# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 2 Buffer-Rod Method
The couplant thickness was measured using pairs of well-polished discs and plates of fused silica, sapphire, Pyrex glass, hardened steel and PMMA, with a 1-\(\upmu\)m resolution micrometer (as nm-resolution devices were unavailable). For ten combinations, the sum of thickness measurements of individual discs or plates and combined thickness with couplant were identical. That is, the couplant thickness was undetectable in all the combinations tested. Thus, the physical thickness was estimated to be 1 \(\upmu\)m, which was the resolution of the micrometer used. Consequently, all seven apparent d values exceeded the estimated couplant thickness by the factor of 4 to 40. The factor was the lowest for sapphire and the highest for PMMA. Large differences between the physical and apparent d-values are difficult to attribute to errors in micrometer measurements when the difference is more than several \(\upmu\)m. It is improbable for the case of PMMA. Thus, it is needed to explore other causes for the differences. A possible source of higher R(_f_) is the non-planar nature of waves arriving at the buffer-sample interface. Theoretical R(_f_) calculations [1,49] relied on the planar wave assumption. For the sound field from a circular transmitter (also called a piston source), it is necessary to include the contributions of non-planar reflection and wave refraction when Z value changes at the interface. Thus, the use of R(_f_) in attenuation testing per the ASTM C1332 demands more detailed analysis of waves impinging on the buffer-sample interface. No such analysis is available at present. In all the cases examined, R(_f_) values can be attributed to the interfacial structure even though exact analysis is unavailable. The apparent d-values were deduced from the observed R(_f_) but appear to exceed the estimate of physical d values, which were within the micrometer resolution. As shown in Figure 1b,c, reflections from the back surface of a sample can be detected and were designated as PE1, PE2, etc. The ASTM C1332 method calculates power spectra of PE1 and PE2 reflections, B\({}_{1}\)(_f_) and B\({}_{2}\)(_f_) (in dB). Then, ultrasonic attenuation, \(\alpha\) (in dB/m) is given by  
\[\alpha=\{\text{B}_{1}(f)+\text{1}\,\text{D}_{1}(f)\,\text{|}+\text{R}(f)-\text {B}_{2}(f)-\text{1}\,\text{D}_{2}(f)\,\text{|}\}/2\text{X} \tag{5}\]where X is sample thickness, and D\({}_{1}\)(\(f\)) and D\({}_{2}\)(\(f\)) are diffraction correction terms (in dB). The D terms were omitted in ASTM C1332 method since it was intended for the high-frequency range above 10 MHz, where these can be ignored. D is given by  
\[\text{D}=20\log[[\cos(2\pi/\text{s})-\text{J}_{1}(2\pi/\text{s})]^{2}+[\sin(2 \pi/\text{s})-\text{J}_{1}(2\pi/\text{s})]^{2}]^{0.5} \tag{6}\]  
where s = xv/\(f\) a\({}^{2}\), with x being the propagation distance from a transmitter of radius, a J\({}_{1}\) represents the Bessel function of the first kind. This is due to [PERSON] [47], who integrated the Lommel integral for a circular piston motion, received over the same sized area, located at distance x away. For pulse-echo tests, the propagation distance corresponds to a multiple of twice the sample thickness, or 2X. It should be noted that the available calculation of diffraction correction cannot be applied to cases with a buffer-sample interface with a discontinuity in Z. In fact, waves pass the interface twice. When attenuation is high, and transducer size is much larger than the wavelength, effects can be minimized, but proper consideration is required. The diffraction correction analysis also assumed that the propagation medium is uniform. That is, no interface exists where Z value changes stepwise, resulting in wave refraction. Thus, the D terms can only be used as approximation. Even when diffraction effects are small, wave refraction persists and this needs to be included in pulse-echo analysis, although this point has not been considered previously.