# Bulk and Rayleigh Waves Propagation in Three-Phase Soil with Flow-Independent Viscosity
## 4 Numerical Examples and Parametric Analysis  
This section will utilize numerical calculation and parametric analysis to analyze the dependence of physical parameters for the soil on the motion behaviors (mainly for the speed and attenuation) of bulk and Rayleigh waves. The value of the soil parameter in the following calculation examples refers to Table 1 [2] unless otherwise specified. When \(t_{s}\) is set to zero, the unsaturated porous viscoelastic model developed in this work can be reduced to the traditional, unsaturated poroelastic model. Thus, the results obtained in this work can be validated through comparing them with the results developed in previous work. The comparison results for the wave velocities of P\({}_{1}\), S, and Rayleigh waves between this work and the work developed by [PERSON] [2] are presented in Figure 3a, which shows that the analytical solutions obtained by this work and [PERSON] [2] have a good agreement. Additionally, [PERSON] [37] carried out acoustic measurements of partial gas saturation in tight sandstones employing a pulse transmission technique, obtaining the wave speeds of P\({}_{1}\) and S waves. Accordingly, this work introduces the physical parameters of tight sandstones [37] and calculates the corresponding wave speeds of P\({}_{1}\) and S waves. The calculated results from this work are compared with the experimental measurement results observed by [PERSON] [37], as portrayed in Figure 3b. The comparison results in Figure 3 have good consistency both quantitatively and qualitatively. Figures 4-6 depict the variations for \(v_{p1}\), \(v_{b}\), \(v_{R}\), \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\) under various values of \(S^{\prime}\) and \(t_{s}\), where \(S^{\prime}\) ranges from 10% to 100% and \(t_{s}\) is taken to be 0 s, 0.5 \(\times\) 10\({}^{-3}\) s, and 1.0 \(\times\) 10\({}^{-3}\) s, respectively. As depicted in Figures 4-6, \(v_{p1}\) is the largest, followed by \(v_{s}\), and then \(v_{R}\). In contrast, the order of attenuation coefficients runs entirely counter to the order of wave speed. As shown in Figures 4a, 5a and 6a, \(v_{p1}\), \(v_{s}\), and \(v_{R}\) decrease obviously with the increase of \(S^{\prime}\) when the foundation soil is partially saturated. Although, there is a sudden sharp increase in \(v_{p1}\) and \(v_{R}\) but not in \(v_{s}\) when the foundation soil reaches complete saturation state. At the same time, as clarified in Figures 4b, 5b and 6b, the variations of \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\) with \(S^{\prime}\) run diametrically counter to that of \(v_{p1}\), \(v_{s}\), and \(v_{R}\) with \(S^{\prime}\). On the other side, the effects of \(t_{s}\) on \(v_{p1}\), \(v_{s}\), \(v_{R}\), \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\) are conspicuous from the variations in Figures 4-6. For P\({}_{1}\), S, and Rayleigh waves, the curves reveal that \(v_{p1}\), \(v_{s}\), \(v_{R}\), \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\) enlarge with the increase of \(t_{s}\) value. In practical engineering, P\({}_{1}\), S, and Rayleigh waves are the main application object, whereas P\({}_{2}\) and P\({}_{3}\) waves (not shown graphically in this work) are challenging to observe in engineering practice due to their slow speed and fast attenuation. Therefore, for the works of wave propagation in unsaturated or saturated foundation soil, it is essential to account for the effect of flow-independent viscosity from the solid skeleton. Additionally, the soil-water characteristic curve (SWCC) is depicted in Figure 7 (\(\phi\) represents the matric suction), illustrating that the increase of \(S^{\prime}\) will reduce the matric suction \(\phi\) of soil. When soil is completely saturated (\(S^{\prime}\) = 100%), \(\phi\) will decrease to 0. Undoubtedly, the effect of \(\phi\) and \(S^{\prime}\) on the motion behaviors (speed and attenuation) of bulk and Rayleigh waves is opposite.