## 5 Results and Discussion
### Aluminum Alloys
#### 5.2.1 Al-1100-H18  
Two Al-1100 plates (G2a and G2b) have Vickers hardness numbers of 41 and 40 and are in fully cold-worked condition. The thicker plate was tested for its bulk-wave attenuation coefficients, resulting in \(\alpha_{\rm L}=15.7\) dB/m/MHz and \(\alpha_{\rm T}=26.5\) dB/m/MHz, as listed in Table 1. These values were used to calculate Lamb-wave attenuation coefficients using the Disperse-based parametric method with [PERSON]'s ratio of 0.34. As the Poisson's ratio of Al-1100 was 0.33, the parametric method gives accurate values of \(\alpha\)-S\({}_{\rm o}\) and \(\alpha\)-A\({}_{\rm o}\). Figure 6a,b show experimental \(\alpha\)-S\({}_{\rm o}\) and \(\alpha\)-A\({}_{\rm o}\) values for an Al-1100 sheet of 2.1-mm thickness in blue and red symbols, respectively. In Figure 6a, two sets of \(\alpha\)-S values are plotted in dots and + symbols from 200 to 1000 kHz, and calculated values of \(\alpha\)-S\({}_{\rm o}\) are plotted in a solid blue curve. Blue and red dashed lines represent the bulk-wave attenuation coefficients, \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\). The velocity range of these datasets was from 5.9 to 3.2 mm/us. The A\({}_{\rm o}\) mode contribution was absent on WT spectrograms with the two-side summation processing and was also cut off in WT intensity summation. Both of the observed \(\alpha\)-S\({}_{\rm o}\) data matched well with the calculated \(\alpha\)-S\({}_{\rm o}\) curve (in solid blue). The maximum deviation was 2.4 dB/m, or 34%, of the calculated value at 427 kHz. Figure 6b gives \(\alpha\)-A values from four tests. Calculated values of \(\alpha\)-A\({}_{\rm o}\) in a solid red curve and the \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\) dashed lines in blue and red are also plotted. The velocity range of these datasets was from 3.3 to 1.1 mm/us, excluding the faster S\({}_{\rm o}\) mode. Differences of the two-sided signals were used to minimize the symmetric components. The \(\alpha\)-A\({}_{\rm o}\) attenuation data showed gradual increase straddling the calculated \(\alpha\)-A\({}_{\rm o}\) (red curve), which is slightly above the red-dashed \(\alpha_{\rm T}\) line. The observed attenuation values ranged from 160 to 450 kHz. These deviated from the predicted values by as much as 5.2 dB/m (or 57% at 320 kHz), but the overall trend followed the calculated attenuation curve. Deviation was higher between 300 and 400 kHz. The matching between theory and experiment is deemed moderate. In order to evaluate the accuracy of attenuation values obtained, deviation and percent deviation were calculated. For this Al sheet, the predicted \(\alpha\)-S\({}_{\rm o}\) values were fitted to a second order polynomial in frequency, and \(\alpha\)-A\({}_{\rm o}\) values were fitted to a fourth order polynomial. Averaged results are given below: \begin{tabular}{c c c c} \hline Wave Mode & Deviation (dB/m) & Percent Deviation (\%) & Data Count \\ \hline Symmetric case: & 0.84 & 11.3 & 26 \\ \hline Asymmetric case: & 1.74 & 19.1 & 36 \\ \hline \end{tabular}   
Because the attenuation values were lower than in the PS case, deviation values were smaller, but percent deviation was 11 and 19%. The higher percent deviation for the asymmetric case resulted from large deviation at the mid-frequency range. For the 2-mm Al-1100 sheet, Lamb-wave attenuation is found to match the theoretical prediction. Agreement was slightly poorer than it was in the PS sheets in Section 5.1. The second Al-1100 plate was thicker, with 6.4-mm thickness (G2b). Figure 7a,b show \(\alpha\)-S and \(\alpha\)-A for the 6.4-mm thick plate in blue and red symbols, respectively. In Figure 7a, a set of \(\alpha\)-S values are plotted in square symbols from 200 to 460 kHz, and calculated values of \(\alpha\)-S\({}_{0}\) are plotted in a solid blue curve. Blue and red dashed curves represent the bulk-wave attenuation coefficients, \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\). The velocity range of the dataset was from 5.9 to 2.2 mm/us.