## 4 Results and Discussion
### Attenuation Behavior of Cast Iron  
Twenty-one cast iron samples were tested and their attenuation parameters are given in Table 5, using the same format as in Tables 2 and 3. Here, two columns are added. One is for C\({}_{2}\), since 11 samples had the quadratic spectra. Another is for damping factor, which is calculated using Equation (1). However, when the C\({}_{d}\) term vanishes, it is improper to use damping factors. For these cases, apparent damping factor, [n], is obtained assuming a linear attenuation spectrum below 1 MHz. These values are in brackets to distinguish them from normally obtained \(\eta\) values. All but two samples were supplied by the Iron Casting Research Institute (Columbus, OH, USA). Additionally, one sample was a continuously cast gray iron bar of 490 MPa class (FC-50; Nippon Chuzo, Kawasaki, Japan) and another was a common gray iron from a broken view. These represent four types of cast iron, namely gray iron, ductile (nodular) iron, malleable iron, and compacted graphite iron [88]. They all have a ferrite matrix with varying morphologies of graphite. In gray iron, graphite appears as irregular, interconnected flakes. During slow cooling, carbon in molten iron separates and forms graphite flakes that are interconnected within each eutectic cell, resulting in the characteristic flake shape. In ductile iron, graphite is changed into nodular shapes by the addition of controlled amounts of Mg and Ce, while compacted graphite iron is of the intermediate type between gray and ductile irons. The malleable iron sample was of the ferritic type and was heat-treated from white cast iron, forming irregular graphite nodules in the ferrite matrix. Twenty other samples were in as-cast condition. Two of the as-cast samples were mislabeled; that is, ductile A and ductile B showed the majority of their graphite to be in flakey forms, so they should be grouped with gray irons. In fact, their wave velocities were comparable to the gray iron of class 20 to 40. The hardness values were low to moderate, with Vickers values ranging from 111 to 322. Their \(\alpha\) values at 5 MHz were in the range of 147 to 1325 dB/m, with low values found in ductile, malleable, and compacted graphite types, while high values were observed in gray irons. [PERSON] [89] conducted attenuation measurements of ductile iron with \(\alpha\) values of 200 to 500 dB/m at 10 MHz. These are in the same range as the lower \(\alpha\) values found in malleable and ductile irons (tests I17-I21), but much lower than \(\alpha\) values observed for some gray irons (tests I1-I4). Representative attenuation spectra for cast iron are shown in Figure 15a,b. Two types of spectral behavior were observed. One followed the Mason-McSkimin relation (Equation (4)), as given in Figure 15a. Plotted in Figure 15a are the spectra for ferritic malleable iron (blue), ductile iron (60-45-10 grade, red; 80-55-06 grade, green), compacted graphite (dark blue), and high-strength gray iron (dark red). Each plot is accompanied by a modeled spectrum according to the Mason-McSkimin relation, given as a dashed curve of the same color. Matches between the observed and model spectra were good. The other spectral behavior was the quadratic relation (Equation (5)), as plotted in Figure 15b. Similar to spectral plots of Figure 11b, the abscissa of Figure 15b is frequency squared (in MHz\({}^{2}\)) and the slope is equal to C\({}_{2}\) (in dB/m/MHz\({}^{2}\)). The plots are listed in order of increasing steepness, as follows: gray class 50 (blue), gray class 60 (dark blue), ductile A (dark red), ductile B (green), gray class 20 (red), and gray piston ring (purple). Each observed spectrum is accompanied by a dotted line for the linear regression (with R\({}^{2}\) values of 0.992-0.998, except for 0.971 for class 20). The quadratic fits were very good, with nine spectra showing R\({}^{2}\) values 0.992 or higher.