## 3 Materials and Experimental Procedures  
Attenuation measurements were conducted on solid samples with two parallel surfaces. Over 300 tests were completed and their results will be reported in the following sections. Most materials tested were on hand from previous research or instructional work, with detailed characteristics for some, but without for most. About a dozen samples were newly acquired for testing, mainly of rocks and wood materials to supplement existing stock. While some had known sources, chemical compositions, and heat treatment history, most were only identifiable by material types. In order to provide indications of material conditions, hardness tests (Vickers and Rockwell) were conducted for metal samples. For others, density was measured. The longitudinal wave velocity was determined for all, giving additional clues to material conditions. In limiting cases when multiple samples are available, heat treatment was applied to find the effects of microstructural changes on attenuation. However, metallography was not used, as suitable sample preparation facilities were unavailable. Attenuation measurements were utilized through transmission setups. This method uses damped, wideband transducers, as in the buffer rod methods of [PERSON] et al. [16], and relies on two transducer setups, as in [26]. However, the present method avoids using the reflectivity parameter in the attenuation calculation [16; 26]. The present approach uses setups shown schematically in Figure 7a. Also included is a photograph of one of the jigs used to hold samples (with an Al rod shown) and transducers (Figure 7b). 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 often called face-to-face testing. Setup 2 contains sample 1 between the transmitter and receiver, while Setup 3 contains 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 dBat 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  
\[R_{1}=T+R \tag{14a}\] \[R_{2}=T-\alpha\,X_{1}-D_{1}-T_{c}+R\] (14b) \[\text{and}\ R_{3}=T-\alpha\,X_{2}-D_{2}-T_{c}+R, \tag{14c}\]  
where \(X_{1}\) and \(X_{2}\) are the thickness of samples 1 and 2, \(D_{1}\) and \(D_{2}\) (= -20 log D) are the diffraction corrections of samples 1 and 2 (D is given in Equation (15) below), and \(T_{c}\) is the transmission coefficient of the sample going from the transmitter to the receiver (Equation (16)). The diffraction loss D for sample thickness x using two circular transducers of the active radius, a, is given by [69]  
\[D=\{[\cos(2\pi/s)-J_{1}(2\pi/s)]^{2}+[\sin(2\pi/s)-J_{1}(2\pi/s)]^{2}\}^{0.5}, \tag{15}\]  
where s = x v\({}_{L}\)/f a\({}^{2}\), with v\({}_{L}\) being the longitudinal wave velocity. This is due to [PERSON] [69], who integrated the Lommel integral for a circular piston motion received over the same sized area, located at distance x. D is frequency-dependent and was first obtained numerically by [PERSON] et al. [70]. Notice that the D calculations relied on the isotropic elasticity theory. In materials with highly anisotropic elastic moduli, such as unidirectional fiber composites, this condition is absent and D is omitted, as it leads to overcorrections. Diffraction correction in more complicated conditions can be obtained by combining multiple Gaussian beams from a piston source [71].