# Dynamic Viscosity and Transverse Ultrasonic Attenuation of Engineering Materials
## 2 Experimental Methods  
The experimental approach used for transverse wave attenuation is identical to that employed in [13] for longitudinal wave attenuation except for the use of transverse wave transducers. It utilizes through transmission setups as shown in Figure 1, using damped, wideband transducers as both transmitter and receiver. Setup 1 is used for direct contact of the transmitter and receiver, giving the voltage output \(\rm V_{1}\) from the receiver as a result of transmitter excitation by a pulser. This is also called fac \(\times\) to-face testing. Setup 2 places sample 1 between the transmitter and receiver, while Setup 3 places sample 2 between the transmitter and receiver. These yield \(\rm V_{2}\) and \(\rm V_{3}\), respectively, in response to pulser excitation. By applying a fast Fourier transform (FFT) on \(\rm V_{1}\), \(\rm V_{2}\), and \(\rm V_{3}\), one gets the corresponding frequency domain spectra \(\rm R_{1}\), \(\rm R_{2}\), and \(\rm R_{3}\), expressed in dB (in reference to 0 dB at 1 V). Expressing the transmitter output (in reference to 0 dB at 1 nm) and receiver sensitivity (in reference to 0 dB at 1 V/nm) as T and R (also in dB), one obtains:  
\[R_{1}=T+R \tag{5}\]  
\[R_{2}=T-\alpha_{t}\,X_{1}-D_{1}-T_{c}+R \tag{6}\]  
\[R_{3}=T-\alpha_{t}\,X_{2}-D_{2}-T_{c}+R \tag{7}\]  
where \(\rm X_{1}\) and \(\rm X_{2}\) are the thickness of sample 1 and 2, \(\rm D_{1}\) and \(\rm D_{2}\) are the diffraction correction of sample 1 and 2 given by Equation (8) below and \(\rm T_{c}\) is the transmission coefficient of the sample going from the transmitter to the receiver (Equation (9)), respectively. The diffraction loss \(\rm D\) for sample thickness \(\rm x\) using two circular transducers of active radius, \(\rm a\), is given by:  
\[D=\{[\cos(2\pi/s)-J_{1}(2\pi/s)]^{2}+[\sin(2\pi/s)-J_{1}(2\pi/s)]^{2}\}^{0.5} \tag{8}\]  
where \(\rm s=X\,v_{t}\)/\(f\) a\({}^{2}\), with \(\rm v_{t}\) being the transverse wave velocity. This is due to [PERSON] and [PERSON] [19], who integrated the Lommel integral for a circular piston motion, received over the same sized area, located at distance \(\rm X\) away. In the UT field, Equation (8) has mostly been applied for the cases of longitudinal waves, but [PERSON] [20] showed that it is also applicable to transverse waves since the original Fresnel formulation was for light diffraction. \(\rm T_{c}\) arises from the differences in acoustic impedances of the transducer face (\(\rm Z_{t}\)) and sample (\(\rm Z_{s}\)) and is given by:  
\[T_{c}=(4Z_{t}\,Z_{s})/(Z_{t}+Z_{s})^{2} \tag{9}\]In the present setup, transducers (Olympus V221, 10 MHz, 6.4-mm diameter, Olympus NDT, Waltham, MA, USA) have an integral buffer made of fused quartz with \(Z_{t}=12.91\) Mrays. \(|T_{c}|\) values ranged from 0.01 dB for BK7 glass to 9.07 dB for ultra-high molecular weight polyethylene (UHMWPE). By using two of three Equations (3a-c), one gets three expressions for the transverse wave attenuation coefficient, \(\alpha_{t}\):  
\[\alpha_{t}=(R_{2}+D_{1}-R_{3}-D_{2})/(X_{2}-X_{1}) \tag{10}\]  
\[\alpha_{t}=(R_{1}-R_{2}-D_{1}-T_{c})/X_{1} \tag{11}\]  
\[\alpha_{t}=(R_{1}-R_{3}-D_{2}-T_{c})/X_{2} \tag{12}\]  
Equation (10) leads to the first method, called transmission difference method 1 (TDM-1), using Setups 2 and 3. This approach cancels out the effects of transmission coefficient due to acoustic impedance mismatch and the coupling layer effects also cancel. Another is called transmission difference method 2 (TDM-2) using either Equations (11) or (12). That is, TDM-2 uses one sample of thickness X with Setups 1 and 2 (or 3). This is more convenient than TDM-1 because only one sample is needed, but it requires the \(T_{c}\) term and a coupling layer enters into the obtained \(\alpha_{t}\) value. For many of the sample materials in the present study, only one thickness was available and TDM-2 had to be used. A predecessor of TDMs has been commonly used in geophysics [14; 15; 21; 22; 23].