# Bulk and Rayleigh Waves Propagation in Three-Phase Soil with Flow-Independent Viscosity
## 3 Wavefield Solution for Bulk and Rayleigh Waves
### Bulk Waves  
For the bulk wave propagation in the triphase unsaturated soil, the displacement potentials of solid, liquid, and air phases in Equations (29) and (30) can be expressed as  
\[\mathbf{\varphi}^{ph}=A^{ph}\exp\big{[}i\big{(}\omega t-k_{p}\mathbf{r}\big{)}\big{]} \tag{32}\]  
\[\mathbf{\psi}^{ph}=\mathbf{B}^{ph}\exp[i(\omega t-k_{s}\mathbf{r})] \tag{33}\]  
where \(\omega=2\pi f\), and \(i=\sqrt{-1}\).  
The following formulas are obtained by combining Equations (29)-(32), as \[\begin{pmatrix}\rho\omega^{2}-[(C_{11}+2\mu_{e})+i\omega(C_{14}+2\mu_{v})]k_{p}^{2}& \rho^{l}\omega^{2}-C_{12}k_{p}^{2}&\rho^{a}\omega^{2}-C_{13}k_{p}^{2}\\ \rho^{l}\omega^{2}-(C_{21}+C_{24}i\omega)k_{p}^{2}&v_{2u}\omega^{2}-b_{2u}i \omega-C_{22}k_{p}^{2}&-C_{23}k_{p}^{2}\\ \rho^{a}\omega^{2}-(C_{31}+C_{34}i\omega)k_{p}^{2}&-C_{32}k_{p}^{2}&v_{g} \omega^{2}-b_{g}i\omega-C_{33}k_{p}^{2}\end{pmatrix}\begin{pmatrix}A^{s}\\ A^{w}\\ A^{g}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix} \tag{33}\]  
\[\begin{pmatrix}\rho\omega^{2}-(\mu_{e}+\mu_{p}i\omega)k_{s}^{2}&\rho^{l} \omega^{2}&\rho^{a}\omega^{2}\\ \rho^{l}\omega^{2}&v_{2u}\omega^{2}-b_{u}i\omega&0\\ \rho^{l}\omega^{2}&0&v_{g}\omega^{2}-b_{g}i\omega\end{pmatrix}\begin{pmatrix} \mathbf{B}^{s}\\ \mathbf{B}^{w}\\ \mathbf{B}^{s}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix} \tag{34}\]  
Finally, the characteristic equations for bulk waves in triphase unsaturated viscoelastic soil are derived from Equations (33) and (34) as  
\[\begin{vmatrix}L_{11}&L_{12}&L_{13}\\ L_{21}&L_{22}&L_{23}\\ L_{31}&L_{32}&L_{33}\end{vmatrix}=0 \tag{35}\]  
\[\begin{vmatrix}J_{11}&J_{12}&J_{13}\\ J_{21}&J_{22}&J_{23}\\ J_{31}&J_{32}&J_{33}\end{vmatrix}=0 \tag{36}\]  
with  
\[\begin{vmatrix}L_{11}=\rho\omega^{2}-[(C_{11}+2\mu_{e})+i\omega(C_{14}+2\mu_{v} )]k_{p}^{2},\ L_{12}=\rho^{l}\omega^{2}-C_{12}k_{p}^{2},\ L_{13}=\rho^{l}\omega ^{2}-C_{13}k_{p}^{2},\\ L_{21}=\rho^{l}\omega^{2}-(C_{21}+C_{24}i\omega)k_{p}^{2},\ L_{22}=v_{u}\omega ^{2}-b_{u}i\omega-C_{22}k_{p}^{2},\ L_{23}=-C_{23}k_{p}^{2},\\ L_{31}=\rho^{a}\omega^{2}-(C_{31}+C_{34}i\omega)k_{p}^{2},\ L_{32}=-C_{32}k_{p} ^{2},\ L_{33}=v_{g}\omega^{2}-b_{g}i\omega-C_{33}k_{p}^{2},\\ J_{11}=\rho\omega^{2}-(\mu_{e}+\mu_{p}i\omega)k_{s}^{2},\ l_{12}=\rho^{l} \omega^{2},\ l_{13}=\rho^{l}\omega^{2},\ l_{21}=\rho^{l}\omega^{2},\ l_{22}=v_{2u} \omega^{2}-b_{u}i\omega,\\ J_{23}=J_{32}=0,\ J_{31}=\rho^{a}\omega^{2},J_{33}=v_{g}\omega^{2}-b_{g}i \omega.\end{vmatrix}\]  
For the complex wavenumber of the longitudinal wave, Equation (35) can be solved into three meaningful solutions. This means that three kinds of longitudinal waves exist in unsaturated soil. The three longitudinal waves are usually signed as P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\) waves, and the speed of the P\({}_{1}\) wave is the fastest, the P\({}_{2}\) wave is in the middle, whereas the P\({}_{3}\) wave is the slowest. Similarly, Equation (36) can be solved to one meaningful solution for the complex wavenumber of the transverse wave. That is, only one kind of transverse wave exists in unsaturated soil, usually labeled as S wave. Additionally, the following formulas can be yielded from Equations (33)-(36) as  
\[p^{\text{as}}=\frac{A^{w}}{A^{s}}=\frac{L_{11}L_{23}-L_{21}L_{13}}{L_{22}L_{13 }-L_{12}L_{23}},\ p^{\text{gs}}=\frac{A^{g}}{A^{s}}=\frac{L_{11}L_{22}-L_{21} L_{12}}{L_{23}L_{12}-L_{13}L_{22}} \tag{37}\]  
\[s^{\text{us}}=\frac{\mathbf{B}^{w}}{\mathbf{B}^{s}}=-\frac{J_{21}}{J_{22}},\ s^{\text{gs}}=\frac{\mathbf{B}^{g}}{\mathbf{B}^{s}}=- \frac{J_{31}}{J_{33}} \tag{38}\]  
Generally, the wave speed and attenuation coefficient of bulk waves are defined as  
\[v_{p1}=\frac{\omega}{\text{Re}(k_{p1})},\ v_{p2}=\frac{\omega}{\text{Re}(k_{p2 })},\ v_{p3}=\frac{\omega}{\text{Re}(k_{p3})},\ v_{s}=\frac{\omega}{\text{Re} (k_{s})} \tag{39}\]  
\[\delta_{p1}=\text{Im}(k_{p1}),\ \delta_{p2}=\text{Im}(k_{p2}),\ \delta_{p3}=\text{Im}(k_{p3}),\ \delta_{s}=\text{Im}(k_{s}) \tag{40}\]