## 4 Results and Discussion
### Attenuation Behavior of Ceramics and Rocks  
Ceramics and rocks were examined for their attenuation characteristics. The results are summarized in Table 8. A broad range of materials, including both engineered ceramics and natural rocks, were included. Attenuation spectra were mostly linearly dependent, eight of which showed the Rayleigh scattering term. Two cases showed the Datta-Kinra scattering. Six tests showed very low attenuation, while five samples had high attenuation of over 1000 dB/m at 1 MHz (porous mortar, brick, emery disc, graphite, and granite). The first three samples (tests C1-C3) were mortar blocks made of rapid setting cement, known as belitic calcium sulfoaluminate cement (BCSA). One was a control block that did not contain extra voids. Besides sand and water, two of them contained 0.05% and 0.1% of an air-entraining agent, Hostapur, which produced 13.5% and 22.7% voids. Increased void contents resulted in higher attenuation. Three attenuation spectra are shown in Figure 20. Red, blue, and green curves correspond to tests C1 to C3, with increasing void contents. These three curves can be modeled well by  
\[\alpha\left(\text{dB/m}\right)=252\text{ f}+6.20\text{ f}^{4}, \left(\text{R}^{2}=0.983\right) \tag{26a}\] \[\alpha\left(\text{dB/m}\right)=394\text{ f}+304\text{ f}^{4}, \left(\text{R}^{2}=0.997\right)\] (26b) \[\alpha\left(\text{dB/m}\right)=235\text{ f}+1273\text{ f}^{2}, \left(\text{R}^{2}=0.995\right) \tag{26c}\]  
The model curves are shown by dashed curves of the same color, except blue curves overlap completely. When the void content is below 15%-20%, the Rayleigh scattering term accounts for the observed deviation from the linear spectrum. When the void content reaches 22.7%, the Datta-Kinra relation for multiple scattering becomes the appropriate model equation. Fitting was good for all three spectra, with R\({}^{2}\) values ranging from 0.983 to 0.997. The transition from the Rayleigh to Datta-Kinra relation was unexpected, but it was also observed for cast iron attenuation; that is, the Mason-McSkimin relation was applicable for ductile iron with nodular graphite particles and the Datta-Kinra relation was applicable for multiple scattering from interconnected flaky graphite in gray iron. The transition observed here should be of interest for evaluating concrete mixes containing gravel. Figure 20: Attenuation spectra of mortar samples without and with Hostapur foaming agent. Red: control block, with Mason–McSkimin fit (dashed curve); blue: 13.5% void content; green: 22.7% void with Datta–Kinra fit (dashed curve). The attenuation of mortar samples was reported by [PERSON] et al. [26; 110] as a linear spectrum, with a C\({}_{\rm d}\) value of 141 dB/m/MHz above 1 MHz, which is 60% of the test C1 result. Other studies showed varying ranges of attenuation coefficients at 1 MHz of 230-400 dB/m [111] or 522-1500 dB/m [112]. Testing conditions vary and direct comparisons are difficult for this material. Damping of concrete at low frequencies has always been an important topic [113]. A recent report [114] attempted to increase damping by adding polymers and various aggregates, and also provided a baseline damping factor of 0.038 for a mortar sample. This result is comparable to test C1. Considering the variations in the cement that was used, these data indicate our test results to be in the same range as other cementitious materials. The next nine tests were on various ceramics and glass; that is, tungsten carbide (WC) cermet (probably with cobalt-binder of unknown source), lead-zirconate-titanate (PZT-5A) transducer element (Valpey-Fisher, Hopkinton, MA, USA), silicon nitride (Si\({}_{3}\)N\({}_{4}\)), Si\({}_{3}\)N\({}_{4}\) + 10% SiC, barium titanate (BaTiO\({}_{3}\), from a high voltage capacitor), graphite (two kinds), and BK7 glass samples (two cuboids, Idea for Life, Guangdong, China). Five of these samples (tests C4-C8) showed linear spectra, with C\({}_{\rm d}\) values ranging from 42 to 490 dB/m/MHz.