## 2 Materials and Methods
### Samples Used for Measurements
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Powder** & **Powder Name** & **Grain Size** & **Bulk Density** & **Areal Density** & **Standard** & **Powsity** & **Plot on** \\
**Code** & **Powder Name** & **[mm]** & **[g/cm\({}^{3}\)]** & **Layer Ig/cm\({}^{2}\)]** & **Grain Size** & **Grain Size** & **Grain Size** & **Figures** \\ \hline \multirow{3}{*}{C-2} & Micro glass beads & & & & & & \\  & (EMB-20) & 0.009158 & 0.9885 & 9.05 \(\times\) 10\({}^{-4}\) & 4.08 \(\times\) 10\({}^{-3}\) & 0.620 & \(\blacktriangle\) \\  & (Potters-Ballotti Co., Ltd., Jbaraki, Japan) & & & & & & \\ \hline \multirow{3}{*}{C-3} & PTFE beads (3M Japan & 0.1786 & 0.907 & 1.62 \(\times\) 10\({}^{-2}\) & 7.16 \(\times\) 10\({}^{-2}\) & 0.578 & \(\blacktriangle\) \\  & Limited. Tokyo, Japan) & & & & & & \\ \hline \multirow{3}{*}{C-4} & Solid glass beads q0.05 & & & & & & \\  & (Toshin Riko Co., Ltd., Tokyo, Japan) & & & & & & \\ \hline \multirow{3}{*}{C-5} & PP beads (Sumitomo & & & & & \\  & Seika Chemicals & & & & & & \\  & Company, Limited. & & & & & & \\  & Tokyo, Japan) & & & & & & \\ \hline \multirow{3}{*}{C-6} & PMMA beads 38 \(<\) d \(\leq\) 53 & & & & & & \\  & \(\mu\)m (Seksiusi Kasei Co., Ltd., Tokyo, Japan) & & & & & & \\ \hline \multirow{3}{*}{C-7} & PMMA beads d \(\leq\) 38 \(\mu\)m & & & & & & \\  & (Seksiusi Kasei Co., Ltd., Tokyo, Japan) & & & & & & \\ \hline \multirow{3}{*}{C-8} & PMMA beads d \(>\) 53 \(\mu\)m & & & & & & \\  & (Seksiusi Kasei Co., Ltd., Tokyo, Japan) & & & & & & \\ \hline \end{tabular} \({}^{*}\) Apparent porosity could not be determined for powders for which the true density of the material was unknown and for hollow powders.  
\end{table} 
Table 1: _Cont._  
Figure 2: Powder layer with a depth of one grain. Therefore, the longitudinal elastic modulus of the powder layer can be calculated as follows:  
\[E=\rho\Big{(}4l_{\mathrm{p}}f_{\mathrm{peak}}\Big{)}^{2} \tag{7}\]  
where \(E\) is the longitudinal elastic modulus, \(f_{\mathrm{peak}}\) is the first peak frequency, \(\rho\) is the bulk density, and \(l_{\mathrm{p}}\) is the thickness of the powder layer. As shown in Figure 3, the powders were classified into three main groups according to their sound absorption curves: group A had gentle sound absorption curves, group B had sharp sound absorption curves, and group C demonstrated only slight vibration. The sharpness of the sound absorption curves for groups A and B were quantified by the half-width method, which requires measurement of the sound absorption coefficient. Figure 4 shows the experimental setup used to measure the sound absorption coefficient. Each sample was mounted in a Bruel & Kjaer Type 4206 two-microphone impedance measuring tube (Naerum, Denmark). Powders maintained at a constant temperature of 20 \({}^{\circ}\)C and humidity of 35% in a storage area were placed into sample holders (inner diameter: 29 mm, aluminum alloy) and tapped until no further volume change was observed. Preliminary experiments confirmed that the acoustic properties of the sample holder were identical to those of a genuine Bruel & Kjaer 4206 sample holder. The following procedure was used to measure the sound absorption coefficient. A sinusoidal sweep signal was output from the signal generator built into the fast Fourier transform (FFT) analyzer ([PERSON], Kanagawa, Japan), and a sound wave was output from the loudspeaker. The transfer function between the sound pressure signals measured by two microphones attached to the impedance tube was then measured by the FFT analyzer. The distance between the microphones was 20 mm. The measured transfer function was then used to calculate the normal incident sound absorption coefficient according to ISO 10534-2 (2002) [24]. Sixteen measurements were averaged to derive  
Figure 3: Examples of sound absorption curves for each powder group: gentle (A1-3 Hollow plastic beads d > 38 μm, \(l\) = 20 mm), sharp (B-2 Hollow glass beads (Sphericl® (110P8)), \(l\) = 20 mm), and slight vibration (C-4 Solid glass beads \(\varphi\) 0.05, \(l\) = 20 mm).