## 4 Conclusions  
The experiments were conducted using powders of various grain sizes and bulk densities with a low areal density per grain layer. Powders were classified according to the difference in sound absorption curves at the first peak frequency of the sound absorption  
Figure 17: Comparison of theoretical and experimental values using estimated primary peak frequency at 20 mm powder layer. (**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 (Xonotolite powder (X)). (**g**) A2-3 Calcium silicate (Tobermorite powder (TK)). (**h**) A2-4 Calcium silicate (Tobermorite powder (TJ)).  
coefficient. Changes in the half-width, bulk density, grain size, and areal density per grain layer were investigated. The following conclusions were obtained.  
A threshold value for the areal density per grain layer was identified where lightweight powders at 0.0006 g/cm\({}^{2}\) or less demonstrated useful sound absorption characteristics via longitudinal vibration. Powders with smooth (i.e., useful) and sharp (i.e., not useful) sound absorption curves could be further identified by the half-width value at 0.0974 \(<\)\(\log f_{2}\)\(-\)\(\log f_{1}\) < 0.119 decade. The bulk density can also be used to identify powders with useful sound absorption characteristics less than 0.0868 g/cm\({}^{3}\). The results demonstrated that powders with useful sound absorption characteristics can be identified based on their powder specifications. A regression analysis was performed to obtain empirical equations expressing the relationship between the areal density per grain layer and the first sound absorption peak frequency normalized by the layer thickness. Separate empirical equations were obtained for solid and hollow powders. The range of areal densities \(\sigma\) for which each equation is applicable is 0.000117 \(<\)\(\sigma\) \(<\) 0.00024 g/cm\({}^{2}\) for solid powders and 0.000126 \(<\)\(\sigma\) \(<\) 0.000569 g/cm\({}^{2}\) for hollow powders. The powder specifications and empirical equations can then be used to predict the sound absorption coefficients of lightweight powders via longitudinal vibration. The difference in the predicted and experimental peak sound absorption frequencies was \(<\)100 Hz for all group A powders.  
Conceptualization, S.S.; Software, K.J., K.I. and H.S.; Formal analysis, K.J., K.I. and Y.K.; Data curation, Y.K. and H.S.; Supervision, S.S.; Project administration, S.S. All authors have read and agreed to the published version of the manuscript.  
This work was supported by JSPS KAKENHI Grant Number 20K04359.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors would like to thank Japan Filite Co., Ltd., Sekisui Kasei Co., Ltd., Japan Insulation Co., Ltd., Potters-Ballotini Co., Ltd., Asahi Kasei Corporation, Sumitomo Seika Chemicals Co., Ltd., Taiheiyo Cement Corporation, 3M Japan Ltd. for providing the powders for this study.  
The authors declare no conflict of interest.