## 4 Results and Discussion
### Attenuation Behavior of Ferrous Alloys
Linear regression and visually estimated fits provided the following expressions of  
\[\alpha\left(\text{dB/m}\right)=170.9\text{ f}. \tag{19a}\]  
Figure 11: Attenuation coefficients vs. frequency. (**a**) The 4340 steel curve shown in blue with a linear fit (dotted line) and 1020 steel shown in red with Mason–McSkimin fitted dashed curve and a linear fit (dotted line). Klinman data [73] for ferritic-pearlitic medium-C steels are plotted in green. (**b**) A286 austenitic stainless steel in solution-treated (blue) and hardened (red) conditions. The \(\alpha\) spectra are plotted with a linear fit (blue and red dotted lines) and Mason–McSkimin dashed curve (red). A possible quadratic fit is also shown as the purple dashed curve. \[\alpha\left(\text{dB/m}\right)=98.68\;\text{f}+0.0788\;\text{f}^{4}. \tag{19b}\]  
Note that at 10 MHz, both equations give similar \(\alpha\) values (1709 and \(987+788=1775\) dB/m). These are plotted in blue dots and red dashes, with the linear part of Equation (19b) given in red dots. The C\({}_{\text{R}}\) value for the hardened A286 alloy was approximately ten times larger than that of 1020 steel. For Equation (19a), the regression fit gave R\({}^{2}=0.990\), while the R\({}^{2}\) value for visually fitted Equation (19b) was 0.974. Although the meaning of R\({}^{2}\) for the latter is complex, the fitted expression models the observed frequency dependence well from 3.5 to 12.5 MHz. These examples represent two different frequency dependencies occurring in nearly all ferrous materials. For the hardened A286 alloy, another spectral function can also represent the observed response. This is a quadratic-type function (Equation (5)), given by  
\[\alpha\left(\text{dB/m}\right)=\text{C}_{2}\;\hat{P}^{2}=18.24\;\hat{P}^{2}, \tag{19c}\]  
with an R\({}^{2}\) value of 0.985, indicating a better fit than the Mason-McSkimin relation. This quadratic spectrum is shown in Figure 8b by a purple dashed curve. This quadratic relation is of the form of the Datta-Kinra scattering [50; 51; 52]. Both solution-treated and aged A286 steel samples typically showed dispersed Laves phase particles, with average spacing of under 20 \(\upmu\)m [74]. Thus, the Datta-Kinra scattering of the Laves particles can possibly explain the quadratic spectrum. However, the solution-treated samples showed the linear spectrum without the scattering term but with intermetallic particles still present. More work is needed here. The quadratic spectrum was also found to describe observed spectra in several more cases of low-C steel, Cu, and brass plates. This is a seldom observed frequency dependence for metallic alloys that will be discussed further. Table 2 presents the attenuation data for 101 tests. For each test, the test number and results for C\({}_{\text{d}}\), C\({}_{\text{R}}\), \(\alpha\) value at 5 MHz, wave velocity (v\({}_{\text{L}}\)), thickness (or length), and Vickers hardness number are listed, along with the condition of the sample. When the C\({}_{\text{R}}\) column is blank, only the linear frequency dependence of \(\alpha\) was observed. The C\({}_{2}\) value is given here in parentheses when applicable. As the Rayleigh scattering term can be ignored at 1 MHz, the C\({}_{\text{d}}\) value equals the \(\alpha\) value at 1 MHz. Observed \(\alpha\) values at 1 MHz ranged from 1.5 dB/m for 304 stainless steel (test F85; UNS S30400) to 170.9 dB/m for A286 steel (test F96, see Figure 11b above). Two other tests (pure iron) showed \(\alpha\) values at 1 MHz exceeding 100 and ten more tests gave \(\alpha>50\) dB/m at 1 MHz. In three cases, \(\alpha\) values were low and only estimated ranges were given (tests F49, F50, and F70). Most structural steels showed \(\alpha\) values at 1 MHz of 5 to 30 dB/m. Most of the austenitic stainless steels belonged to this group of relatively low attenuation steels. Tests on cold-worked plates revealed anisotropic and unusual attenuation responses. Tests F12 and F13 of a low-carbon steel plate produced different attenuation spectra.