# Dynamic Viscosity and Transverse Ultrasonic Attenuation of Engineering Materials
## 2 Experimental Methods
Following transmission tests of PMMA samples of various thicknesses and FFT processing, spectral differences were calculated to produced plots of transverse attenuation versus frequency. Four examples are shown in Figure 2. These include three transverse attenuation spectra of TDM-1 tests, giving the low (blue curve) and high (red) limits and the middle C\({}_{\rm dt}\) value (green). The red curve shows the worst linear fit (R\({}^{2}\) = 0.989). C\({}_{\rm dt}\) values were obtained from the least squar \(\times\) fit with the average R\({}^{2}\) = 0.993. One TDM-2 test curve is shown in purple with a better R\({}^{2}\) of 0.997. Twenty-one other spectra fall within the blue and red curves. \[R_{2}=T-\alpha_{t}\,X_{1}-D_{1}-T_{c}+R \tag{6}\]0 Transverse attenuation coefficient of polymethyl methacrylate (PMMA) vs. frequency. Three transmission difference method 1 (TDM-1) spectra for low, middle, and high C\({}_{\rm dt}\) values are shown by blue, green, and red curves, accompanied by their linear fits in dotted lines. One TDM-2 spectrum is plotted in purple. Test results of transverse attenuation coefficients, \(\alpha_{t}\), of PMMA are tabulated in \[R_{2}=T-\alpha_{t}\,X_{1}-D_{1}-T_{c}+R \tag{6}\]1 All the data were fitted to the linear frequency dependence and the least-square fitted slopes of \(\alpha_{t}\) vs. \(f\) curves, C\({}_{dt}\), are listed with R\({}^{2}\) values. The averages of TDM-1 and TDM-2 test results are C\({}_{dt}\) = 253.2 and 238.6 dB/m/MHz, respectively. The TDM-2 result is 5% lower than the TDM-1 value. A similar difference was previously observed for the case of longitudinal attenuation coefficients of PMMA [13]. It was attributed to the T\({}_{c}\) term requiring a correction. The observed C\({}_{dt}\) of 253 \(\pm\) 16.5 dB m/MHz from 17 TDM-1 tests agrees to that found in [PERSON] et al. [37], within 6% of [PERSON] et al. [35] and within 14% of the Hartmann-Jarzynski data [36]. The observed agreement of C\({}_{dt}\) values indicates that the test methods used here, TDM-1 and TDM-2, can be used to estimate the transverse attenuation behavior of materials with an estimated error of \(\pm\) 20%. This error estimate is taken to be three times higher than the best-case error of PMMA, considering typical attenuation tests used TDM-2 and sample conditions were less favorable. PMMA transverse attenuation data was further separated into two parts. The first group (T1 through T11) is for plate thicknesses under 18 mm including T9 to T11 with 27.3-mm thickness, which were glued together from 9.1-mm plates. The second group (T12 to T17) used plates of nominally 25-mm thickness. The average values of C\({}_{dt}\) were 245 and 269 dB/m/MHz for the two groups. Thicker plates showed about 10% higher \(\alpha_{t}\) value. This difference appears to come from the internal stress of PMMA as thinner plates have higher residual stress from faster cooling during manufacturing. The damping factor of PMMA was shown to decrease with the applied stress [38]. Taking the average C\({}_{dt}\) value of 253 dB/m/MHz to be representative, this is 2.8 times that of C\({}_{d}\) value of 91.4 dB/m/MHz for the longitudinal waves. This ratio is larger than the inverse ratio of wave velocities of 1.6. The average transverse damping factor was 0.013, compared to 0.008 for the