## 2 Dislocation damping  
Dislocation damping has been analyzed based on the Koehler vibrating string model [33]. Koehler-Granato-Lucke (KGL) formulation showed a linear frequency dependence of damping factor \(\eta\) below the resonance frequency of around 20 MHz [34; 35; 36]. In terms of attenuation coefficient \(\alpha\), this becomes quadratic frequency dependence through Equation (1). [PERSON] and [PERSON] [57] derived an explicit equation for this linear part between logarithmic decrement (\(\Delta=\pi\,\eta\)) and angular frequency. The logarithmic decrement for Cu was measured in multiple studies at kHz frequencies. At 20 to 25 kHz, \(\Delta\) was found to vary on the low side, ranging from 0.001 [58] to 0.0037 [57] and as high as 0.08 [59], with a median value of 0.007. This \(\Delta\) range agreed with the prediction of the KGL model using the dislocation drag factor obtained from theory, experiments and atomistic simulations [60; 61; 62]. Low-frequency internal friction studies used a strain amplitude range of \(10^{-7}\) to \(10^{-5}\)[58; 59; 60], but one study used an amplitude of up to \(3\times 10^{-4}\)[58]. In ultrasonic attenuation studies, the strain amplitude is slightly lower at \(10^{-6}\) to \(10^{-7}\), since most transducers can only generate 1 to 10 nm peak output [63] and samples are over 10 mm. The two approaches cover comparable strain amplitude ranges. These kHz data were extrapolated to 1 MHz and converted to \(\alpha\) values to compare them with ultrasonic attenuation. The \(\alpha\) values from internal friction studies were 0.22 to 8.65 dB/m and the median \(\alpha\) was 1.5 dB/m at 1 MHz. In contrast, ultrasonically obtained \(\alpha\) values for Cu were 3-100 times higher and typically exhibited the linear frequency dependence (except in two cases in a slightly cold-worked plate). The linear frequency dependence of \(\alpha\) was the dominant feature of the present study for various solids. Thus, it is necessary to explore another source of dislocation damping that depends linearly on frequency. When pinned dislocation segments are cyclically stressed, they oscillate, as shown in Figure 4a. It is assumed that they have a constant segment length, L, and are distributed uniformly in space. Low lattice resistance for dislocation motion in solids with non-local bonding is also assumed, as shown by [PERSON] [64]. Starting from a straight line at stress position 0 and at time 0, each segment bows out to the left in the first quarter-cycle under applied stress (Figure 4b) and becomes a series of circle segments at position 1 and time 1. Between 0 and 1, stress increases to the positive peak and work is added by moving the dislocation segments while lengthening the segments, which increases the elastic energy stored in the solid medium. The added work, shown in Figure 4c, is lost as heat. Between times 1 and 2, the dislocation returns to the straight line (position 2). This is driven by the shrinkage of the lengthened dislocation as the dislocation segments are moving against positive stress. This is repeated from positions 2 to 4 in reverse manner, again generating energy loss in the form of work done by the moving dislocation segments, pushed by negative applied stress. As will be shown below, dislocation motion is quasi-static, but the dislocation drag term still acts against the motion. For each cycle, energy loss occurs twice during the positive and negative half-cycles, contributing to damping, which is proportional to frequency. This linear behavior originates from the work caused by dislocation motion during each half-cycle. The amount of work caused by the bow-out process can be calculated from the basic dislocation theory [65]. A bowed-out segment with Burgers vector magnitude, b, is shown in Figure 5 under shear stress \(\tau\) acting on the slip plane. The distance of the bow-out is equal to length from A to B, defined as [AB]. Here, square brackets indicate length.