## 4 Calculation of Lamb-Wave Attenuation  
Two methods can be used to obtain theoretical values of attenuation coefficients of Lamb-wave modes, as noted earlier [19; 25; 51]. Both methods utilize complex wavenumbers or elastic moduli by adding damping terms as their imaginary components. The damping terms are inserted as bulk-wave attenuation coefficients, \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\). The methods also require the plate thickness and longitudinal and transverse wave velocities, \(\
u_{\rm L}\) and \(\
u_{\rm T}\). For the Viktorov method [19], Lamb-wave attenuation coefficients \(\alpha\)-S\({}_{\rm n}\) and \(\alpha\)-A\({}_{\rm n}\) can be calculated using A and B parameters for each mode, with the two expressions below. \[\alpha\text{-}S_{\rm n}=A_{\rm Sn}\ \alpha_{\rm L}+B_{\rm Sn}\ \alpha_{\rm T}. \tag{3}\]  
\[\alpha\text{-}A_{\rm n}=A_{\rm An}\ \alpha_{\rm L}+B_{\rm An}\ \alpha_{\rm T}. \tag{4}\]  
These parameters are given graphically for the cases of n = 0 and 1 and in equation forms. These equations apparently contain typographical errors and could not be used to reproduce \(\alpha\)-S\({}_{\rm n}\) and \(\alpha\)-A\({}_{\rm n}\) values (private communication, Barat, V., 2021). The A and B parameters for n = 0 and 1 were digitized, and \(\alpha\)-S\({}_{\rm n}\) and \(\alpha\)-A\({}_{\rm n}\) were computed for an Al plate of 6.4 mm thickness. For these calculations, the bulk-wave attenuation coefficients, \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\), of the longitudinal and transverse waves were taken from Disperse, namely, \(\alpha_{\rm L}\) = 4.125 dB/m/MHz and \(\alpha_{\rm T}\) = 27.76 dB/m/MHz for Al. It is important to recognize that both Viktorov and Disperse calculations require experimentally obtained values of \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\). Results are shown by solid curves in Figure 4a,b. Using the same input values, a calculation with Disperse software [51] (ver. 2.0.20a, Imperial College, London, UK, 2013) was conducted as well, and results are shown by dotted curves together with the Viktorov calculations. Here, blue curves are for S\({}_{\rm o}\) and S\({}_{\rm 1}\) modes, and red curves are for A\({}_{\rm o}\) and A\({}_{\rm 1}\), respectively. In all cases, the trends of corresponding curves matched well, especially for the zeroth modes. Viktorov S\({}_{\rm o}\) showed higher attenuation than Disperse S\({}_{\rm o}\), whereas the attenuation of the A\({}_{\rm o}\) mode was reversed. For the first mode, deviations were larger, with the Viktorov curves 5 to 20 dB/m higher. In both figures, blue and red dashed lines represent the bulk-wave attenuation coefficients, \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\), of the longitudinal and transverse waves. In Viktorov calculations, the contributions of the longitudinal and transverse components are obtained separately, then they are summed for the final attenuation value. For the selected \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\), both \(\alpha\)-S\({}_{\rm n}\) and \(\alpha\)-A\({}_{\rm n}\) consisted mostly of the transverse components, which are plotted in green and purple dashed curves. These are plotted in Figure 4a, but these overlap almost completely, being barely visible only below 250 kHz. Here, the contribution of the longitudinal component is minimal. This is partly due to low \(\alpha_{\rm L}\), which was only 15% of \(\alpha_{\rm T}\). Even when \(\alpha_{\rm L}\) is larger with respect to \(\alpha_{\rm T}\), however, the dominance of \(\alpha_{\rm T}\) component is expected. The A\({}_{\rm o}\) curve from Disperse almost overlaps the \(\alpha_{\rm T}\) line, whereas Viktorov A\({}_{\rm o}\) is approximately 20% below the \(\alpha_{\rm T}\) line. All Lamb-wave attenuation curves are also within a factor of three to the \(\alpha_{\rm T}\) line (excluding those near the low frequency cut-offs). These are always above the \(\alpha_{\rm L}\) line by a factor of two or more.