## 4 Results and Discussion
### Comparison of TDM-1 and TDM-2
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Material** & **x\({}_{1}\)** & **x\({}_{2}\)** & **C\({}_{4}\)-TOM-1** & **C\({}_{4}\)-TDM-2** & **C\({}_{4}\)-TDM-2 with TCC** & **TCC** & **R\({}^{2}\)-TDM-1** & **R\({}^{2}\)-TDM-2** & **R\({}^{2}\) TDM-2** & **Notes** \\ \hline  & mm & mm & dB/m/MHz & dB/m/MHz & dB/m/MHz & dB & & & with TCC \\ PMMA & 9.0 & 18.2 & 94.0 & 74.6 & 93.3 & 2.00 & 0.947 & 0.929 & 0.777 \\ PMMA & 9.0 & 24.3 & 93.8 & 86.8 & 94.2 & 2.50 & 0.949 & 0.952 & 0.852 \\ PMMA & 9.0 & 27.1 & 86.2 & 75.5 & 86.6 & 1.90 & 0.959 & 0.952 & 0.916 \\ PMMA & 9.0 & 38.6 & 89.4 & 81.3 & 89.3 & 2.30 & 0.935 & 0.96 & 0.901 \\ PMMA & 9.0 & 43.2 & 98.7 & 88.3 & 98.6 & 3.00 & 0.971 & 0.927 & 0.852 \\ PMMA & 9.0 & 45.6 & 92.3 & 83.7 & 92.9 & 2.80 & 0.962 & 0.937 & 0.878 \\ PMMA & 9.0 & 48.1 & 91.0 & 81.3 & 90.6 & 3.00 & 0.985 & 0.964 & 0.938 \\ PMMA & 9.0 & 47.4 & 95.0 & 85.7 & 94.6 & 2.90 & 0.941 & 0.96 & 0.924 \\ PMMA & 9.0 & 38.6 & 82.3 & 59.8 & 82.7 & 4.70 & 0.947 & 0.928 & 0.78 \\ Average & & & 91.41 & 79.67 & 91.42 & & 0.955 & 0.945 & 0.869 \\ Std Dev * & & & 4.92 & 8.81 & 4.74 & & 0.955 & 0.945 & 0.869 \\ PVC & 4.5 & 17.5 & 224 & 206 & 223 & 1.84 & 0.991 & 0.956 & 0.929 \\ PVC & 4.5 & 13.5 & 245 & 200 & 245 & 5.03 & 0.993 & 0.971 & 0.936 \\ PVC & 4.3 & 13.5 & 248 & 163 & 248 & 7.25 & 0.93 & 0.918 & 0.664 & 12.7 mm * \\ Average & & & 239.0 & 189.7 & 238.7 & & 0.963 & 0.947 & 0.865 \\ PC & 2.3 & 16.5 & 638 & 581 & 639 & 3.20 & 0.983 & 0.991 & 0.981 \\ \hline \hline \end{tabular}  
* A pair of larger transducers with a 12.7-mm radius was used for these two tests:
* Std Dev indicates standard deviation. PMMA, polymethyl methacrylate; PVC, polyvinylchloride, PC, polycarbonate; C\({}_{d}\), damping coefficient; TDM, transmission difference method; TCC, transmission coefficient correction.  
\end{table} 
Table 1: Comparison of attenuation coefficients between TDM-1 and TDM-2 methods. \(\rm C_{d}=92.3\) (R\({}^{2}=0.962\)) for TDM-1 and \(\rm C_{d}=83.7\) (R\({}^{2}=0.937\)) for TDM-2. Both TDM-1 results for \(\rm C_{d}\) and R\({}^{2}\) were higher than those of TDM-2. The difference of R\({}^{2}\) values was small, but deviations were much higher in TDM-2, implying that careful slope estimation is needed based on FFT spectral behavior. In the present case, both R\({}_{2}\) and R\({}_{3}\) spectra showed broad reduction over 2 to 6 MHz, producing a positive deviation for the green curve. In this case, the regression fit for TDMA-2 appears justified, although \(\alpha\) for TDM-2 was lower than \(\alpha\) for TDM-1. Figure 9b shows a similar graph, again for the two methods, but this time the thick sample came from a different plate with a thickness of 43.2 mm. TDM-2 results for the thick sample used the \(\rm T_{c}\) term as required. For these cases, \(\rm C_{d}\) values (in dB/m/MHz) were obtained as follows:  
\(\rm C_{d}=98.7\) dB/m (R\({}^{2}=0.971\)) for TDM-1 and \(\rm C_{d}=88.3\) (R\({}^{2}=0.927\)) for TDM-2. Results were slightly improved compared to those in Figure 9a. In both comparisons, \(\rm C_{d}\) values were within 9 to 10 dB/m/MHz. The second comparison also indicates that when their properties are similar, TDM-1 can still be used even with different raw stocks. Seven additional comparison tests were conducted and all the results are summarized in Table 1. From these nine sets of attenuation spectra, the average \(\rm C_{d}\) value was 91.4 \(\pm\) 4.9 dB/m/MHz for TDM-1 and 79.7 \(\pm\) 8.8 dB/m/MHz for TDM-2, with an 11.4% higher \(\rm C_{d}\) value for TDM-1. One of the PMMA tests used a set of larger transducers (12.7 mm diameter), which produced a much lower \(\alpha\) value for TDM-2. The obtained TDM-1 \(\alpha\) value was close to the average of three recent studies [24; 25; 26], which was 103.3 dB/m/MHz. The averaged TDM-2 value was between the average data in Figure 1 (85.9 dB/m/MHz) and averaged \(\rm C_{d}\) value of 75.1 dB/m/MHz from the older studies collected by [PERSON] [19].