## 5 Results and Discussion
### Glass-Fiber Reinforced Plastic (GFRP)  
A 2.3-mm thick GFRP sheet with cross-ply reinforcement was tested for Lamb-wave attenuation. As this is anisotropic, theoretical predictions for \(\alpha\)-S\({}_{\rm o}\) and \(\alpha\)-A\({}_{\rm o}\) values are not available, even though its bulk-wave attenuation coefficients were measured using laminated test samples made from the same sheet as listed in Table 1. Figure 18a,b show observed \(\alpha\)-S\({}_{\rm o}\) and \(\alpha\)-A\({}_{\rm o}\) values for the GFRP sheet in blue and red symbols, respectively. Lamb waves propagated along the 0\({}^{\circ}\)-fiber direction. In both figures, the experimental \(\alpha\)-S\({}_{\rm o}\) and \(\alpha\)-A\({}_{\rm o}\) values were near the lines representingts bulk-wave attenuation coefficients, \(\alpha_{\rm L}\) and \(\alpha_{\rm T}\), plotted in dashed blue and red lines. These were given by \(\alpha_{\rm 33}\) and \(\alpha_{\rm 32}\). Here, the surface normal, 90\({}^{\circ}\), and 0\({}^{\circ}\) fiber directions are designated as the 1, 2, and 3 axes, respectively, following [PERSON] [46]. In Figure 18a, a dotted blue line is added, which represents \(\alpha_{\rm L}=\alpha_{\rm 22}\). Initially, 0\({}^{\circ}\) and 90\({}^{\circ}\) directions were assumed to be equivalent, but subsequent measurements revealed clear differences reaching nearly twice in both longitudinal and shear attenuation. This is reflected in dashed and dotted blue lines of \(\alpha_{\rm L}\). In order to make approximate comparisons with experiment, the Disperse-based parametric method was also used for the calculation of \(\alpha\)-S\({}_{\rm o}\) and \(\alpha\)-A\({}_{\rm o}\) values (solid blue and red curves), assuming isotropy. Experimentally obtained \(\alpha\)-S\({}_{\rm o}\) values in Figure 18a showed an increasing trend with frequency above 100 kHz (as they were too low to measure at frequencies below 100 kHz). Results of two test series are given, and both followed the same trend. At 250 kHz, \(\alpha\)-S\({}_{\rm o}\) reached the \(\alpha_{\rm L}\) line and continued to increase at a similar rate to 430 kHz, where the value was about 50% higher than the \(\alpha_{\rm L}\) value. Figure 18b gives \(\alpha\)-A\({}_{\rm o}\) plots of three test series. Data points followed the predicted \(\alpha\)-A\({}_{\rm o}\) curve, which almost overlapped the \(\alpha_{\rm T}=\alpha_{\rm 32}\) line. Most of the observed \(\alpha\)-A\({}_{\rm o}\) values exceeded the \(\alpha\)-A\({}_{\rm o}\) prediction, with more than half by 10 to 20 dB, but remained below another \(\alpha_{\rm T}=\alpha_{\rm 31}\) line (red dotted), except one point. Despite these differences, the overall trend is recognized as a modest match between approximate theory and experiment. Although the significance of this theory-experiment matching is obscure because theory for isotropic medium was used, this finding nonetheless coincides with all other tests conducted in this study for isotropic plate samples. Because theory to deal with Lamb-wave attenuation in anisotropic solids has been published e.g., [4], exact predictions for \(\alpha\)-S\({}_{\rm o}\) and \(\alpha\)-A\({}_{\rm o}\) values are expected to become accessible, as was the case of Disperse, making an improved comparison of theory and experiment possible. Note that the \(\alpha_{\rm T}\) value used in the calculation was that for \(\alpha_{32}\) = 445 dB/m/MHz (transverse wave traveling along the fiber direction, with the polarization along the crossing in-plane fiber direction). See Table 1 for complete listings. With the polarization along the surface normal, \(\alpha_{31}\) is 72% higher at 861 dB/m/MHz. Thus, the experimental \(\alpha\)-A\({}_{\rm o}\) values fall between these two \(\alpha_{\rm T}\) lines. This may point to a possible resolution through the anisotropic calculation, which incorporates contributions to attenuation from different vibration modes, e.g., \(\alpha_{32}\) and \(\alpha_{31}\).