# Bulk and Rayleigh Waves Propagation in Three-Phase Soil with Flow-Independent Viscosity
## 1 Introduction  
The wave motion behavior in natural earth foundation is a fundamental scientific problem for theoretical study (e.g., soil dynamics) and practical engineering (e.g., non-destructive examination of natural and artificial materials in foundation soil). Generally, bulk and surface waves are captured in the natural earth foundation. Bulk waves, including longitudinal and transverse waves, are the seismic waves generated by source vibrations and are propagated in the media. Surface waves, such as Rayleigh waves, are secondary waves derived from bulk waves on the surface or layered surface of media. On the other side, the natural earth foundation can be divided into entirely and partially saturated states. The wave propagation behavior in partially saturated soil is significantly more complex than that in completely saturated soil because the capillary pressure and coupling effect will affect the motion behaviors of elastic waves in unsaturated soil [1, 2].  
Undoubtedly, the emergence of Biot theory [3, 4] and mixture theory [5] not only give the sound basis for the research of wave theory but accelerate the research process. For the bulk and Rayleigh waves propagation in saturated soil, researchers including [PERSON] [6], [PERSON] [7], [PERSON] [1], [PERSON] [8, 9], [PERSON] and [PERSON] [10], [PERSON] et al. [11], [PERSON] et al. [12], [PERSON] [13], and [PERSON] et al. [14] implemented several theoretical and experimental works. They drew that the longitudinal, transverse, and Rayleigh waves depend on not only the frequency but the soil parameters such as permeability and soil mass types. Additionally, the applied research on the dynamic response of natural foundation also aroused broad concern from researchers [15, 16, 17, 18], which undoubtedly promotes the basic research on wave propagation in completely saturated soils. In recent years, with the improvement of mathematics solving ability and engineering precision requirements,several researchers, including [PERSON] [19], [PERSON] et al. [20], and [PERSON] et al. [21; 22; 23], have implemented works on the bulk and Rayleigh waves propagation in three-phase partially saturated soil. These works concluded that a kind of longitudinal wave (typically named P\({}_{3}\) wave) exists in three-phase soil, and the liquid saturation has a significant effect on the bulk and Rayleigh waves.  
Additionally, it is usually recognized that the deformation of the soil, even dry soil, is related to time. That is, the soil viscosity is dependent on the soil skeleton viscosity. Nonetheless, most of the existing works on wave propagation in unsaturated soils only take into account the fluid viscosity but seldom mention the effect of solid skeleton viscosity. Appreciatively, the relationship between soil-structure response and the soil skeleton viscosity has been proposed by researchers [24; 25]. Moreover, recently, the dependence of soil skeleton viscosity on the motion behaviors of elastic waves in saturated soils has been studied with an analytical solution [26]. To overcome the lack of previous research on the relationship between the soil skeleton viscosity (flow-independent viscosity) and the elastic wave propagation in unsaturated soil, this work simplifies the unsaturated saturated soils to the multiphase mixture consisting of soil particles, water, and air. The viscoelastic wave equations are established with the Biot model, mixture theory, and the generalized Kelvin-Voigt model. Then, the characteristic formulas of longitudinal, transverse, and Rayleigh waves are yielded with theoretical derivation. Finally, the relationships between the motion behaviors (speed and attenuation) of each wave and the soil parameters are graphed and discussed by utilizing calculation examples and parametric analysis.