# Dynamic Viscosity and Transverse Ultrasonic Attenuation of Engineering Materials
## 4 Discussion  
Ultrasonic attenuation coefficients for the transverse mode were measured and tabulated in Section 3 for 223 test samples. Their corresponding longitudinal attenuation coefficients were also given. The frequency dependence of the attenuation coefficients followed four patterns, linear, Mason-McSkimin (linear +\(f^{4}\)), linear +\(f^{2}\) (or\(f^{2}\)), and linear +\(f^{3}\) (or/\(f^{3}\)). The distributions of attenuation spectra among the four patterns are summarized in Table 8 in the first five columns. Values shown are transverse test counts for each pattern separated into five material groups. Moreover, given in the parentheses are longitudinal test counts from the previous study [13]. Total test counts of the tabulated were 222 (fused silica test was omitted due to non-detection) and 348 for the transverse and longitudinal modes. The distribution of spectral patterns of transverse attenuation were generally similar to that of longitudinal attenuation; 65% of the transverse tests exhibited the linear frequency dependence (vs. 67.5% for longitudinal). For the Mason-McSkimin relation with Rayleigh scattering contribution, these percentages were 28 and 22%, respectively. These distributions indicate that the linear damping component is dominant in the majority of the attenuation spectra in both modes of wave propagation. The percentages of two power-law distributions,\(f^{2}\) or\(f^{3}\), were reduced in the transverse mode. The next two columns give the range of the linear coefficient, \(\mathrm{C_{dt}}\). On the low side, \(\mathrm{C_{dt}}\) is zero when a power-law component describes the attenuation spectrum. For these cases, the next low value is listed (in parentheses). Metal groups showed lower maximum \(\mathrm{C_{dt}}\) values and polymers and FRP groups the highest, over 1800 dB/m/MHz. When attenuation coefficients of two modes of wave propagation are compared, most of the polymers and FRP groups showed higher \(\mathrm{C_{dt}}\) than \(\mathrm{C_{d}}\) values. In contrast, the ferrous and ceramics and rocks group gave more even distributions of \(\mathrm{C_{dt}}\)/\(\mathrm{C_{d}}\) ratios. These are shown in columns 8 to 10. \(\mathrm{C_{dt}}\)/\(\mathrm{C_{d}}\) ratios were separated into three: \(\mathrm{C_{dt}}\)/\(\mathrm{C_{d}}\geq 1.5\), \(1.5>\mathrm{C_{dt}}\)/\(\mathrm{C_{d}}\geq 0.67\), and \(0.67>\mathrm{C_{dt}}\)/\(\mathrm{C_{d}}\). About 2/3 of the non-ferrous group had high \(\mathrm{C_{dt}}\)/\(\mathrm{C_{d}}\) ratios, but the rest were of comparable or lower \(\mathrm{C_{dt}}\) than \(\mathrm{C_{t}}\). Over all the test results, 61% showed \(\mathrm{C_{dt}}\)/\(\mathrm{C_{d}}\) ratio high than 1.5, indicating that transverse attenuation coefficients are higher than longitudinal ones in the majority of materials. However, the opposite or comparable cases are not negligible, especially in the ferrous group or in the ceramics and rocks group. The final two columns present the average values of damping factors, \(\eta\) and \(\eta_{\mathrm{t}}\). These were calculated for each group. Three groups of FRP, polymers, and ceramics and rocks have high \(\eta\) and \(\eta_{\mathrm{t}}\) values above 0.012. Recall that PMMA had \(\eta\) and \(\eta_{\mathrm{t}}\) values of 0.008 and 0.013. That is, these three groups showed on average comparable or higher damping factors than one of the representative viscoelastic solids. Both metal groups had lower damping factors, but the average \(\eta\) values were only 20% lower than that of PMMA and \(\eta_{\mathrm{t}}\) values 45 and 70% down. The only one that is clearly outside of the viscoelastic solid category is fused silica. Its attenuation was not detected with the measurement sensitivity of the methods used in this work. Thus, most test samples used in this study may be designated to be viscoelastic solids. For such solids, [PERSON] [1] derived a connecting equation between the viscosity term of the Kelvin-Voigt solid and \(\mathrm{Q^{-1}}=\eta=2\pi\mathrm{f}\eta_{\mathrm{K}}\)/E', as noted in Section 1.