## 3 Materials and Experimental Procedures
T\({}_{c}\) arises from the differences in acoustic impedances of the transducer face (\(Z_{t}\)) and sample (\(Z_{s}\)), and is given by [1]  
\[T_{c}=(4Z_{t}\,Z_{s})/(Z_{t}+Z_{s})^{2} \tag{16}\]  
Figure 7: (**a**) Schematics of the attenuation experiment, including from top, pulser, 1: direct contact test, 2 and 3: through-transmission test 1 and 2. (**b**) Photograph of one of the test jigs with an Al sample, using two 10 MHz transducers. Transducers are manually aligned and pressed together using a 12.7-mm diameter, 0.8-mm pitch bolt on top of a steel hemisphere. Finger-tightening of the bolt generates \(\sim\)200 N force. This expression for \(T_{c}\) assumes the usual condition of the planar wave front. Because the transducer face material is alumina (\(Z_{t}\) = 38 Mrayl), \(T_{c}\) may be omitted when a sample has \(Z_{s}\) larger than 30 Mrayl (for an error of 0.1 dB), unless attenuation values are low. Both \(D_{n}\) (n = 1 or 2) and \(T_{c}\) are expressed in dB. \(D_{n}\) is frequency-dependent, while \(T_{c}\) is assumed to be a constant. Note that \(D_{n}\) increases linearly with frequencies below 0.1 to 2 MHz for the typical experimental conditions used in the present study. Here, \(D_{n}\) values often exceeded observed attenuation effects and useful attenuation data were only obtained in the near-field region, where \(D_{n}\) values varied slowly with frequency (e.g., at f \(>\)2 MHz for 66 mm Al and \(>\)0.2 MHz for 10 mm PMMA). By using two of the three equations (Equations (14a), (14b), and (14c)) given above, one gets  
\[\alpha=(R_{2}+D_{1}-R_{3}-D_{2})/(x_{2}-x_{1}) \tag{17a}\] \[\alpha=(R_{1}-R_{2}-D_{1}-T_{c})/x_{1}\] (17b) \[\alpha=(R_{1}-R_{3}-D_{2}-T_{c})/x_{2}. \tag{17c}\]  
As Equations (17b) and (17c) are equivalent, two distinct methods can be utilized. The first uses setups 2 and 3 with Equation (17a). This approach cancels out the effects of the transmission coefficient due to acoustic impedance mismatch. Coupling layer effects are also canceled. The second method utilizes setups 1 and 2 with Equation (17b). These two approaches will be called transmission difference methods 1 and 2, with TDM-1 and TDM-2 as abbreviated terms. That is, TDM-1 uses two sample thicknesses, x\({}_{1}\) and x\({}_{2}\), with setups 2 and 3, and obtains the value of \(\alpha\) from Equation (17a). In contrast, TDM-2 uses one sample thickness, x, with setups 1 and 2 (or 3). The value of \(\alpha\) comes from Equations (17b) or (17c). This is more convenient than TDM-1 because only one sample is needed, but requiring the use of \(T_{c}\). Preparing two samples of different thicknesses is not practical for materials that are difficult to machine, but the effects of one extra coupling layer remain. There is no direct way to measure the value of \(T_{c}\), but when attenuation results are obtained by using both TDM-1 and TDM-2, the value of \(T_{c}\) can be estimated. By comparing the estimated and calculated \(T_{c}\), a transmission coefficient correction (TCC) is obtained. This requires adjusting the \(T_{c}\) value in the determination of \(\alpha\) by TDM-2, so that its result matches the TDM-1 result. TCC can be applied to TDM-2 results from similar test conditions. In Section 4.1, three materials will be tested using TDM-1 and TDM-2. Results will be compared between them. If these two sets provide good agreement in the values of \(\alpha\), TDM-2 may be used in lieu of TDM-1, utilizing calculated \(T_{c}\) values. It was found that comparison provided no matching \(\alpha\) values, but TDM-2 gave \(\alpha\) values of 10% to 25% lower than those from TDM-1 tests. The differences were corrected by applying a TCC term of 2 to 7 dB. Even when TCC is not used, \(\alpha\) values from TDM-2 can be treated as approximations. Thus, \(\alpha\) values of most materials will be measured using TDM-2. Whenever multiple samples are available, TDM-1 will be used, since its results were less sensitive to the inadequacy in sample preparation and provided more reliable attenuation spectra with minimal couplant effects.