## 2 Method
### Effect of the \"re\" Feature on the Performance of the Cell  
In order to provide a quantitative description of a cell with the \"re\" form, this paper introduces a new variable, denoted as e, which is referred to as the \"re\" factor. The \"re\" factor represents the ratio between the inward bending amount of the outer side of the straight-edge-form skeleton and its original length before bending. Figure 1b illustrates the outer side of the initial straight-edge-form skeleton using a red dotted line, while concaving each of its six outer sides inward according to the direction indicated by the red dotted line and arrow generates the \"re\"-form skeleton. By varying e, it has been observed that while there is only a gradual increase in unit cell equivalent density, there is a rapid decrease in the equivalent volume modulus, as depicted in Figure 3a. Prior to introducing e, an individual cell possesses an equivalent density value of 1; however, as e increases from 0 to 0.20, this equivalent density gradually rises to approximately 1.2. Simultaneously, there is also a reduction in the equivalent bulk modulus from 1 down to around 0.6. Herein we analyze a cell with a \"re\" factor using the structure located at e = 0.086, marked by the red arrow in Figure 3a, as an example case study. Calculation results indicate that this particular unit cell exhibits an equivalent density value of 1.07 and an equivalent compression wave velocity value of 0.78; furthermore, its relative equivalent velocity appears smaller than that exhibited by ss1 based on a comparison analysis conducted through band curves for both cases (Figure 2b). The vibration mode of point a in Figure 2a is compared with the band curve of the unit cell (Figure 2b) and the vibration mode of point B in Figure 3c. In Figure 3c, it can be observed that the cell undergoes contraction-expansion deformation while vibrating along the direction of \(\Gamma\)M, indicating that the oblique line represented by the black dashed line in Figure 2b corresponds to the compression wave vibration mode curve of the cell. To compare this cell's equivalent compression wave velocity with that of ss1, we have superimposed and aligned Figure 2b, with a red dotted line representing ss1's position adjustment. By comparing these two dotted lines, it becomes evident that this \"re\"-factor-containing cell exhibits a lower compressed wave velocity.  
The phenomenon of slow sound velocity caused by the \"re\" factor is then analyzed. As the \"re\" factor increases, the air portion within the cell gradually diminishes, leading to a gradual increase in the relative equivalent density of the cell. If the equivalent sound velocity of a unit cell remains constant with an increasing \"re\" factor, it would be expected that the equivalent volume modulus of a unit cell should also increase slowly. However, in reality, there is a rapid decrease in the equivalent volume modulus of a unit cell. This suggests that as the \"re\" factor increases, there is a gradual decrease in the equivalent compression wave velocity of a unit cell. Consequently, introducing a \"re\" factor will result in slower sound propagation within individual cells. For designing underwater metafluid cells, it becomes necessary to reduce skeleton thickness within each cell to achieve a reduction in their equivalent volume modulus and consequently induce slow sound effects within these cells. It can be observed that altering skeleton depression levels by incorporating a \"re\" factor reduces both the equivalent volume modulus and the sound velocity within individual cells. Therefore, when aiming for an equalized equivalent volume modulus among different types of cells, it can be inferred that skeletal structures in \"re\"-form cells should be thicker compared to those without the \"re\" design.