## 4 Calculation of Lamb-Wave Attenuation
Because pairs of attenuation curves of the same modes from these two methods are smooth-varying, Viktorov's A and B parameters can be modified to represent the Disperse results. Two cases were tried, and Disperse results can be expressed in the same manner as the Viktorov method. Disperse calculations were made using zero or non-zero \(\alpha_{\rm T}\) input for glass and PMMA plates with Poisson's ratios of 0.28 and 0.34. Polynomial curve-fitting is applied sectionally to the results of the Disperse calculations. Frequency shifts need to be used in some sections to achieve better fits. By comparing Lamb-wave attenuation with zero and non-zero \(\alpha_{\rm T}\) inputs, two complete sets of A and B parameters for materials were constructed. Because Disperse provides attenuation values up to the fourth order, the tables of A and B parameters enable calculating Lamb-wave attenuation up to S4 and A4 in the absence of Disperse software. For the case of Poisson's ratio of 0.34, results generally matched well to the direct Disperse calculation for HDPE, with a Poisson's ratio of 0.36. Deviation was mainly in frequency and the maximum shift was 0.4%. Thus, the tables of A and B can cover a Poisson's ratio of 0.25 to 0.36, with practical precision. This method may be called the Viktorov-Disperse parametric method or the Disperse-based parametric method. (See Appendix A for example calculation and parts of tables.) Relative contributions of the longitudinal and transverse components to Disperse calculations were examined using the Disperse-based parametric method. In this case, \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\) values of PMMA (Sample G7b) from Table 1 were selected, i.e., 106 and 216 dB/m/MHz. Figure 4c,d plotted S\({}_{\rm o}\) and A\({}_{\rm o}\) cases, respectively. The curves of S\({}_{\rm o}\) and A\({}_{\rm o}\) are given as solid blue and red curves, whereas the transverse components are shown by blue and red dotted curves. Reflecting a higher \(\alpha_{\rm L}\), relative to \(\alpha_{\rm T}\), solid and dotted curves are separated in both figures. The A\({}_{\rm o}\) curve from Disperse again overlaps the \(\alpha_{\rm T}\) line, except it is higher than the \(\alpha_{\rm T}\) line. This is likely due to higher \(\alpha_{\rm L}\). The contributions of the transverse components in percent are plotted by dash-dotted curves in Figure 4c,d in blue and red, respectively. The dominant roles of transverse-wave attenuation are clearly illustrated, as 80% or higher transverse contributions were observed above 100 or 50 kHz, respectively. This is an unexpected finding, as Lamb-wave modes have always been considered to be synthesized by the interactions of reflecting longitudinal and transverse waves. In retrospect, [PERSON]'s work [19] had already predicted the theoretical basis for the dominance of the transverse components on Lamb-wave attenuation. This feature is now confirmed with Disperse calculations [25; 51] as well. It is difficult to suggest any plausible physical mechanisms to account for the dominance of transverse attenuation, but it is worthy of further study. Figure 4: \(Cont.\)  
In this study, Disperse was used for eight cases, two with zero transverse attenuation. For most of the Lamb-wave tests with varied attenuation values, A-B parameter tables were used, and thickness changes were incorporated by factoring the attenuation and frequency scales. Unlike the velocity dispersion curves, which only need frequency rescaling for a thickness change, attenuation dispersion curves must use attenuation rescaling as well. Some published attenuation dispersion curves label abscissa as \"Frequency-thickness\", while casually labeling ordinate as \"Attenuation\" e.g., [4; 32]. The latter needs to be \"Attenuation-thickness\", as given in the Disperse Manual as an example [51] (p. 148). Figure 4: **(a)** Theoretical calculations of Lamb wave attenuation from Viktorov [19] and Disperse program [25; 51] for an isotropic Al plate (6.4-mm thickness) with Poisson’s ratio of 0.34.