## 1 Introduction  
With the wide application of pile foundations in transportation and civil and energy engineering, more and more researchers have paid attention to the topics of pile foundation vibration [1, 2, 3, 4, 5, 6]. The transmission of dynamic loads in an elastic half-space is the theoretical basis for studying specific problems such as pile vibration. This topic has received much attention and stimulated research efforts, which have resulted in beneficial findings [7, 8, 9, 10, 11, 12]. From the perspective of the saturated porous media theory, some researchers have studied the vibration of piles in saturated soil and investigated the important influence of the liquid phase of the soil layer on pile vibration [4, 13, 14]. In the published studies, the piles were treated as a one-dimensional elastic rods [3, 15] or beams [6, 16, 17, 18]. The [PERSON]'s elastic model [19, 20] and poroelastic model [21, 22, 23], as well as the viscoelastic model [11, 24, 25, 26] were adopted to simulate the saturated soil around a pile. The influences of the parameters of the saturated soil, the dimensions of the pile, and the Poisson's ratio of the pile vibration were further analyzed [6, 19, 20, 21, 22]. Some studies also analyzed the relationship between the soil layer structure around the pile and the vibration response of the pile [27, 28, 29]. The above works have greatly promoted the development of pile dynamics and provided theoretical support for the dynamic design of pile foundations. It is noted that in alpine and high-latitude regions, the upper soil layers are always frozen; this is a special kind of soil, with complex properties. When the soil layers are affected by the environmental temperature, especially by a temperature below the freezing point, frozen soil is formed because part of the water it contains turns into ice. Since changes in the ice content of soil have a significant impact on the soil properties, they could also affect the dynamic response of a pile foundation. Therefore, many scholars have also paid attention to the vibration of a pile foundation in a frozen soil layer covering the upper layer of an elastic half-space. [PERSON] et al. [30; 31] adopted [PERSON]'s frozen porous medium and [PERSON]'s porous medium models to simulate frozen and unfrozen soil layers, respectively, realizing a dynamic analysis of end-bearing piles in single- and double-layer frozen porous media. They found that the properties of frozen soil have a great influence on the dynamic response of a pile foundation. [PERSON] et al. [32] assumed that the water in soil gradually condenses into ice when the temperature is close to the freezing point and analyzed the influence of the change in soil ice content on the cementation between solid particles and the dynamic characteristics of a pile foundation. Due to the difference in the thickness of frozen soil layers, there is a large number of non-end-bearing piles that are utilized in frozen soil areas. The engineering importance of non-end-bearing piles has promoted a significant research progress in recent years [33; 34; 35]; however, to the authors' knowledge, there are few research results available on the vertical vibration of non-end-bearing piles in frozen soil layers. To investigate the vertical vibration characteristics of a floating pile in frozen soil areas, this study examined the vertical vibration of a pile embedded in a frozen poroelastic half-space based on the theory of frozen saturated porous media. At first, vertical vibration solutions for the pile in the frozen porous medium were derived by using analytical approaches. Then, the influence of the frozen soil parameters on the dynamic response of the pile were discussed in detailed. The innovation of this paper is the derivation of a theoretical solution for the vertical vibration of a single pile with any type of pile-end support condition in frozen soil.