# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 3 Experimental Procedures  
The experimental methods for longitudinal and transverse wave attenuation measurements used pairs of damped, wideband transducers as both transmitter and receiver. A \begin{table}
\begin{tabular}{c c c c} \hline \hline Sample Coupling & gel under compression & gel under compression & dry contact \\ \hline Transducer Input & pulse & pulse & frequency sweep \\ \hline Transducer Count & 2 & 1 & 2 \\ \hline Diffraction Correction & always used & not used at high & not used \\ \hline Best method for & ultrasonic attenuation & high frequency & elastic constants of \\  & including transverse & attenuation, longitudinal & anisotropic solids \\  & mode & mode only & ultrasonic attenuation \\ \hline Can be used for & elastic constants & & \\ \hline Problem Areas & sample parallelism & sample/buffer & sample preparation, \\  & & coefficients, refraction at & interface \\ \hline References & [43; 44] & [30; 31; 32; 33] & [17; 18; 19] \\ \hline \multicolumn{4}{l}{T and R stand for transmitter and receiver.} \\ \end{tabular}
\end{table} 
Table 2: Comparison of Test Methods. pair each is needed for the longitudinal and transverse wave modes as noted previously in [43; 44], which also described the methods in detail. A summary of the methods follows. Two through-transmission setups are used as shown in Figure 6a. Setup 1 is for direct contact of the transmitter and receiver, giving the voltage output \(V_{1}\) from the receiver as the transmitter is excited by a pulser. Setup 2 places a sample between the transmitter and receiver (see Figure 6b), yielding \(V_{2}\). By applying a fast Fourier transform (FFT) on \(V_{1}\) and \(V_{2}\), one gets the corresponding frequency domain spectra, \(R_{1}\) and \(R_{2}\), 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 \(T_{x}\) and \(R_{x}\) (also in dB), one obtains:  
\[R_{1}=T_{x}+R_{x} \tag{8}\]  
\[R_{2}=T_{x}-\alpha\chi-\mid D\mid-Tc+R_{x} \tag{9}\]  
where \(\alpha\) is the attenuation coefficient in dB/m, \(X\) is the thickness of the sample (in m), D is the diffraction correction given by Equation (6) and \(T_{c}\) is the transmission coefficient of the sample going from the transmitter to the receiver (Equation (10)), respectively. Both D and \(T_{c}\) are expressed in dB. Tc arises from the differences in acoustic impedances of the transducer face (\(Z_{t}\)) and sample (\(Z_{s}\)) and is given by:  
\[T_{c}=-20\log{(4Z_{t}Z_{s})}/(Z_{t}+Z_{s})^{2}, \tag{10}\]  
From Equations (8) and (9), attenuation coefficient \(\alpha\) can be obtained by  
\[\alpha=(R_{1}-R_{2}-\mid D\mid-T_{c})/\chi. \tag{11}\]  
The above formulas apply to the longitudinal mode, and t suffixes are needed for \(\alpha\) and \(\
u\) in the case of the transverse mode. Note that explicit conversion steps to dB magnitude values of D and \(T_{c}\) were previously not included in [43; 44]. Note that attenuation coefficient, \(\alpha\), expressed in dB/m is equal to \(\alpha=8.69\alpha^{\prime}\), where \(\alpha^{\prime}\) is expressed in Nepers  
Figure 6: (**a**) Schematics of the attenuation experiment. Top: pulser with output waveform. 1: direct contact test. 2: through-transmission test. (**b**) Photograph of the test jig with malachite sample between two 10-MHz transducers. Transducers are manually aligned and pressed together using a 12.7 mm diameter bolt and a hemisphere. Finger-tightening of the bolt generates \(\sim\)200 N force. unit (1 Np = 8.69 dB) as Np/m. \(\alpha\) is commonly used in NDE, while \(\alpha^{\prime}\) is related to damping factor n (= Q\({}^{-1}\)) as \(\alpha^{\prime}=\pi\,\eta\,f\)/v. Transverse wave attenuation tests used a pair of transducers (Olympus V221, 10 MHz, 6.4 mm diameter, Olympus NDT). These have an integral buffer made of fused quartz with Z\({}_{4}\) = 12.91 Mrayl. For longitudinal wave attenuation tests, a pair of transducers (Olympus V111, 10 MHz, 12.7 mm diameter, Olympus NDT) were used.