## 4 Results and Discussion
### Attenuation Behavior of Ferrous Alloys
These spectra can be fitted to \(\alpha=\text{C}_{2}\hat{P}^{2}\) or the quadratic spectrum that was seen for A286 steel (see Figure 11b). This quadratic dependence, which is well-known for water, now has two mechanisms to explain the attenuation in solids; that is, dislocation-based theory [33; 34; 35; 36; 37] and Datta-Kinra scattering [50; 51; 52], as introduced in Section 1. It was also used to describe the attenuation spectra of two plates of Cu, a brass rod and a brass plate, Al-SiC composites, and in ten cast iron cases, as will be shown later. Tests F12 and F13 corresponded to the longitudinal (L) and transverse (T) directions of the lightly cold-worked steel plate (Vickers hardness 164), giving  
\[\alpha\left(\text{dB/m}\right)=4.84\;\hat{P}^{2},\qquad\left(\text{R}^{2}=0.975\right)\] (20a) and \[\alpha\left(\text{dB/m}\right)=5.91\;\hat{P}^{2}.\qquad\left(\text{R}^{2}=0.990\right) \tag{20b}\]For these two equations, small linear terms can be added without changing R\({}^{2}\) values much. However, the thickness (S) direction of the same plate sample (test F14) yielded the usual Mason-McSkimin relation of  
\[\alpha\left(\text{dB/m}\right)=18.4\text{ f}+0.0306\text{ f}^{4}.\quad\left( \text{R}^{2}=0.977\right) \tag{20c}\]  
When this plate was annealed (850 \({}^{\circ}\)C, 1 h), reducing the Vickers hardness to 126, the quadratic spectrum for the T direction (test F15) disappeared, becoming the Mason-McSkimin type, except with a weak Rayleigh term. The spectrum for the S direction (test F16) showed the linear frequency dependence, losing the Rayleigh scattering contribution, as given below:  
\[\alpha\left(\text{dB/m}\right)=6.8\text{ f}+0.0016\text{ f}^{4}.\ \left(\text{R}^{2}=0.652\right) \tag{21a}\] \[\alpha\left(\text{dB/m}\right)=16.0\text{ f}+0.03\text{ f}^{4}. \tag{18b}\]0  
Four of these spectra are shown in \[\alpha\left(\text{dB/m}\right)=16.0\text{ f}+0.03\text{ f}^{4}. \tag{18b}\]1 Model equations gave moderate to good fits (R\({}^{2}\) values of 0.915 to 0.990), except for Equation (21a), as the attenuation levels were very low. It appears that the observed behaviors of the in-plane directions, L and T, come from quadratic damping effects introduced by cold rolling [20; 35; 36; 57], but more work is needed to clarify the underlying dislocation mechanisms. Reduction in attenuation in the annealed condition was higher for the T direction and it increased with frequency. The S direction also showed frequency-dependent reduction in \(\alpha\) values above 6 MHz, but the amount of the reduction was about one-third that observed in the T direction. These changes in \(\alpha\) can be attributed to reduced dislocation densities after annealing, in accordance with the findings of [39; 40]. However, the observed anisotropy in the amount of reduction in \(\alpha\) values requires additional effects for reconciliation. Another apparent disagreement was noted for pure iron cases. Tests F1 and F2 for pure Fe gave comparable high attenuation. Apparently, annealing for test F2 (800 \({}^{\circ}\)C, 1 h) was inadequate, unlike the 1080 \({}^{\circ}\)C annealing temperature used in [40]. In the above examples, cold working (CW) increased attenuation levels, corresponding to a higher hardness number and higher \(\alpha\) value. Annealing decreased both hardness and attenuation. This will be referred to as the type-p CW effect. \[\alpha\left(\text{dB/m}\right)=16.0\text{ f}+0.03\text{ f}^{4}. \tag{18b}\]2 Anisotropic attenuation spectra of a cold-worked, low-carbon steel plate with curve fitting. Dark red: transverse (T) direction, cold working (CW) condition (Equation (20b)); dark blue: thickness (S) direction, CW condition (Equation (20c)); blue: transverse direction, annealed condition (Equation (21a)); red: thickness direction, CW condition (Equation (21b)). In 302 and 304 stainless steels, the opposite effect of cold working was observed; that is, CW decreased the attenuation effect (higher hardness number and lower \(\alpha\) value), which will be called the type-n CW effect.