## 1 Introduction  
Studies on heat transfer in media subjected to moving heat sources have been motivated by a broad range of applications. The research directions mainly focus on the heat transfer in various relative-scale media subjected to heat source with different shapes and moving paths. Heat transfer characteristics may appear differentiated among media with the irregular shapes and discrete heating area [1]. Generally, the studies can be classified by relative scales including the infinite [2], the semi-infinite [3] and the finite [4] media. [PERSON] et al. [5] presented a mapped transient infinite element in the global coordinate system, of which heat transfer functions were derived to solve thermal problems in the infinite media. [PERSON] et al. [6] revealed the temperature evolution mechanism of infinite/finite-length cylindrical solids and helped successfully predict the temperature rise in the ball screw for engineering applications. [PERSON] et al. [7] proposed an analytical semi-infinite body model for temperature field description of multi-pass arc weld surfacing. Discussed in a rectangular coordinate system, when the sizes are negligible compared with these of the heat transfer media in all three directions, the heat source can be regarded as a point heat source [8]. Division of the line and the surface heat source [9] obeys the similar rule. In addition, studies defined the spherical [10], the circular [11] and the cylindrical [12] heat sources in their respective coordinate systems. [PERSON] et al. [13] established the common point and linear heat source model to simulate the temperature fields of electron beam welding and predict the weld shapes. [PERSON] et al. [14] utilized analytical and numerical approaches to model a gas metal arc welding process with the discretely distributed point heat source. According to the various heat generation laws, the temperature solutions to a moving point heat source can also be divided into periodic-type [15], gamma-type [16] variable power and so on. The temperature field distribution in media subjected to straight [17], alterable-direction [18] and specific included angle motion heat sources have also been investigated; however, few studies considering the reciprocating characteristics of heat source can be seen in our visible references. For the specific media subjected to an excitation heat source with constant unidirectional velocity, the transient heat conduction equation is commonly utilized to derive the corresponding temperature-response function, of which the simplified solution has been demonstrated effective by transforming into a moving coordinate whose origin moving at the coincident velocity with the heat source [16]. [PERSON]'s function method is one of the most significant tools for applying field theory in different branches of physic systems [19]. In the light of heat sources with the reciprocating characteristics, [PERSON]'s functions are no doubt able to construct a suitable mathematical description for the temperature field distribution. [PERSON]'s functions can be utilized to deal with all kinds of unsteady-state heat conduction equations with heat source, non-homogeneous boundary and specific initial conditions. [PERSON]'s function method is also known as pulse decomposition or point source superposition method, which decomposes the physical quantities into pulses according to the time and space instead of the spectrum. The decomposed active factors including sources, boundary and initial disturbances are uniformly regarded as point sources. Sources generate fields and the heat conduction equations formed as partial differential are used to describe the distribution and variation of temperature fields. The early investigations mainly focused on heat generation laws, shapes of heat sources, boundary conditions, etc., barely touching on the reciprocating motion of heat sources. In this work, periodically reciprocating characteristics of point heat source has been taken into prior consideration.