## 6 Analysis of the _u-p_ Equation
\tag{70}\]  
From Equation (69), the potential functions of displacements of P1 and P2 waves have the forms of wave equations. The wave equation of the S wave is the same as Equation (28). The velocities of the three waves refer to Equations (62) and (63). To discuss the application of the _u-p_ formulation, the product of \(k\cdot\omega\) of the permeability and the frequency is used as a variation. To explain the accuracy of the simplified _u-p_ formulation, Figure 3 shows the differences in the velocities of P1, P2, and S waves, obtained from the simplified _u-p_ formulation and the completed _u-w-p_ formulation, respectively. In addition, comparisons of the velocities are conducted in saturated gravel soil, sand soil, clay silt and silt, respectively. In Figure 3, blue shade fields represent the actual range of \(k\cdot\omega\). By discuss the relative errors of the velocities of the P1, P2 and S wave in the actual range, the scope of the application of the _u-p_ formulation can be gotten preliminarily. Figure 3: _Cont._  
In clay silt and silt, value of \(k\cdot\omega\) is less than \(10^{-3}\) and \(10^{-6}\), respectively. In this scope, three velocities obtained from \(u\)-\(p\) formulation are same with them from the \(u\)-\(w\)-\(p\) formulation. The relative errors of the velocities between the two formulations are zeros. In gravel and sand soils, the values of \(k\cdot\omega\) are \(10^{-5}\)-\(10^{2}\) and \(10^{-7}\)-\(1\), respectively. In these scopes, both of the relative errors of the velocities between the two formulations are beyond the relative error limit of 3% in the two soils. Therefore, the application of the \(u\)-\(p\) formulation would be unsuitable in soils with high permeability, like sand and gravel. We mainly discuss the influence of the permeability and frequency on the wave velocities gained from the \(u\)-\(p\) formulation as the value of \(k\cdot\omega\) is between \(10^{-7}\)-\(10^{2}\). In this range, the primary range of the loading frequency is \(10^{-2}\)-\(10^{2}\) Hz, so the range of the permeability is \(10^{-5}-1\) m/s. Figure 4 shows the relations of the permeability, frequency, and the velocities in gravel. By comparing the relative errors of the velocities of P1, P2, and S waves, obtained from the \(u\)-\(p\) and \(u\)-\(w\)-\(p\) formulations, the variation of scope of the \(u\)-\(p\) formulation with increasing permeability and frequency are further discussed. In Figure 4, the relative errors of the P2 wave velocities are the largest of the three waves. Therefore, taking the relative errors of the P2 wave velocities in sand soils as an example (in Figure 4), the relative errors of the three wave velocities obtained, respectively, from the \(u\)-\(p\) formulation and the \(u\)-\(w\)-\(p\) formulation are less than 3% as the permeability \(k\) is less than \(10^{-2}\) m/s and frequency \(\omega\) is less than 1 Hz. In addition, the permeability \(k\) and frequency \(\omega\) satisfy the relation of \(\omega=-100k+101\). Within this field, the wave velocities are mostly closed to the real velocities, and the \(u\)-\(p\) formulation can be safely used. Figure 3: Comparisons of the velocities obtained from \(u\)-\(p\) and \(u\)-\(w\)-\(p\) formulation in the four saturated soils.