## 4 Analytical Solutions of Semi-Infinite Media
### Non-Dimensionalization and Parameter Discussion  
By introducing one more dimensionless parameter \(\omega=(2\alpha H)/U\), the non-dimensionalized forms of each parts can be expressed as  
\[\Theta_{\rm amb}=\Theta_{0}\bigg{[}\mathrm{erf}\bigg{(}\frac{\xi}{\sqrt{2 \kappa\beta}}\bigg{)}+e^{(\alpha\varsigma+\frac{\kappa\beta\kappa^{2}}{2})} \mathrm{erfc}\bigg{(}\frac{\xi^{2}}{2\kappa\beta}+\omega\sqrt{\frac{\kappa \beta}{2}}\bigg{)}\bigg{]} \tag{38}\]  
\[\Theta^{N}_{12}= \sqrt{\pi}\omega\mathrm{e}^{\omega\varsigma-\xi-\beta(N-\kappa)} \tag{39}\] \[\times\left[\begin{array}{c}\chi\bigg{(}0,\frac{\varsigma}{ \sqrt{2\beta(\kappa-N+1/2)}};\frac{[\xi+\beta(\kappa-N)]^{2}+\mu^{2}}{\varsigma ^{2}},\frac{(1-\omega^{2})\varsigma^{2}}{4},\frac{\omega\varsigma}{2}\bigg{)} \\ -\chi\bigg{(}0,\frac{\varsigma}{\sqrt{2\beta[\kappa-(N+1)]}};\frac{[\xi+\beta( \kappa-N)]^{2}+\mu^{2}}{\varsigma^{2}},\frac{(1-\omega^{2})\varsigma^{2}}{4},\frac{\omega\varsigma}{2}\bigg{)}\end{array}\right]\] \[\Theta^{N}_{22}= \sqrt{\pi}\omega\mathrm{e}^{\omega\varsigma-\xi-\beta(N-\kappa)}\] (40) \[\times\left[\begin{array}{c}\chi\bigg{(}0,\frac{\varsigma}{ \sqrt{2\beta[\kappa-(N+1/2)]}};\frac{[\xi+\beta(\kappa-N+1)]^{2}+\mu^{2}}{ \varsigma^{2}},\frac{(1-\omega^{2})\varsigma^{2}}{4},\frac{\omega\varsigma}{2} \bigg{)}\end{array}\right]\] \[-\chi\bigg{(}0,\frac{\varsigma}{\sqrt{2\beta[\kappa-(N+1)]}}; \frac{[\xi+[\kappa-(N+1)]]^{2}+\mu^{2}}{\varsigma^{2}},\frac{(1-\omega^{2}) \varsigma^{2}}{4},\frac{\omega\varsigma}{2}\bigg{)}\end{array}\right]\]  
If \(\kappa-K\leq 1/2\),  
\[\Theta_{\rm r2}= \Theta^{K}_{12}+\sqrt{\pi}\omega\mathrm{e}^{\omega\varsigma-\xi- \beta(K-\kappa)} \tag{42}\] \[\times\bigg{[}\chi\bigg{(}0,\frac{\varsigma}{\sqrt{2\beta[\kappa -(K+1)]}};\frac{[\xi+\beta[\kappa-(K+1)]]^{2}+\mu^{2}}{\varsigma^{2}},\frac{(1 -\omega^{2})\varsigma^{2}}{4},\frac{\omega\varsigma}{2}\bigg{)}\bigg{]}\]  
specifying when \(K=0\), \(\Theta=\Theta_{\rm amb}+(\Theta_{\rm r1}-\Theta_{\rm r2})\).  
It is easy to see that \(\beta\) characterizes the relative speed and motion range of the heat source, \(\omega\) characterizes the relative intensity of boundary convective heat transfer, and \(\Theta_{0}\) characterizes the difference between initial and ambient temperature. These three comprehensively describe the influence factors of temperature field from various angles.  
Finally, two points are discussed about the temperature field.  
1. If the initial temperature is equal to the ambient temperature, which means \(\Theta_{\rm amb}=0\), then \[\Theta=\sum_{N=0}^{K-1}\Big{[}\left(\Theta^{N}_{11}-\Theta^{N}_{12}\right)+ \left(\Theta^{N}_{21}-\Theta^{N}_{22}\right)\Big{]}+(\Theta_{\rm r1}-\Theta_{ \rm r2})\] (43) Hence, the initial temperature only has an influence on the component \(\Theta\)amb of \(\Theta\) but has nothing to do with the other components.
2. The situation of \(\omega=0\).  
With the two conditions listed as  
1. \(h=0\), which means the adiabatic boundary. \(\omega=0\) leads \(\Theta^{N}_{12}=\Theta^{N}_{22}=\Theta_{r2}=0\).
2. Neglecting the influence of initial temperature, \(\Theta_{\rm amb}=0\).  
Then, the expression of dimensionless surplus temperature degenerates into the form as  
\[\Theta=\sum_{N=0}^{K-1}\left(\Theta_{11}^{N}+\Theta_{21}^{N}\right)+\Theta_{\rm r1} \tag{44}\]  
The equation is the surplus temperature solution in the region \(0<z<+\infty\) of an infinite medium induced by a \(2Q\)-strength point heat source moving along \(x\) axis. Therefore, if \(\Theta_{\rm amb}=0\), it can be inferred that the temperature field of a semi-infinite medium subjected to a periodic-motion point heat source is similar to that in an infinite medium. However, the former temperature values are lower than the latter, due to surface cooling.