## 5 Discussions of Wave Velocities, Dispersion and Attenuation Characteristics
### Comparison of Velocities and Frequency Dispersion in Different Saturated Soils  
Four saturated porous medium, saturated gravel soil, sand soil, clay silt, and silt, were selected to analyze propagation characteristics. The mechanical parameters are listed in detail in Table 2.  
Figure 1 shows the relation of the frequency, permeability, and velocities of the P1, P2, and S waves in the case of finite permeability. From Figure 1, as the permeability \(k<1\times 10^{-6}\) m/s, the velocities of the three waves do not change with the increase in permeability and loading frequency. The values always maintain the minimum value, which is close to the minimum velocity under the condition of zero permeability. As the permeability \(k>1\times 10^{-6}\) m/s, with the increase in permeability, the velocities of the three waves increase gradually to their maximum velocities, which are close to the velocities in the condition of infinite permeability. As the permeability is in the range of \(k<1\times 10^{-6}\) m/s, the velocities of the three waves increase with the increase in the loading frequency. This indicates that the three waves are dispersed in the propagation procedure. Moreover, the permeability is proportional to the velocities, and the influence of loading frequency on the velocities is less than the permeability.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Saturated Gravel Soil** & **Saturated Sand Soil** & **Saturated Clay Silt** & **Saturated Silt** \\ \hline Elastic modulus \(E\) (MPa) & 2000 & 700 & 166.8 & 64.7 \\ Shear modulus \(G\) (MPa) & 854.7 & 277.8 & 64.15 & 21.7 \\ Density of fluid \(\rho_{f}\) (kg/m\({}^{3}\)) & 1000 & 1000 & 1000 & 1000 \\ Density of soil \(\rho_{s}\) (kg/m\({}^{3}\)) & 2150 & 2100 & 1970 & 1700 \\ Poisson ratio \(\upsilon\) & 0.17 & 0.26 & 0.3 & 0.49 \\ Bulk modulus of fluid \(K_{f}\) (GPa) & 1.85 & 1.85 & 1.85 & 1.85 \\ Bulk modulus of solid \(K_{S}\) (GPa) & 45 & 30 & 20 & 12 \\ Porosity \(n\) & 0.4 & 0.4 & 0.594 & 0.625 \\ Permeability \(k\) (m/s) & \(3.24\times 10^{-3}\) & \(4.6\times 10^{-4}\) & \(5.79\times 10^{-6}\) & \(6\times 10^{-10}\) \\  & or \(1\times 10^{-12}-1\) & or \(1\times 10^{-12}-1\) & or \(1\times 10^{-12}-1\) & or \(1\times 10^{-12}-1\) \\ Frequency \(f=\omega/2\pi\) (/s) & 0–200 & 0–200 & 0–200 & 0–200 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material parameters of four saturated soils.  
Figure 1: Relations of permeability, frequency and the velocities of three waves in four saturated soils.  
In Figure 1, by contrasting the relations of permeability, loading frequency, and the velocities in the four saturated soils, we can see that different saturated soils do not change the shapes of the velocity curves, but vary the amplitudes of the curves. The harder the soil, the faster the propagation wave is. The permeability coefficients of the grey planes in Figure 1 represent the actual permeability coefficients of the four saturated soils. They are \(k=3.24\times 10^{-3}\), \(4.6\times 10^{-4}\), \(7.9\times 10^{-6}\) and \(6\times 10^{-10}\) m/s, respectively. For the saturated soils with low permeability, such as clay silt and silt, the permeability coefficients are less than \(1\times 10^{-6}\) m/s. The corresponding velocities are close to the minimum velocities and do not change with the loading frequency. Therefore, the minimum wave velocities can be used to replace the actual wave velocities in this condition. For the saturated soils with a high permeability, such as sand and gravel, the velocities fall in between the maximum and minimum velocities. The velocities of the three waves should be calculated by the actual velocity equations in this condition.