## 4 Results and Discussion
### Attenuation Behavior of Fiber-Reinforced Materials
Seven other tests exhibited this Biwa relation. As noted earlier, this term comes from the scattering from parallel fibers. The next five tests (tests R5-R9) were for another purltruded GFRC measuring 20 mm in diameter (Goldworthy Engineering, El Segundo, CA, USA; 61.5% volume of E-glass fibers, epoxy matrix). Four samples of varying lengths were used, giving three TDM-1 and two TDM-2 test results. The C\({}_{\text{d}}\) values ranged from 99.1 to 121, averaging 111.7 (\(\pm\)9.2) dB/m/MHz. R\({}^{2}\) values averaged 0.931 (ranging from 0.837 to 0.971). Due to size limitation, the T direction was not tested. Two GFRC plate samples were tested (tests R10-R14). One was a UD type (vinylester resin, Showa R802) [101] and the other was a woven roving XP-GFRC with an epoxy matrix (Electrophy, El Segundo, CA, USA). The former GFRC showed higher attenuation than epoxy-based GFRCs, as it used a more viscous resin matrix. It also exhibited the Biwa relation for both T and S directions. The latter was made for printed circuit boards. Both had lower fiber contents and higher attenuation was observed, as listed in Table 7 The attenuation ratio between the fiber and normal fiber directions was 2.2 to 2.9. Perhaps because GFRCs are not used in aerospace applications, reliable GFRC attenuation data based on accepted ultrasonic methods are scarce [7]. [PERSON] [102] examined void effects on attenuation and reported 230 dB/m at 1 MHz for GFRC with a 3% void. This composite was made of woven  
Figure 18: Fast Fourier transform (FFT) output and attenuation spectra of uni-directional glass-fiber-reinforced composite (UD-GFRC) from TDM-1 (purple) and TDM-2 (green). Solid red curve: FFT output of 33.5 mm sample; blue: FFT output of 152 mm sample; red dotted line: diffraction correction (not used); green dash: diffraction correction added to TDM-2 spectrum (not used). rovings of E-glass fibers with a polyester matrix. Measurement details were not given in [102], but it seems that this test was in the S direction, so the \(\eta\) value was estimated to be 0.024. Another work gave 275 dB/m at 1 MHz for GFRC, except details were again unavailable [103]. Using complex elastic moduli measurements, two test results for UD-GFRC were reported [104,105]. In the fiber direction, the observed \(\eta\) values ranged from 0.033 to 0.056. These agree with the results of tests R10 and R11, but are two to four times larger than those of pultruded UD-GFRC samples (tests R1-R9). Since the \(\eta\) values for PMMA using the complex elastic moduli measurements were also three to six times higher than the results from the usual ultrasonic methods [7], this comparison may be of questionable merit. Thus, GRFC results cannot be compared directly to previous test results. However, these attenuation values appear reasonable in comparison to CFRC results published [7] and the present data. Plates of four CFRC materials were used for this study. CFRC with AS-4 fiber was supplied as plates by NASA Langley Research Center, Langley, VA, USA. The matrix was a 3501-6 epoxy. A UD plate was used for this work. The PMR-15 polyimide matrix AS-4 CFRC was obtained in plate form in the 1980s, but details are unavailable. Two other GFRC plates were made in-house using prepregs of G50 and T700 fibers (see [106] for detailed information for G50 CFRC, which was hot-pressed and had 60% fiber volume). General purpose T700 prepregs were obtained from Torayca (Tokyo, Japan), which contained 2501 epoxy resin and were out-of-autoclave cured at 130 \({}^{\circ}\)C. From its density, samples contained 0.41 volume fraction of T700 carbon fibers. AS-4 CFRC showed \(\mathrm{C_{d}}\) values ranging from 91 in the L direction to 247 dB/m/MHz in the T direction. The attenuation spectra are given in Figure 19. The L-spectrum is linear while the T-spectrum fits the Biwa relation. \[\alpha=90.6\ \mathrm{f} \tag{25a}\] \[\alpha=247\ \mathrm{f}+4.94\ \mathrm{f}^{3}. \tag{25b}\]  
The L-spectrum is more discontinuous than usual, but linear regression results in R\({}^{2}\) = 0.883.