## 1 Introduction
This is a combination of Equations (2) and (5), or  
\[\alpha=C_{\mathrm{d}}\;\mathrm{f}+C_{2}\;\mathrm{f}^{2}, \tag{6}\]  
with \(C_{\mathrm{d}}=36.3\) dB/m/MHz and \(C_{2}=545\) dB/MHz\({}^{2}\). This will be referred to as the Datta-Kinra relation (with or without \(C_{\mathrm{d}}\) term), as it is based on theoretical works by [PERSON] [51, 52]. These types of spectra were seen for gray cast iron data, in addition to 26 other cases. For unidirectional fiber-reinforced composites, [PERSON] and coworkers [53, 54] modeled longitudinal wave attenuation in the normal fiber direction and provided graphical data of a scattering cross-section vs. frequency (both in normalized values). Reading selective data points from [53] and converting the frequency for the case they analyzed, Figure 2b shows the results. Biwa data can be fitted well (\(\mathrm{R}^{2}=0.998\)) to a cubic frequency dependence, or  
\[\alpha=C_{3}\;\mathrm{f}^{3}, \tag{7}\]  
with \(C_{3}=550\) dB/m/MHz\({}^{3}\). Again, the damping term is usually non-zero, giving the Biwa relation given by  
\[\alpha=C_{\mathrm{d}}\;\mathrm{f}+C_{3}\;\mathrm{f}^{3}. \tag{8}\]  
In the present study, this Biwa relation fitted to eight out of 13 observed composite attenuation datasets in the surface normal direction. In one of them (Kevlar composite), the cubic spectrum (Equation (7)) best described the attenuation. In early attenuation studies of metals, viscous damping effects were shown to be low by [PERSON] and [PERSON] [8] and others (see [7, 14] for earlier works). [PERSON]'s book [1] listed \(\alpha\) values as less than 10 dB/m at 2 MHz for most structural alloys (except for copper-based alloys) in the first edition in 1969. In subsequent years, research activity on this topic, especially below 5 MHz, has been sparse. In addition to absorption studies noted above [37, 38, 39], [PERSON] et al. [18] separated \(\alpha_{\text{d}}\) terms of steels of various carbon (C) contents. The levels of attenuation were low, with C\({}_{\text{d}}\) values of 0.7 to 3.5 dB/m/MHz. In most other attenuation studies, only the scattering terms have been examined. However, some works reported results showing that it is unwise to ignore the damping term. Figure 3 gives some examples. Observed attenuation coefficients are plotted as circles (in blue or red) for a low-C steel (Figure 3a, with data from [55]) and for pure niobium (Figure 3b, with data from [56]). For steel, [PERSON] and [PERSON] [55] normalized 0.2% C steel samples at 900 and 1100 \({}^{\circ}\)C, and obtained \(\alpha\) values for 5 to 9 MHz, fitting them using a power law, with n measuring approximately 1.5. However, such power law fits have no rational basis from existing theories. These data sets can be modeled using the Mason-McSkimin relation (Equation (4)), assuming that the low-frequency region possesses linear frequency dependence from damping, while the Rayleigh scattering law is obeyed at higher frequencies. The fitted data points are shown by blue or red + symbols, while blue or red dots represent the extension of the linear fit at low frequencies. In the niobium (Nb) case, the data from [PERSON] et al. [56] is plotted in Figure 3b, indicating \(\alpha\) values for two grain sizes (blue for 32 \(\upmu\)m and red for 60 \(\upmu\)m). Here, available Nb data jumps from 9 to 20 MHz, giving two frequency bands. [PERSON] et al. [56] compared their data sets with the Stanke-Kino theory [47], but no match was achieved. Using the Mason-McSkimin relation, it is possible to combine damping and scattering components, as shown by blue or red + symbols in Figure 3b. The match was poorer than for steel, however, as the frequency range extends to  
Figure 2: (**a**) Attenuation coefficient vs. frequency for a composite of 0.3-mm diameter glass spheres in epoxy matrix. The volume fraction of glass sphere was 0.451. Data reads from [PERSON] et al. [50]. Quadratic frequency dependence of \(\alpha\) (blue dotted curve) fits the data.