent sound velocity is 1/2, and the unit cell is 1/2 of the impedance-matching slow cell. **(b)** re3 acquisition process. Before the “re” factor is introduced, the relative equivalent density is slightly less than 3, and the relative equivalent sound velocity is 0.36. With the increase in the “re” factor, the equivalent density gradually increases, and the equivalent sound velocity gradually decreases. When it is 0.080 (red arrow), the equivalent density of the unit cell is 3, the relative equivalent sound velocity is 1/3, and the unit cell is 1/3 of the impedance-matching slow cell. except for a \"bulge\" near 6000 Hz. The slow wave sound speed in the metafluid is less than that of the background medium. As the frequency increases, the effective wavelength (\(\lambda_{\rm eff}=c_{\it off}/f\)) in the metafluid gradually becomes smaller compared to the size of the unit cell, and it gradually fails to meet the requirement of \(\lambda_{\rm eff}>\) 5a (a is the side length of the unit cell), which causes the phononic crystal effect [44] of the unit cell to manifest, thereby destroying the broadband effectiveness of its slow sound speed. This is consistent with the sub-wavelength condition that the unit cell needs to satisfy in the effective medium theory [34,45]. Therefore, as the equivalent sound speed decreases, the effective frequency band of its slow sound speed gradually narrows. To emphasize the advantages of cells with a \"re\" feature in terms of skeleton thickness, straight-edge cells ss2 and ss3 are also designed. Their band curves and structural forms are shown in Figure 8a,c, along with the corresponding illustrations in the lower right corner. These four cells can be divided into two groups; ss2 and re2 represent cells with equivalent sound velocity but differing forms (\"ss\" vs. \"re\"), while ss3 and re3 belong to another group. The detailed structural parameters for these two groups of cells are presented in Table 2. Analyzing the table's variable h representing skeleton thickness reveals that compared to ss2, the skeleton thickness of re2 increases by 28.7%. Similarly, relative to ss3, the skeleton thickness of re3 experiences a growth rate of 17.5%. Furthermore, Table 1 demonstrates that compared to its counterpart ss1, the skeleton thickness of re1 increases by approximately 16%.