## 1 Introduction
Data from both groups exhibited linear frequency dependence in the MHz range. However, the damping factor started to increase below 1 MHz by a factor of 3 to 5, showing a maximum at 50 Hz (3 Hz for torsional vibration) [7]. For rocks, attenuation studies covered similarly wide frequency ranges, and \(\alpha_{\text{d}}\) was found to show linear frequency dependence from 1 kHz to 1 MHz; that is, Q values were constant [45]. Scattering is another cause of wave attenuation. Rayleigh scattering that depends on the fourth power of frequency, \(\ell^{4}\), arises from random scattering centers, such as grain boundaries and distributed second phase particles [8]. This occurs when the distance between scattering centers, d, is much less than \(\lambda\), or \(\lambda>>\) d. For single phase alloys, polycrystalline grain size is used as d, which is of the order of 10 to 100 \(\upmu\)m for most structural metallic alloys. Thus, this Rayleigh scattering effect, denoted by the attenuation coefficient due to scattering, \(\alpha_{\text{s}}\), becomes significant above 5 to 10 MHz. Because of the 4 th power frequency dependence, this attenuation effect shows a steep decrease with decreasing frequency, often becoming negligible in the low MHz region. This behavior was reported by [8; 18; 20; 46]. Using a constant C\({}_{\text{R}}\) for Rayleigh scattering, we have  
\[\alpha_{\text{s}}=C_{\text{R}}\,\ell^{4}. \tag{3}\]  
Figure 1: Attenuation coefficient vs. frequency for polymethyl methacrylate (PMMA) reads from the literature. Blue circle: [PERSON] [19]; red +: [PERSON] and [PERSON] [24]; green circles: [PERSON] et al. [25]; purple triangles: [PERSON] et al. [26]. Blue dotted line represents the linear regression fit to all the data points. [PERSON] and [PERSON] [8] used the sum of \(\alpha_{\mathrm{d}}\) and \(\alpha_{\mathrm{s}}\) to represent their attenuation data for large-grained Al. This is given as  
\[\alpha=\alpha_{\mathrm{d}}+\alpha_{\mathrm{s}}=C_{\mathrm{d}}\;\mathrm{f}+C_{ \mathrm{R}}\;\mathrm{f}^{\mathrm{d}}, \tag{4}\]  
and will be referred to as the Mason-McSkimin relation. [PERSON] and [PERSON] [47] formulated a unified theory, which provides a smooth transition between three types of scattering in Rayleigh, stochastic, and geometric regions. Stochastic scattering gives \(\mathrm{f}^{2}\)-dependence, and is applicable when the wave length \(\lambda\) is of the order of the mean spacing of scattering centers, such as grain boundaries and second-phase particles; this is above 20 MHz for typical metals. The geometric scattering types appear to be insignificant in usual NDE applications below 20 MHz and are not considered in this work. Other power law attenuation effects may exist and can be described as \(\alpha_{\mathrm{n}}=C_{\mathrm{n}}\;\mathrm{f}^{\mathrm{n}}\). This has been used when it is needed to fit experimental observation. It is known that water attenuation follows the quadratic form, while other liquids exhibit slightly lower n values [1, 48, 49]. For the case of n = 2, attenuation can also arise from dislocation damping in crystalline solids [20, 33, 34, 35, 36], with \(\alpha\) written as  
\[\alpha=C_{2}\;\mathrm{f}^{2}. \tag{5}\]  
That is, the KGL theory predicts quadratic frequency dependence, and it is necessary to investigate the mechanisms of dislocation damping, which predict the linear frequency dependence, as in the case of polymer damping. This will be discussed in the next section. Another quadratic frequency dependence of attenuation was observed in particulate-epoxy composites. [PERSON] et al. [50] observed a sharp rise in attenuation with frequency using 0.3-mm diameter glass spheres finely distributed in epoxy matrix. The \(\alpha\) data for the case of 0.451 volume fraction is plotted in Figure 2a after converting the unit for \(\alpha\) into dB/m. A second-order polynomial curve fits the data well, with \(\mathrm{R}^{2}=0.987\).