# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 2 Buffer-Rod Method
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Z** & **Vobserved** & **1 Rtheory** & **Robserved** \\ \hline BK7 glass Buffer & 14.6 & 0.87 & & \\ Water & 1.50 & 0.75 & 0.813 & 0.862 \\ PMMA & 3.20 & 0.65 & 0.640 & 0.747 \\ PVC & 3.26 & 0.69 & 0.634 & 0.793 \\ Mortar 31.3\% void & 3.56 & 0.60 & 0.607 & 0.690 \\ Mortar 26.2\% void & 4.08 & 0.60 & 0.562 & 0.690 \\ Mortar 22.8\% void & 5.10 & 0.59 & 0.481 & 0.678 \\ Mortar 10.7\% void & 7.42 & 0.40 & 0.325 & 0.460 \\ Pyrex glass & 12.3 & 0.32 & 0.084 & 0.362 \\ Fused silica & 12.9 & 0.19 & 0.060 & 0.218 \\ Soda-lime glass & 14.2 & 0.19 & 0.013 & 0.218 \\ Borosilicate glass & 14.2 & 0.24 & 0.013 & 0.276 \\ Granite & 14.3 & 0.26 & 0.009 & 0.299 \\ BK7 glass & 14.6 & 0.24 & 0.000 & 0.276 \\ Marble & 17.7 & 0.32 & 0.097 & 0.368 \\ Al 2024 & 17.8 & 0.27 & 0.100 & 0.310 \\ Ti-6-4 & 26.8 & 0.39 & 0.296 & 0.448 \\ Ti-Beta-III & 27.4 & 0.40 & 0.306 & 0.460 \\ Alumina & 32.1 & 0.45 & 0.376 & 0.517 \\ Brass 360 & 36.4 & 0.58 & 0.429 & 0.667 \\ Sapphire & 43.6 & 0.60 & 0.499 & 0.690 \\ High-C Steel & 46.2 & 0.68 & 0.521 & 0.782 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Acoustic Impedance and Reflection Coefficients. Figure 3: Calculated R\({}_{\rm th}\) magnitude using Equation (2) (shown by blue dots) and experimentally obtained R using BS and IF signal amplitude (red dots) for 21 different materials from water to hardened steel. Data is listed in Table 1.  
Figure 4: Reflection coefficient, R, vs. frequency, \(f\), with glass buffer using BS and IF signals (blue curve) and calculated R\({}_{\text{th}}\) using Equation (2) (red dotted line). (**a**) PMMA sample. Additionally, shown  
by orange and green curves are calculated R\({}_{\rm th}\) using Equation (4) with couplant thicknesses of 35 and 45 \(\upmu\)m. (**b**) Mortar sample without entraining agent, plus R\({}_{\rm th}\), with Equation (4) for couplant thicknesses of 10, 12 and 15 \(\upmu\)m (purple dashed, red, and green dashed curves). (**c**) Granite sample, plus R\({}_{\rm th}\) with Equation (4) for couplant thicknesses of 9 and 10 \(\upmu\)m (purple and green curves). (**d**) BK7 glass sample, plus R\({}_{\rm th}\) with Equation (4) for couplant thicknesses of 7 and 8 \(\upmu\)m (light-blue and purple curves). (**e**) Alumina sample, plus R\({}_{\rm th}\) with Equation (3) for couplant thickness of 8 \(\upmu\)m (orange curves). (**f**) Sapphire sample. Additionally, shown by orange curve is calculated R\({}_{\rm th}\) using Equation (4) with couplant thickness of 4 \(\upmu\)m.  
For samples with differing Z from that of the buffer, an equation for R(_f_) is given as,  
\[\text{R}(f)=\{((1-p)^{2}\cos^{2}(2\uppi df/v_{\rm c})+(n-m)^{2}\sin^{2}(2\uppi df /v_{\rm c}))/((1+p)^{2}\cos^{2}(2\uppi df/v_{\rm c})+(n+m)^{2}\sin^{2}(2\uppi df/ v_{\rm c}))\}^{\alpha,5}, \tag{4}\]  
where n = Z\({}_{\rm c}\)/Z\({}_{\rm s}\) and p = Z\({}_{\rm b}\)/Z\({}_{\rm s}\), along with m = Z\({}_{\rm b}\)/Z\({}_{\rm c}\), d = couplant thickness, and v\({}_{\rm c}\) = wave velocity of the couplant, respectively [34,49]. Two cases for PMMA and sapphire samples were calculated and their R values were plotted in Figure 2c,d for various couplant thicknesses of 2 to 45 and 1 to 15 \(\upmu\)m, respectively. For PMMA, the observed R(_f_) was between calculated R(_f_) values for couplant thicknesses of 35 and 45 \(\upmu\)m, as shown in Figure 4a. This indicated that d = 40 \(\upmu\)m provided a good match between the observed and calculated R(_f_). Figure 4f shows a good match exists with d = 4 \(\upmu\)m for the case of glass-sapphire interface. For four other cases of mortar, granite, glass and alumina, calculated R(_f_) curves of nearly matching d values were plotted with observed R(_f_) in Figure 4b.e. For mortar, granite, glass and alumina, matching apparent d-values were 12, 10, 7 and 8 \(\upmu\)m, while the corresponding d values for fused silica, PMMA and sapphire were 6, 40 and 4 \(\upmu\)m, respectively.