## 4 Results and Discussion
### Attenuation Behavior of Cast Iron
Because of the presence of interconnecting flakey graphite in gray iron (including two samples that were incorrectly labeled as \"ductile\" iron), the effective size of graphite flakes may be larger than the optically measured flake length by a few tenths of a millimeter. A possible explanation of the observed f\({}^{2}\)-dependence for gray iron is the Datta-Kinra relation [50; 51; 52] without damping (Equation (5)). Here, attenuation is caused by multiple scattering from inclusions, namely interconnected graphite flakes. Acoustic impedance of graphite is 12.5 times smaller than iron, making graphite flakes good scattering sources. Repeated wave reflections are expected due to graphite connectivity. However, no quantitative analysis exists and further study is needed to verify this mechanism. Another possible source of attenuation is absorption effect. [PERSON] and [PERSON] [91] demonstrated that the surface reflection spectrum from random granular surfaces follows a quadratic relation. When graphite flakes cover the wave  
Figure 15: Representative attenuation spectra for cast iron. (**a**) Ductile and malleable iron with Mason–McSkimin relation: ferritic malleable iron (blue); ductile iron (60-45-10 grade, red); ductile iron (80-55-06 grade, green); compacted graphite (dark blue); high-strength gray iron (dark red). (**b**) Gray iron with quadratic spectra: gray class 50 (blue); gray class 60 (dark blue); ductile A (dark red); ductile B (green); gray class 20 (red); gray piston ring (purple). path across the planar wavefront, the reflection from an acoustic impedance mismatch produces transmission loss or attenuation. Within a gray iron sample, multiple graphite flake layers contribute to successive reflection and transmission at many interfaces. Experimentally, such an interface can be viewed in fractographs or on etched surfaces [88; 92]. The laser profilometer result for a gray iron fracture surface was given in Nakae [93] and is reproduced in Figure 16. As the intervals of height oscillations are 0.2-0.5 mm, [PERSON] theory is relevant for gray iron cases as well. While quantitative analysis needs to follow, this can account for the quadratic attenuation spectra observed for gray irons. Some correlations between hardness and attenuation levels may exist. For gray irons, attenuation at 5 MHz trends down with increasing hardness (shown by red circles in Figure 17). For stronger gray irons, microstructures are refined and higher amounts of flakes are converted to nodular forms by faster cooling. In contrast, non-gray irons showed a different trend, with \(\alpha\) values at 5 MHz rising slightly with higher hardness (indicated by + marks in Figure 17). In the non-gray irons, attenuation changes were small and were not affected strongly by microstructural refinements. Quantitative analysis of nodular effects is required for clarification. Next, it is important to compare the observed damping factors with those determined at low frequencies [94], since high damping properties are important in many cast iron applications [88]. For gray irons, \(\eta\) values ranged from 0.0019 to 0.016, while ductile iron showed \(\eta\) values ranging  
Figure 16: Surface contour of the fracture surface of gray cast iron using a high-resolution laser profilometer. Reproduced from [PERSON] [93] with the permission of Japan Foundry Engineering Society, Tokyo, Japan. Figure 17: Attenuation coefficients at 5 MHz vs. Vickers hardness of cast iron. Red circles: gray iron; black +: non-gray iron. from 0.00014 to 0.00063. Additionally, \(\eta\) values of 0.006 to 0.031 were reported at a higher strain amplitude [94]. These values are generally comparable to the \(\eta\) and [\(\eta\)] values in Table 5, indicating the frequency effect on \(\eta\) is small. This section reported the attenuation behavior of four types of cast iron samples. For high-strength nodular irons of ductile and ferritic malleable types, attenuation was low and was found to be comparable to some steels.