## 4 Results and Discussion
### Attenuation Behavior of Cast Iron
It should be noted that when attenuation spectra deviate from linear frequency dependence, the damping factor and other related terms such as Q\({}^{-1}\) are no longer independent of frequency. One needs to use caution when estimating ultrasonic attenuation from damping factors or the Q\({}^{-1}\) values obtained at low kHz frequencies reported in references [37; 41; 42; 43]. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Test** & **Material** & **C\({}_{\textbf{d}}\)** & **C\({}_{\textbf{2}}\)** & **C\({}_{\textbf{R}}\)** & **\(\alpha\) at 5 MHz** & **Damping** & **Velocity** & **Thickness** & **Vickes HN** & **Notes** \\ \hline No. & dB/m/MHz & dB/m/MHz\({}^{2}\) & dB/m/MHz\({}^{4}\) & dB/m & factor & mm/μs & mm & & \\ I1 & Common Gray Iron & 7.15\(\times\) 10\({}^{1}\) & 1787.5 & (0.10150) & 4.39 & 12.2 & 158 & As-cast \\ I2 & Highly Star 50 & 1.20\(\times\) 10\({}^{1}\) & 255.0 & (0.00181) & 4.83 & 304.0 & 208 & Continuous cast \\ I3 & Gray class 20 & 3.50\(\times\) 10\({}^{1}\) & 875.0 & (0.00573) & 4.47 & 25.0 & 145 & As-cast \\ I4 & Gray class 30 & 2.57\(\times\) 10\({}^{1}\) & 642.5 & (0.00474) & 4.75 & 25.8 & 250 & As-cast \\ I5 & Gray class 40 A & 1.34\(\times\) 10\({}^{1}\) & 335.0 & (0.00230) & 4.68 & 25.5 & 214 & As-cast \\ I6 & Gray class 40 B & 1.42\(\times\) 10\({}^{1}\) & 355.0 & (0.00253) & 4.86 & 25.4 & 257 & As-cast \\ I7 & Gray class 50 & 1.34\(\times\) 10\({}^{1}\) & 335.0 & (0.00245) & 5.00 & 25.1 & 322 & As-cast \\ I8 & Gray class 60 & 1.87\(\times\) 10\({}^{1}\) & 467.5 & (0.00434) & 5.00 & 24.8 & 242 & As-cast \\ I9 & Gray piston ring & 3.96\(\times\) 10\({}^{1}\) & 990.0 & (0.00624) & 4.30 & 25.3 & 166 & As-cast \\ I10 & Double A & 2.87\(\times\) 10\({}^{1}\) & 592.5 & (0.00404) & 4.65 & 24.4 & 171 & As-cast \\ I11 & Dottle B & 2.73\(\times\) 10\({}^{1}\) & 682.5 & (0.00453) & 4.53 & 26.9 & 314 & As-cast \\ I12 & Fine Gray & 59.3 & \(5.92\times\) 10\({}^{1}\) & 666.5 & \(9.34\times\) 10\({}^{1}\) & 4.30 & 25.3 & 111 & As-cast \\ I13 & High-Strength Gray A & 40.2 & \(4.62\times\) 10\({}^{1}\) & 498.9 & \(6.92\times\) 10\({}^{1}\) & 4.70 & 25.5 & 267 & As-cast \\ I14 & High-Strength Gray B & 24.7 & \(6.70\times\) 10\({}^{1}\) & 542.3 & \(4.16\times\) 10\({}^{3}\) & 4.60 & 28.3 & 240 & As-cast \\ I15 & Dottle piston ring & 42.8 & \(3.82\times\) 10\({}^{3}\) & 216.4 & \(8.86\times\) 10\({}^{3}\) & 5.65 & 25.6 & 220 & As-cast \\ I17 & Dottle 60-45-10 & 31.8 & \(8.93\times\) 10\({}^{3}\) & 164.6 & \(6.49\times\) 10\({}^{3}\) & 5.57 & 25.7 & 168 & As-cast \\ I18 & Dottle B0-55-06 & 41.0 & \(8.38\times\) 10\({}^{3}\) & 210.2 & \(8.47\times\) 10\({}^{3}\) & 5.64 & 25.0 & 282 & As-cast \\ I19 & Compact Graphite A & 31.9 & \(1.86\times\) 10\({}^{2}\) & 171.8 & \(6.42\times\) 10\({}^{3}\) & 5.34 & 25.0 & 197 & As-cast \\ I20 & Connected Graphite B & 31.8 & \(7.05\times\) 10\({}^{2}\) & 203.1 & \(6.00\times\) 10\({}^{3}\) & 5.15 & 28.3 & 265 & As-cast \\ I21 & Maleable-ferrite & 28.5 & \(7.90\times\) 10\({}^{3}\) & 147.4 & \(5.76\times\) 10\({}^{3}\) & 5.52 & 24.0 & 145 & Heat treated \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Attenuation coefficients and material parameters for cast iron. Rayleigh scattering due to nodular graphite, amounting to 16% volume at the eutectic composition of 4.3% C, was verified quantitatively by [PERSON] [89]. Note that [PERSON] and [PERSON] [90] treated the case of independent scattering of spherical particles and derived the same f\({}^{4}\)-dependence of Rayleigh scattering. In addition, the damping from the ferrite matrix contributes to the linear part, with a slope of 25 to 60 dB/m/MHz, as tabulated in Table 5. This part is expected to include transmission loss through graphite particles, since their volume cannot be ignored at 16%. The attenuation behavior shown in Figure 15b must be separately considered from those quadratic spectra found earlier in cold-worked steel, Cu, and brass, since no cold work exists in cast iron.