# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 4 Results and Discussion
### Other Ceramics, Mortars and Composites
Zeolite composites were designed for sound absorbing applications and contained 46-60% zeolite, 30% polyurethane and 10-24% isocyanate by weight ([PERSON], M., private communication). These are designated as Zeolite composite 1 to 3, with the density changing from 1.11 to 0.90 Mg/m\({}^{3}\). That is, porosity increased by 18% as density decreased. C-C composite was a brake disc of 424 mm outside diameter, 272 mm inside diameter and 10.4 mm thickness. Manufacturer's information is unavailable. It was reinforced using carbon fiber tows in three dimensions along the radial (R), circumferential (C) and thickness (S) directions. Comparing its density to the theoretical value of 2.269 Mg/m\({}^{3}\), the porosity is estimated to be 15.6%. Thin samples were prepared with surface normal in these three directions. For the R and C directions, two plates were aligned and glued. The SiC-SiC composite sample had nominally [\(\pm\)45\({}^{\circ}\)]\({}_{6}\) fiber reinforcement, but fiber alignments were poor. Its surfaces were ground using a diamond disc. Attenuation coefficients of longitudinal mode of mortar samples in dry and saturated conditions can be fitted to the linear plus frequency-cubed dependence. Transverse attenuation coefficients in both conditions followed the Mason-McSkimin relation or the linear plus fourth-power frequency dependence. Regardless of the frequency dependence type, attenuation coefficients were large and also increased with the void contents, except for the longitudinal attenuation in the saturated condition. Both C\({}_{d}\) and damping factors were especially high for dry samples with void additions (Tests 18 and 19) in the longitudinal mode. In comparison, transverse attenuation was less affected by the presence of the additional void. This behavior in the dry state is reflected in sharp decreases in C\({}_{dt}\)/C\({}_{d}\) and in \(\eta_{t}\)/\(\eta\). In the saturated condition, the values increased. As is well-known (1992; 1993), the longitudinal and transverse wave velocities of the mortar samples decreased as the void content increased. Relative velocity reduction due to void was comparable between longitudinal and transverse waves. These changes affected the elastic moduli, as tabulated in Table 10. Young's modulus decreased by 40 and 64% and the shear modulus similarly dropped. These changes were much higher than the density reductions. This effect on elastic moduli has been studied extensively as summarized in a review (2017). Early studies also established that increased void contents cause higher attenuation (or lower Q values) in rocks (2017; 2018). In order to examine the void effects, the attenuation coefficients of the mortar samples at 1 MHz, the \(\alpha_{\rm 1M}\) and \(\alpha_{\rm tIM}\) were obtained by using C\({}_{d}\) + C\({}_{2}\) and C\({}_{3}\) for the longitudinal mode and C\({}_{dt}\) + C\({}_{\rm Rt}\) for the transverse mode. Results are plotted with blue dots and + symbols for \(\alpha_{\rm tIM}\) and red dots and + symbols for the \(\alpha_{\rm tIM}\) against the void content on a log-log scale in Figure 11. Here, the + symbols are used for the saturated conditions. As discussed above, the sample without air-entrainment agent has a 10.7% void (as measured by the immersion method) and the samples of Test No. 18 and 19 have a V of 22.8% and 31.3%. While the data is limited, it appears that \(\alpha_{\rm tIM}\) in the dry condition increases with the second power of %void, V\({}^{2}\), and the \(\alpha_{\rm tIM}\) shows a linear relation, \(\alpha_{\rm tIM}\propto\) V. Blue and red lines represent these relations, while the blue dashed line is for the \(\alpha_{\rm tIM}\) of the saturated samples. A theoretical prediction of attenuation based on the scattering from spherical voids was given [83; 84] as  
\[\alpha=\text{Ar}^{3}f^{4}\text{ V}, \tag{15}\]  
where A is a constant inclusive of the elastic constants and wave velocities and \(r\) is the void radius.