## 3 Results
### Regression Analysis
#### 3.2.1 Peak Sound Absorption Frequency  
Figure 12 shows the relationship between the areal density per grain layer and the first sound absorption peak frequency of the lightweight powders. The peak frequency varies with the layer thickness.  
Figure 11: Relationship between areal density per grain layer and bulk density of the powders.  
Figures 13 and 14 plot the relationships between the layer thickness and the first sound absorption peak frequency of group A and group B, respectively. Increasing the layer thickness decreased the first sound absorption peak frequency. Here, \(R^{2}\) denotes the determination coefficient of the regression curve. In Figures 13 and 14, the coefficients of determination exceed 0.8, indicating a satisfactory fit of the regression curves to the observed values. The least-squares method was applied to deriving regression equations from the plots in Figures 13 and 14. For group A, the change in the first sound absorption peak frequency with the layer thickness was generally close to \(-1\), which indicated an inversely proportional relationship that is similar to observations of the change in the peak frequency of general porous sound-absorbing materials. Thus, the first sound absorption peak frequency could be normalized by multiplying it by the layer thickness. In contrast, the exponent of the regression equation was not close to \(-1\) for group B, and the normalization by layer thickness was not as coherent as for group A.  
Figure 15 plots the relationship between the areal density per grain layer and the first sound absorption peak frequency normalized by the layer thickness of the group A powders. The regression curve from Figure 15 was used to derive an empirical equation for determining the first sound absorption peak frequency from the areal density per grain layer. Two quadratic curves were obtained for group A1 and had a normalized first sound absorption peak frequency of higher than 15,000, and group A2 had a normalized first sound absorption peak frequency of lower than 15,000. As illustrated in Figure 15, the determination coefficients \(R^{2}\) for groups A1 and A2 were approximately 0.7 and 0.8, respectively, suggesting an acceptable reproducibility of the regression curves to the actual values. This approach was not suitable for the group B powders because the normalized first sound absorption peak frequency was not well organized. In addition, no regression  
Figure 12: Relationship between the areal density per grain layer and first sound absorption peak frequency of the powders according to layer thickness.  
curves were attempted because the group B powders were not expected to be useful due to their small half-widths (Figure 7). The regression equation is applicable to areal densities per particle layer ranging from 0.000126 g/cm\({}^{2}\) to 0.000569 g/cm\({}^{2}\) for hollow powders, and from 0.000117 g/cm\({}^{2}\) to 0.00024 g/cm\({}^{2}\) for solid powders. Additional experiments with more samples are required to expand the applicability range of the regression equation.  
Figure 13: _Cont._  
Figure 14: _Cont._  
Figure 13: Relationship between the layer thickness and first sound absorption peak frequency for group A powders: (**a**) A1-1 Hollow glass beads, (**b**) A1-2 Hollow glass beads \(d>38\) μm, (**c**) A1-3 Hollow plastic beads \(d>38\) μm, (**d**) A1-4 Hollow urethane balloons, (**e**) A2-1 Granulated silica, (**f**) A2-2 Calcium silicate (Xonotltite powder (X)), (**g**) A2-3 Calcium silicate (Tobermorite powder (TK)), and (**h**) A2-4 Calcium silicate (Tobermorite powder (TJ)).