# Surface Motion for P-Wave Scattering by an Arbitrary-Shaped Canyon in Saturated Half-Space
## 6 Conclusions
When \(i=1\), \(C_{n}^{(1)}(\cdot)=J_{n}(\cdot)\) and \(i=3\), \(C_{n}^{(3)}(\cdot)=H_{n}^{(1)}(\cdot)\)  
\[E_{11,1}^{(i)}(n,k_{1}r)=\left[\left(n^{2}+n\right)-M_{1}\frac{k_{1}^{2}r^{2}} {2}\right]C_{n}^{(i)}(k_{1}r)-k_{1}rC_{n-1}^{(i)}(k_{1}r) \tag{11}\]  
\[E_{12,1}^{(i)}(n,k_{2}r)=\left[\left(n^{2}+n\right)-M_{2}\frac{k_{2}^{2}r^{2} }{2}\right]C_{n}^{(1)}(k_{2}r)-k_{2}rC_{n-1}^{(1)}(k_{2}r) \tag{12}\]  
\[E_{13,1}^{(i)}(n,k_{1}r)=n\Big{[}k_{t}rC_{n-1}^{(i)}(k_{t}r)-(n+1)C_{n}^{(i)} (k_{t}r)\Big{]} \tag{13}\]  
\[E_{11,2}^{(i)}(n,k_{1}r)=-\Bigg{[}\left(n^{2}+n\right)+N_{1}\frac{k_{1}^{2}r^{ 2}}{2}\Bigg{]}C_{n}^{(i)}(k_{1}r)+k_{1}rC_{n-1}^{(i)}(k_{1}r) \tag{14}\]  
\[E_{12,2}^{(i)}(n,k_{2}r)=-\Bigg{[}n^{2}+n+N_{2}\frac{k_{2}^{2}r^{2}}{2}\Bigg{]} C_{n}^{(i)}(k_{2}r)+k_{2}rC_{n-1}^{(i)}(k_{2}r) \tag{15}\]  
\[E_{13,2}^{(i)}(n,k_{t}r)=n\Big{[}-k_{t}rC_{n-1}^{(i)}(k_{t}r)+(n+1)C_{n}^{(i)} (k_{t}r)\Big{]} \tag{16}\]  
\[E_{21}^{(i)}(n,k_{1}r)=n\Big{[}k_{1}rC_{n-1}^{(i)}(k_{1}r)-(n+1)C_{n}^{(i)}(k_ {1}r)\Big{]} \tag{17}\]  
\[E_{22}^{(i)}(n,k_{2}r)=n\Big{[}k_{2}rC_{n-1}^{(i)}(k_{2}r)-(n+1)C_{n}^{(i)}(k_ {2}r)\Big{]} \tag{18}\]\[E_{23}^{(i)}(n,k_{t}r)=-\left[n^{2}+n-\frac{(k_{t}r)^{2}}{2}\right]C_{n}^{(j)}(k_{ t}r)+k_{t}rC_{n-1}^{(i)}(k_{t}r) \tag{111}\]  
\[E_{11,1}^{(i)}(n,k_{1}r)=\left[\left(n^{2}+n\right)-M_{1}\frac{k_{1}^{2}r^{2}} {2}\right]C_{n}^{(i)}(k_{1}r)-k_{1}rC_{n-1}^{(i)}(k_{1}r) \tag{11}\]0  
\[E_{11,1}^{(i)}(n,k_{1}r)=\left[\left(n^{2}+n\right)-M_{1}\frac{k_{1}^{2}r^{2}} {2}\right]C_{n}^{(i)}(k_{1}r)-k_{1}rC_{n-1}^{(i)}(k_{1}r) \tag{11}\]1  
\[E_{11,1}^{(i)}(n,k_{1}r)=\left[\left(n^{2}+n\right)-M_{1}\frac{k_{1}^{2}r^{2}} {2}\right]C_{n}^{(i)}(k_{1}r)-k_{1}rC_{n-1}^{(i)}(k_{1}r) \tag{11}\]2  
\[E_{11,1}^{(i)}(n,k_{1}r)=\left[\left(n^{2}+n\right)-M_{1}\frac{k_{1}^{2}r^{2}} {2}\right]C_{n}^{(i)}(k_{1}r)-k_{1}rC_{n-1}^{(i)}(k_{1}r) \tag{11}\]3  
\[E_{11,1}^{(i)}(n,k_{1}r)=\left[\left(n^{2}+n\right)-M_{1}\frac{k_{1}^{2}r^{2}} {2}\right]C_{n}^{(i)}(k_{1}r)-k_{1}rC_{n-1}^{(i)}(k_{1}r) \tag{11}\]4  
\[R^{1}=-\sum\limits_{n=0}^{\infty}\left\{ \begin{array}{l}\left[a_{n}E_{11,1}^ {(1)}(n,k_{1}r)+c_{n}E_{12,1}^{(1)}(n,k_{2}r)-f_{n}E_{13,1}^{(1)}(n,k_{t}r) \right]\sin n\theta\\ +\left[b_{n}E_{11,1}^{(1)}(n,k_{1}r)+d_{n}E_{12,1}^{(1)}(n,k_{2}r)+e_{n}E_{13,1}^{(1)}(n,k_{t}r)\right]\cos(n\theta)\\ -\sum\limits_{n=0}^{\infty}\left\{\begin{array}{l}\left[a_{n}E_{21}^{(1)}(n,k_{1}r)+c_{n}E_{22}^{(2)}(n,k_{2}r)+f_{n}E_{23}^{(1)}(n,k_{t}r)\right]\cos(n \theta)\\ +\left[-b_{n}E_{21}^{(1)}(n,k_{1}r)-d_{n}E_{22}^{(1)}(n,k_{2}r)+e_{n}E_{23}^{(1 )}(n,k_{t}r)\right]\sin(n\theta)\end{array}\right\}\sin(n\theta)\end{array} \right\}\cos\alpha \tag{117}\]  
\[L_{n}^{21}=E_{11,2}^{(3)}(n,k_{1}r)\sin n\theta\sin\alpha+E_{21}^{(3)}(n,k_{1} r)\cos n\theta\cos\alpha \tag{118}\]  
\[L_{n}^{22}=E_{12,2}^{(3)}(n,k_{2}r)\sin n\theta\sin\alpha+E_{22}^{(3)}(n,k_{2} r)\cos n\theta\cos\alpha \tag{119}\]  
\[L_{n}^{23}=E_{13,2}^{(3)}(n,k_{t}r)\cos n\theta\cos\alpha+E_{23}^{(3)}(n,k_{t }r)\sin n\theta\cos\alpha \tag{120}\]  
\[R^{2}=-\sum\limits_{n=0}^{\infty}\left\{ \begin{array}{l}\left[a_{n}E_{11,2}^ {(1)}(n,k_{1}r)+c_{n}E_{12,2}^{(1)}(n,k_{2}r)-f_{n}E_{13,2}^{(1)}(n,k_{t}r) \right]\sin n\theta\\ +\left[b_{n}E_{11,2}^{(1)}(n,k_{1}r)+d_{n}E_{12,2}^{(1)}(n,k_{2}r)+e_{n}E_{13,2}^{(1)}(n,k_{t}r)\right]\cos(n\theta)\\ -\sum\limits_{n=0}^{\infty}\left\{\begin{array}{l}\left[a_{n}E_{21}^{(1)}(n,k_{1}r)+c_{n}E_{22}^{(1)}(n,k_{2}r)+f_{n}E_{23}^{(1)}(n,k_{t}r)\right]\cos(n \theta)\\ +\left[-b_{n}E_{21}^{(1)}(n,k_{1}r)-d_{n}E_{22}^{(1)}(n,k_{2}r)+e_{n}E_{23}^{(1 )}(n,k_{t}r)\right]\sin(n\theta)\end{array}\right\}\cos\alpha\end{array} \right\}\sin\alpha \tag{121}\]  
\[-\sum\limits_{n=0}^{\infty}\left\{ \begin{array}{l}\left[a_{n}E_{21}^{(1)}(n,k_{1}r)+c_{n}E_{22}^{(1)}(n,k_{2}r)+f_{n}E_{23}^{(1)}(n,k_{1}r)\right]\cos(n \theta)\\ +\left[-b_{n}E_{21}^{(1)}(n,k_{1}r)-d_{n}E_{22}^{(1)}(n,k_{2}r)+e_{n}E_{23}^{( 1)}(n,k_{t}r)\right]\sin(n\theta)\end{array}