# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 4 Results and Discussion
### Rocks and Single Crystals  
Attenuation coefficients of longitudinal and transverse modes of rocks and single crystals were next determined. See Table 7, Tests 47 to 56. Most samples were evaluated previously for at least the longitudinal attenuation, but sample thicknesses were reduced and retested. Tektite sample (Test 54) was newly added. Results of Tests C38 to C50 are listed for comparison. The linear or Mason-McSkimin relations were observed for most, except granite and marble, which had the frequency-squared relations (Tests 55 and 56). \(\rm C_{d}\) and \(\rm C_{dt}\) values ranged from low to high, with ADP and Si single crystals (SX) at nil or low attenuation and traverting and granite showing high values. The majority exhibited moderate levels of attenuation. Table 10 listed the elastic constants based on the wave velocity data. [PERSON] and [PERSON] (2018) previously measured the elastic constants of tektite and the present values matched them within 2%. Pyrophyllite sample was hardened by heating it at 900 \({}^{\circ}\)C for 1 h. After the firing, \(\rm C_{d}\) was reduced by a third, but \(\rm C_{dt}\) was down by only 15%. The firing converts pyrophyllite to pyrophyllite dihydroxylate, modifying the Al-O bond configuration. This apparently caused changes in the attenuation behavior. Relatively high \(\rm C_{d}\) value of tektite is surprising as it is glassy, but its lower \(\rm C_{dt}\) value is of the same trend as silicate glasses, listed in Table 5.  
Traverting (Test 53) showed the highest \(\rm C_{d}\) and \(\rm C_{dt}\) values in this group. It has many visible cavities and its density indicates 11% porosity in the matrix in comparison to the density of calcium carbonate (2.71 Mg/m\({}^{3}\)). Structural defects other than void appear to add attenuation, although microstructural analysis could not be conducted in this study. Granite (Test 55) was retested using a sample with its thickness reduced by 81.7%. As the high frequency component was enhanced, the frequency-squared dependence emerged in the longitudinal mode, extending to 3.4 MHz. The previous spectrum (Test C33) can also be fitted to the same \(f^{2}\)-dependence to 1.3 MHz upon re-analysis, but its \(\rm C_{2}\) coefficient was higher by a factor of 2.2. That is, the longitudinal attenuation was reduced by about a half using the thinner sample. (Actually, the thin sample was cut from one side; i.e., it is normal to the thickness direction previously used. Thus, directional effect may be present.) In contrast, the linear part of its transverse attenuation was 1.22\(\times\) higher from the thickness reduction and the \(f^{4}\)-term emerged over 1 to 2 MHz segment. The \(\eta\) and \(\eta_{\rm t}\) values in this work were 0.0905 and 0.135. These are close to those reported recently for an Australian granite of 0.0707 and 0.0819 ([PERSON], 2018). Both \(\eta\) values are several times larger than those measured at MHz frequencies, which were cited in [PERSON]'s classical review ([PERSON], 2018). Sample thickness reduction affected the longitudinal attenuation of marble slightly (or 22% higher), but the transverse attenuation was nearly doubled in terms of \(\eta_{\rm t}\) value. The cause for the effects on the transverse mode is uncertain at present. Comprehensive collections of the elastic and anelastic properties of rocks can be found in [15; 16]. The wave velocities are tabulated for more than 30 minerals along with the densities and Poisson's ratios. Results of the present study generally matched with those listed. [PERSON] [15] plotted the longitudinal attenuation of seven types of minerals against frequency from 10 Hz to 10 MHz, using data from the literature. This attenuation was due to absorption, or the hysteretic component. Linear frequency dependence was found for metamorphic rocks, consolidated sediments, magnetic rocks, limestones and dry sands. Low consolidated or unconsolidated sediments showed \(f^{1.5}\)-dependence.