## 3 Non-Dimensionalization and Parameter Discussions
### Discussions on Dimensionless Parameters  
Non-dimensionalization helps appropriately reduce the number of variables in equations, reveal the mathematical essence of physical formulas, and facilitate the relative value comparisons of various items, so as to simplify the calculation and facilitate the analysis. This is guided by the generalized incomplete Gamma formula [13]  
\[\Gamma\left(\frac{1}{2},x;b\right)=\frac{\sqrt{\pi}}{2}\left[\mathrm{e}^{-2 \sqrt{b}}\mathrm{erfc}(\sqrt{x}-\sqrt{\frac{b}{x}})+\mathrm{e}^{2\sqrt{b}} \mathrm{erfc}(\sqrt{x}+\sqrt{\frac{b}{x}})\right] \tag{22}\]  
where \(\mathrm{erfc}\) is complementary error function. Numerical results with various values of dimensionless parameters are graphically discussed in this section. To reveal the temperature evolution regularities over time, Figure 3a shows that the curves experience a rapid ascending process at the initial period of heat source effect and then gradually flatten out. The dimensionless surplus temperature finally rises to a stable fluctuation state over time. The curves perform the characteristics of almost unchangeable time-averaged quantity but periodically fluctuated instantaneous quantity. The surplus temperature fluctuation mainly results from periodic motion of the point heat source. Dimensionless parameter \(\gamma\) represents the distance away from \(x\) axis. Meanwhile, the smaller the \(\gamma\) values, the higher the time-averaged quantity and the more violent fluctuation the dimensionless surplus temperature develops, the less time used to reach a stable state. As moving along the \(x\) axis, the point heat source impacts much more on the regions nearby (smaller \(\gamma\) values) rather than those far away. The fluctuation amplitude A is defined as the maximum offset quantity compared to the time-average value during oscillation process. Figure 3b clarifies more clearly the relationship between the fluctuation amplitude and the distance parameter. With fixed \(\beta\) values, the smaller the distance away from the \(x\) axis, the larger fluctuation amplitude. The fluctuation amplitudes A over dimensionless velocity and travel parameter \(\beta\) with various \(\gamma\) values show a similar rule, which performs a rapid increase at first and then reaches a stable state when parameter \(\beta\) starts to increase from zero. However, \(\beta\) only affects the fluctuation amplitude within a certain range, in which the larger \(\beta\) values aggravate the oscillation of surplus temperature With smaller \(\gamma\) values, fluctuation amplitudes reach a stable state more rapidly and remain unchanged as parameter \(\beta\) increases. Moreover, the values of A are larger at a stable state, which indicates that the distance parameter is still the main influence factor of fluctuation amplitude compared with parameter \(\beta\). Figure 3c reveals the influence of reduced parameter \(\beta\), the smaller \(\beta\) values, the greater time consumption for surplus temperature to reach a stable state and the lower final time-average value it ascends to. For a reciprocating moving heat source with a fixed travel, the lower velocity means larger time intervals between twice heating effects, and untimely heat compensation for that transferred to low-temperature regions causes the difficulty in temperature rise. Additionally, the value of \(\beta\) is inversely proportional to that of the thermal diffusivity. For regions in the infinite medium with inapparent temperature fluctuation, a lower \(\beta\) value means a faster heat transfer of media, so that the surplus temperature increases more slowly due to the increased heat dissipation. Moreover, more time is needed to reach a steady state, and its steady-state value is higher. Figure 3d displays the evolution process of dimensionless surplus temperature. For suitable \(\beta\) and \(\gamma\) values, the received heat is more than the dissipated.