# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 4 Results and Discussion
### Rocks and Single Crystals
The result showing the dominance of the linear frequency dependence validated the use of frequency-independent damping factor, \(\eta\) (or Q\({}^{-1}\)) for rocks and minerals. The constancy of Q\({}^{-1}\) (preferred term in geoscience) at \(f\) below 1 MHz has withfood the test of time since the early days [13; 95], but additional scattering terms are needed as shown in this study at higher frequencies. The attenuation of granite (Test 55) was within the range of metamorphic rocks (80 to 450 dB/m at 1 MHz), while that of marble (Test 56) was 40% lower. Since both showed the frequency-squared relation, marble's attenuation matched at 2 MHz, while that of granite moved higher. In the case of travertime (a terrestrial limestone), \(\alpha_{\rm 1M}\) of Test 53 was 15 times higher than the corresponding value for limestones (measured at \(f\) = 5.8 MHz) in [15; 16]. This appears to arise from higher porosity. A note of caution is needed here as some attenuation studies in geoscience skipped the correction needed for diffraction losses e.g., [94; 96]. This omission has been justified by making substitutional measurements in a standard sample of the same shape, typically using aluminum, assumed to have a high Q value of 150,000 [71; 72; 94]. However, no aluminum alloy possesses MHz-attenuation lower than fused silica [43] and the diffraction loss can be severe below a few MHz. For example, [PERSON] et al. [94] used a pair of 5 mm diameter transducers (presumed to have a 4 mm diameter piezoelectric element), causing 35 to 21 dB loss at 0.2 to 1 MHz at 76 mm distance. Since they used 38 mm diameter samples, sidewall reflections were dominant in a replication test. Unless only the direct signals were analyzed, their attenuation data contained incorrect propagation distances. In earlier studies on rocks [76; 77], attenuation results were fitted to \(\alpha^{\prime}\) vs. \(f\) relations, but observed data showed non-zero vertical intercepts. That is, arbitrary shifts in observed attenuation levels were applied. Thus, such data sets need further analysis. Additionally, in the geoscience field, there exists efforts to account for multiple sources of attenuation, such as cracks and porosity, e.g., [97]. However, the use of standard linear solids without considering physical mechanisms resulted in damping peaks at 0.1 to 1 MHz. No such peaking behaviors are known, and other approaches need to be explored. Finally, Table 14 listed the elastic constants of various mineral samples that were smaller than the size needed for attenuation measurements. This group includes two carbides, Fe\({}_{3}\)C\({}_{7}\) and Cr\({}_{3}\)C\({}_{7}\), produced by high-pressure syntheses at 1300 \({}^{\circ}\)C under 4 GPa ([PERSON], Nagoya University, Nagoya, Japan), actinolite, jadeite and nephrite samples ([PERSON], Gemological Institute of America, Santa Monica, CA, USA), Santa Monica slate (outcrop, Encino, CA, USA), Mono Lake obsidian (Lee Vining, CA, USA) and a hematite single crystal. Note that a pair of near-square faces of hematite, \(<\)100\(>\) in the pseudo-cubic notation, were used for the velocity determination. The obsidian sample showed a low Poisson's ratio even though its density and E value were similar to most silicate glasses. It has G = 37.7 GPa, which was 23% higher than that of the S-L glass (Test C43). This section covered a limited range of materials, intended to compliment available elasticity data in the literature, e.g., [15; 16]. While E and longitudinal velocity (v) values are usually available, G and transverse velocity (v\({}_{\rm t}\)) values are more limited. Some minerals, such as jadeite, have been studied extensively in connection to seismic anisotropy e.g., [98]. On the other hand, attenuation data is rarely available, especially in the transverse mode.