## 5 Results and Discussion
### Implications for Lamb Wave Applications  
In addition to the comparison between theory and experiment concerning Lamb-wave attenuation, a few more aspects deserve brief discussion. One of the observed features of Lamb-wave attenuation is the fact that the attenuation of both symmetric and asymmetric modes depends substantially on the \(\alpha_{\mathrm{T}}\) value of the propagating medium. In fact, the values of \(\alpha_{\mathrm{T}}\) and \(\alpha\)-A\({}_{0}\) differ less than 10% over most of the frequency range of interest, and it is possible to take \(\alpha\)-A\({}_{0}\approx\alpha_{\mathrm{T}}\). Because \(\alpha\)-S\({}_{0}\) values vary nonlinearly, simple approximation is not feasible. However, observed \(\alpha\)-S\({}_{0}\) values were often bounded by the \(\alpha_{\mathrm{L}}\) and \(\alpha_{\mathrm{T}}\) lines. Thus, approximating \(\alpha\)-S\({}_{0}\) with the average of \(\alpha_{\mathrm{L}}\) and \(\alpha_{\mathrm{T}}\), or \(\alpha\)-S\({}_{0}\approx(\alpha_{\mathrm{L}}+\alpha_{\mathrm{T}})/2\), has merit. This is valid when the S\({}_{0}\) wave velocity is one-third higher than the A\({}_{0}\) wave velocity. Approximate expressions are as follows:  
\[\alpha\)-S\({}_{0}\approx(\alpha_{\mathrm{L}}+\alpha_{\mathrm{T}})/2\text{ and }\alpha\)-A\({}_{0}\approx\alpha_{\mathrm{T}}. \tag{5}\]  
As shown in Section 4, it is possible to obtain the \(\alpha\)-S and \(\alpha\)-A values up to the fourth order modes when \(\alpha_{\mathrm{L}}\) and \(\alpha_{\mathrm{T}}\) values are available by the use of the Disperse-based parametric method. That is, the access to Disperse software is not the prerequisite. Still, simple equations may be useful for the initial step. Transverse wave attenuation had been difficult to find from the literature, but many values of \(\alpha_{\mathrm{T}}\) were recently listed in [45]. For example, two HSLA steels, SM50 and A533B, have \(\alpha_{\mathrm{T}}\) of 13.0 and 5.2 dB/m/MHz, whereas stainless steels, 304 and 17-4 PH, have \(\alpha_{\mathrm{T}}\) of 17.1 and 26.3 dB/m/MHz, respectively. Using SM50 steel as a representative pipeline steel, the \(\alpha\)-S\({}_{0}\) approximation is valid to 120 kHz, when the thickness is assumed to be 15 mm. For this steel, \(\alpha\)-S\({}_{0}\) and \(\alpha\)-A\({}_{0}\) values were calculated and compared to Equation (5) in Figure 19. Both approximations from Equation (5) (green and red dash lines) agree well to the calculated \(\alpha\)-S\({}_{0}\) and \(\alpha\)-A\({}_{0}\) values (blue and red curves). For this steel, \(\alpha\)-S\({}_{0}\) = 0.25 dB/m at 30 kHz, and this value agrees well with the measured pipeline attenuation of 0.17 dB/m [13; 14], which was based on AE wave propagation data for the distances from 2 to 300 m. Often, \(\alpha\)-S\({}_{0}\) values were undetected even with propagation distances of 10 to 100 m [14]. With low attenuation materials and thin plate shapes, it is normally not possible to measure the bulk-wave attenuation coefficients. Although it is also difficult to determine Lamb-wave attenuation coefficients, it is possible to get \(\alpha\)-S\({}_{0}\) and \(\alpha\)-A\({}_{0}\) values under certain conditions. The 2.0-mm thick glass sheet, discussed in Section 5.4, is an example. This allows one to estimate a likely \(\alpha_{\mathrm{T}}\) value. Because of weaker effects of \(\alpha_{\mathrm{L}}\) on \(\alpha\)-S\({}_{0}\) and \(\alpha\)-A\({}_{0}\), it is expected to be difficult to get \(\alpha_{\mathrm{L}}\), however. In this connection, a correlation between \(\alpha_{\mathrm{L}}\) and \(\alpha_{\mathrm{T}}\) values is useful. Figure 20 shows a cross-plot of \(\alpha_{\mathrm{L}}\) and \(\alpha_{\mathrm{T}}\) values from identical samples available from [14; 45]. A total of 173 sets were given, 9 from [14] and the rest from [45]. Of these, 14 were excluded, as they exceeded \(\alpha_{\mathrm{L}}\) = 300 and \(\alpha_{\mathrm{T}}\) = 1200 dB/m/MHz, to avoid their oversized effects. The majority of data points had less than 80 and 200 dB/m/MHz for \(\alpha_{\mathrm{L}}\) and \(\alpha_{\mathrm{T}}\).