## 1 Introduction
Here, \(2\pi\eta\) is the ratio of energy dissipated per cycle to maximum energy stored per cycle; \(\eta\) is often denoted as \(\mathrm{Q^{-1}}\) in seismology [32] and in electrical engineering, and \(\eta\) is also equal to the loss tangent, \(\tan\delta\). This \(\tan\delta\) is defined as the ratio of the imaginary part (E\({}^{\prime}\)) to the real part (E\({}^{\prime}\)) of a complex elastic modulus, \(\mathrm{E^{*}}=\mathrm{E^{\prime}}-i\mathrm{E^{\prime}}\), with \(i2=-1\). The damping factor (or loss tangent) is often used in dealing with vibration damping at lower frequencies, especially in polymers, where attenuation arises from viscous damping or hysteretic behavior. When \(\eta\) is independent of frequency, the attenuation coefficient due to viscous damping, \(\alpha_{\mathrm{d}}\), is given by  
\[\alpha_{\mathrm{d}}=\mathrm{C_{d}}\;\mathrm{f}=\pi\,\eta/\lambda=\pi\,\eta\; \mathrm{f}/\
u_{\mathrm{L}}, \tag{2}\]  
where f is the frequency and \(\
u_{\mathrm{L}}\) is the longitudinal wave velocity; that is, attenuation coefficient \(\alpha_{\mathrm{d}}\) increases linearly with frequency, with \(\mathrm{C_{d}}\) as a constant for attenuation due to damping of longitudinal waves. This \(\alpha_{\mathrm{d}}\), which is proportional to frequency, was initially attributed to elastic hysteresis [8], and early works on damping factors (\(\eta\) = \(\mathrm{Q^{-1}}\)) of metals were tabulated in [32]. The damping phenomena were also called internal friction, for which log decrement \(\Delta=\ln{(\mathrm{A_{o}}/\mathrm{A})}\) was commonly used and \(\Delta=\pi\,\eta\). The origin of \(\alpha_{\mathrm{d}}\) was correlated to dislocation damping in metals [33; 34], known as the Koehler-Granato-Lucke (KGL) theory. Dislocation damping is attributed to phonon and electron drags in dislocation oscillations, thermoelastic damping, and mode conversion losses. This and related topics were thoroughly reviewed in [35; 36; 37], and most available experimental data were compiled in a handbook by [PERSON] et al. [37]. Additional works on ultrasonic absorption not cited in [37] also demonstrated that (i) magnetic damping effects constitute the bulk of \(\alpha_{\mathrm{d}}\) in a steel [38], (ii) dislocation damping increased with plastic strain in Al single crystals [39], and (iii) \(\alpha_{\mathrm{d}}\) in pure Fe diminished with annealing [40]. Material damping at lower frequencies is an important issue in structural design. [PERSON] provided a review with a compilation of the \(\mathrm{Q^{-1}}\) values vs. tensile strength for 25 alloys [41]. More recently, [PERSON] and [PERSON] [42] gave an updated review of this topic and covered high damping metals in depth. In a book on high intensity ultrasound technology, [PERSON] [43] provided damping data and showed that the damping factors of common metals, such as Al, Cu, Ni, Fe, and Zn, exceed 0.01 at high strain amplitudes in excess of \(10^{-4}\); that is, these metals under high intensity ultrasound (below \(\sim\)100 kHz) have damping factors higher than that of polymethyl methacrylate (PMMA) under normal ultrasonic test conditions (i.e., strain amplitude of \(\sim\)\(10^{-7}\)). Methods for testing high damping materials were reviewed in [44]. These works demonstrated that materials with low ultrasonic attenuation, such as Al, Mg, and Fe-Cr alloys, can double as high damping materials at low frequencies and at high strain amplitude [41; 42; 43; 44]. For polymers, this hysteresis effect comes from molecular rearrangements [9; 10]. An example of the frequency dependence of this hysteretic damping is shown in Figure 1 for PMMA. In this plot, data points from multiple sources are combined, including [PERSON] [19], [PERSON] [24], [PERSON] et al. [25], and [PERSON] et al. [26], resulting in a linear regression fit with C\({}_{\text{d}}\) = 85.9 dB/m/MHz (shown by the blue dotted line). The data points formed two groups, with a smaller slope for [PERSON]'s older collected data from six studies and a larger slope for another group that was more recent [24; 25; 26].