## 4 Results and Discussion
### Comparison of TDM-1 and TDM-2
The averages of R\({}^{2}\) values were close, although visually TDM-1 plots were closer to straight lines. In all the attenuation spectra, no effect of scattering induced attenuation was observed for the amorphous polymers evaluated. These test data were utilized to determine values of transmission coefficient corrections (TCC). Once a TCC is obtained, the TDM-1 attenuation coefficient can be estimated from TDM-2 tests. The results in dB were tabulated along with the corrected TDM-2 \(\rm C_{d}\) values (see Table 1). For eight PMMA tests using 6.4-mm diameter transducers, the TCC ranged from 1.9 to 3 dB, while it was 4.7 dB for the larger transducer pair; that is, the TDM-2 test could not approximate the TDM-1 test using 12.7-mm transducers without applying TCC, as its error was almost 40% without correction. However, the use of TCC is always recommended whenever the amount of attenuation through a sample is low. Two more polymers, PVC and PC, were tested similarly by comparing TDM-1 and TDM-2. The attenuation spectra after distance normalization are shown in Figure 10. The attenuation data shown used Equations (17a) and (17c) by dividing amplitude difference spectra ((R\({}_{2}\) + D\({}_{1}\) - R\({}_{3}\) - D\({}_{2}\)) for TDM-1 and (R\({}_{1}\) - R\({}_{3}\) - D\({}_{2}\) - T\({}_{\rm c}\)) for TDM-2 with the propagation distances of (x\({}_{2}\) - x\({}_{1}\)) for TDM-1 and of x\({}_{2}\) for TDM-2. PVC results with x\({}_{1}\) = 4.5 mm and x\({}_{2}\) = 17.5 mm are plotted in dark blue and blue curves extending to 10 MHz, while PC data with x\({}_{1}\) = 4.6 mm and x\({}_{2}\) = 9.4 mm are in dark red and red curves up to 8 MHz. For the PC tests, an NDT System C16 transmitter was used, producing anti-resonance dips at 5 MHz causing oscillations. The attenuation spectra were fitted to linear frequency dependence as before and \(\rm C_{d}\) values in dB/m/MHz (R\({}^{2}\) values) were as follows (also in Table 1):  
PVC: \(\rm C_{d}=239\) (R\({}^{2}=0.963\)) for TDM-1 and \(\rm C_{d}=190\) (R\({}^{2}=0.947\)) for TDM-2. PC: \(\rm C_{d}=638\) (R\({}^{2}=0.983\)) for TDM-1 and \(\rm C_{d}=581\) (R\({}^{2}=0.991\)) for TDM-2. Here, values for PVC are averaged data. For both PVC and PC, results for \(\alpha\) values from the two methods matched from 10% to 25%, which were comparable with PMMA. All four R\({}^{2}\) values for PVC and PC were slightly better than the corresponding R\({}^{2}\) values in PMMA. Thus, these cases also support the use of calculated \(\rm T_{c}\) in the TDM-2 method for approximation. However, the use of TCC, which ranged from 1.8 to 7.3 dB, provides a good estimate of TDM-1 attenuation data. The largest TCC was for a pair of 12.7-mm diameter transducers. This means the TCC application is mandatory when a larger transducer pair is used. From the results for three polymeric materials, linear frequency-dependent attenuation behavior can approximate observed data up to 5 to 10 MHz. This approximation corresponds to a constant damping factor \(\eta\) for these polymers in the low MHz region. It was noted that \(\eta\) for PMMA peaked at 50 Hz and reached an apparent asymptote of \(\sim\)0.01 at 1 to 10 MHz [7]. However, available data are inadequate to conclude a constant \(\eta\). Upon close inspection, all the TDM-1 plots of \(\alpha\) vs. f consistently exhibited positive deviations from the linearity at low frequencies. Possible variation of \(\eta\) with frequency needs to be probed by using a single starting material. This requires varying the thickness and obtaining well-prepared samples. This section shows that TDM-1 is the superior method for measuring attenuation coefficients, but the more convenient TDM-2 can be used for approximate determination of \(\alpha\) values when multiple samples are not readily available. For PMMA, measured \(\alpha\) values were close to the averaged literature values and their frequency dependence was well represented by linear behavior, except systematic deviations were noticeable at lower MHz frequencies. No scattering effects were found.