# Surface Motion for P-Wave Scattering by an Arbitrary-Shaped Canyon in Saturated Half-Space
## 2 Computational Model and Governing Equations
### [PERSON]'s Theory  
Based on [PERSON]'s theory [24; 25; 26], without considering the viscous coupling of solid and liquid two-phase media in saturated soil and the dissipative characteristics of the media, the wave equations of skeleton and fluid in saturated elastic media can be written as follows [35]  
\[\left\{ \begin{array}{c}(\lambda+2\mu)\
abla^{2}\phi+Q\
abla^{2}\Phi=\frac{ \partial^{2}}{\partial t^{2}}(\rho_{11}\phi+\rho_{12}\Phi)\\ Q\
abla^{2}\phi+P\
abla^{2}\Phi=\frac{\partial^{2}}{\partial t^{2}}(\rho_{12} \phi+\rho_{22}\Phi)\end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{c}\mu\
abla^{2}\psi=\frac{\partial^{2}}{\partial t^{2}} (\rho_{11}\psi+\rho_{12}\Psi)\\ 0=\frac{\partial^{2}}{\partial t^{2}}(\rho_{12}\psi+\rho_{22}\Psi)\end{array} \right. \tag{2}\]  
where \(\phi\) and \(\Phi\) are P-wave potential functions in soil skeleton and fluid, and \(\psi\) and \(\Psi\) are SV wave potential functions in soil skeleton and fluid. \(\lambda\), \(\mu\), \(P\), and \(Q\) are the elastic constants of the saturated porous media, which are determined by [PERSON]'s ratio \(v\), porosity \(n_{s}\), critical porosity \(n_{cr}\), critical bulk modulus \(K_{cr}\), solid particle bulk modulus \(K_{g}\), fluid bulk  
Figure 1: The model of P-wave scattering by an arbitrary-shaped canyon in saturated half-space. modulus \(K_{f}\), and soil skeleton bulk modulus \(K_{dry}\). \(K_{dry}=Kcr+(1-n_{s}/n_{cr})(K_{g}-Kcr)\), \(\mu=3(1-2v)K_{dry}/(2+2v)\), \(Q=n_{s}(1-n_{s}-K_{dry}/K_{g})K_{g}/((1-n_{s}-K_{dry}/K_{g})+n_{s}K_{g}/K)\), \(P={n_{s}}^{2}K_{g}/((1-n_{s}-K_{dry}/K_{g})+n_{s}K_{g}/K)\), \(\lambda=3v/(1+v)K_{dry}+Q^{2}/P\). \(\rho_{11}\), \(\rho_{12}\), and \(\rho_{22}\) are the dynamic mass coefficients. \(\rho_{11}=(1-n_{s})\rho_{s}+\hat{\gamma}(1-n_{s})\rho_{f}\), \(\rho_{12}=-\hat{\gamma}(1-n_{s})\rho_{f}\), \(\rho_{22}=n_{s}\rho_{f}+\hat{\gamma}(1-n_{s})\rho_{f}\), where \(\rho_{s}\) is the density of dry soil; \(\rho_{f}\) is the fluid density; \(\hat{\gamma}\) is the interaction coefficient between the reaction soil skeleton and the fluid, depending on the shape of the soil particles, when the soil particles are spherical, \(\hat{\gamma}=0.5\)[35]. By solving Equations (1) and (2), we can derive the wave equations about \(\phi_{1}\), \(\phi_{2}\), and \(\psi_{t}\)[36;37]  
\[\
abla^{2}\phi_{1}+k_{1}^{2}\phi_{1}=0 \tag{3}\]  
\[\
abla^{2}\phi_{2}+k_{2}^{2}\phi_{2}=0 \tag{4}\]  
\[\
abla^{2}\psi_{t}+k_{t}^{2}\psi_{t}=0 \tag{5}\]  
In the above formula, \(\phi_{1}\) and \(\phi_{2}\) are longitudinal waves, corresponding to the potentials of fast and slow P-waves, respectively; \(\psi_{t}\) is the transverse wave, representing the potential of the SV wave. \(k_{1}\), \(k_{2}\), and \(k_{t}\) are the fast P-wave number, slow P-wave number, and SV wave number, respectively, \(k_{i}=\omega/c_{i}\) (\(I=1\), \(2\)), \(k_{t}=\omega/c_{t}\). \(c_{1}\) and \(c_{2}\) are fast P-wave velocity and slow P-wave velocity, respectively, and \(c_{t}\) is SV wave velocity [35]. \[c_{1}=\sqrt{\frac{2A}{B-\left(B^{2}-4 AC\right)^{1/2}}}\,c_{2}=\sqrt{\frac{2A}{B+ \left(B^{2}-4 AC\right)^{1/2}}} \tag{6}\]  
\[c_{t}=\sqrt{\frac{\mu\rho_{22}}{C}} \tag{7}\]  
where \(A=(\lambda+2\mu)P-Q^{2}\), \(B=\rho_{11}P+\rho_{22}(\lambda+2\mu)-2\rho_{12}Q\), \(C=\rho_{11}\rho_{22}-\rho_{12}^{2}\). The P-wave and SV wave potentials in the soil skeleton are expressed as follows  
\[\phi=\sum_{i=1}^{2}\phi_{i} \tag{8}\]  
\[\psi=\psi_{t} \tag{9}\]  
and the P-wave and SV wave potentials in the fluid can be expressed as  
\[\Phi=\sum_{i=1}^{2}\zeta_{i}\phi_{i} \tag{10}\]  
\[\Psi=\xi_{i}\psi_{t} \tag{11}\]  
where \(\xi_{i}=(A/{c_{i}}^{2}-\rho_{11}P+\rho_{12}Q)/(\rho_{12}P-\rho_{22}Q)\) (\(i=1\), \(2\)), \(\zeta_{t}=-\rho_{12}/\rho_{22}\).