# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 3 Experimental Procedures
These have alumina face (Zt = 38 Mrayl). Both transmitters were excited by short pulses (\(-\)195 V peak, 32 ns risetime, 85 ns decay time), covering 0.5 to 16 MHz. A variac was needed to keep the pulse peak voltage constant. Examples of D against frequency \(f\) were shown previously [44], but it is worthy of further examination. Figure 7a shows five D-\(f\) curves for x = 1, 5, 10, 20 and 50 mm with a = 3.2 mm and v = 3.4 mm/us. At x = 1 mm, D initially decreased with \(f\) at steep slopes from the far-field effects, but suddenly D decreased slowly at f > 0.1 MHz. This is due to the increase in the near-field distance with increasing\(f\) value. The transition to reduced D slope occurs when the near-field distance exceeds approximately one-fourth of the propagation distance, or x/4, i.e., (4a\({}^{2}\)-(v/\(f\))\({}^{2}\))/4v > x/4. With x larger than 5 mm, the transition became a gradual decrease of slope and the far-field effect becomes dominant at low MHz frequencies. Another aspect of this effect is shown in Figure 7b, which plots D vs. x at 0.5 MHz (blue dots) and 1 MHz (red+). The far-field dominance started at 5 and 10 mm, respectively, and exhibited linear x-dependence as expected for a spherically expanding wavefront. This has been taken for granted for spherical waves, but this linear dependence has not been recognized explicitly from Equation (6). It is difficult to anticipate this result when it includes two terms of Bessel function. Figure 7: (**a**) Diffraction loss, \(\mid\)D(\(f\)) \(\mid\) (dB), vs. frequency, \(f\) (MHz) for five values of distance, x (mm). Curves for x = 1, 5, 10, 20 and 50 mm are in blue, red, green, black and purple. (**b**) Diffraction loss, \(\mid\)D\(\mid\) (dB), vs. distance, x (mm) for \(f\) = 0.5 MHz (blue dot) and 1 MHz (red +). A linear fit is shown by dotted lines. Attenuation coefficients showed the following relationships with frequencies, \(f\). \[\alpha=\text{C}_{\text{d}}f+\text{C}_{2}f^{2}+\text{C}_{3}f^{3}+\text{C}_{\text{R }}f^{4}\qquad\quad\text{(longitudinal)} \tag{12}\]  
\[\alpha_{\text{t}}=\text{C}_{\text{dt}}f+\text{C}_{2\text{t}}f^{2}+\text{C}_{3 \text{t}}f^{3}+\text{C}_{\text{Rt}}f^{4}\qquad\text{(transverse)} \tag{13}\]  
In all cases, only one or two terms were non-zero. Values of C\({}_{2}\), C\({}_{2\text{t}}\), C\({}_{3}\) and C\({}_{3\text{t}}\) are listed in the columns of C\({}_{\text{R}}\) and C\({}_{\text{Rt}}\), followed by n = 2 or n = 3 in parentheses. Of possible combinations, the linear term only and linear plus _f\({}^{4}\)_ terms (or Mason-McSkimin relation) were most often observed. The well-known Rayleigh scattering produces the _f\({}^{4}\)_ term and has been used to analyze effects of grain boundaries and voids [50]. The _f\({}^{2}\)_ and _f\({}^{3}\)_ terms observed in fiber-reinforced composites and cast iron were discussed in [43; 44]. [PERSON] et al. [33] combined Rayleigh scattering and pore size distribution and showed the frequency-squared relationship. A recent modeling study [51] predicted a frequency-cubed relation, combining elastically anisotropic grain distributions with [PERSON]'s theory of diffuse wave field [52]. Since predicting these frequency dependencies in ceramics requires detailed information regarding their microstructures, calculations applicable to a ceramic sample are unavailable. For fiber-reinforced composites, theories based on scattering have been proposed by [PERSON] and [PERSON] [53] and [PERSON] [54], predicting the square- and cube-frequency dependencies. Materials used for this study ranged from hard and soft ceramics, common ceramics, ceramic-matrix composites, mortars, silicate glasses, rocks, minerals and crystals. The attenuation coefficients of some samples for this study were previously reported [43; 44], but many were tested again using thinner samples. For others, only longitudinal attenuation coefficients were measured earlier. When previous attenuation coefficients were confirmed, these are listed with the previous test numbers starting with C.