## 6 Computation of the Scattered Fields from a Spherical, a Circular, and a Square Cylindrical Cavity
### Circular Cylindrical Cavity  
Consider now a cylindrical (coin-shaped) cavity oriented perpendicular to the surface of an elastic half-space. The cavity has a depth \(D\) and the radius of the cylinder is \(R_{0}\). The tractions on the bottom and lateral surfaces generate the scattered field of the surface waves. Like in the case of a spherical cavity, it is suitable to use cylindrical coordinates. In a similar procedure, the displacement is found using:  
\[\begin{array}{ll}u_{z}=\frac{A_{0}}{4E}W(z)&\bigg{\{}\int_{0}^{\overline{R}_ {0}}\int_{0}^{2\pi}\big{[}iU(0)\Big{(}d_{6}e^{-\overline{\
u}\overline{D}}+d_{ 7}e^{-\overline{\sigma}\overline{D}}\Big{)}e^{\overline{\tau}_{0}\cos\theta_ {0}}\overline{\tau}_{0}H_{1}^{(1)}\Big{(}\overline{r}^{\prime}\Big{)}\cos \theta^{\prime}\\ &+W(0)\Big{(}d_{8}e^{-\overline{\
u}\overline{D}}+d_{9}e^{-4\overline{D}} \Big{)}e^{\overline{\tau}_{0}\cos\theta_{0}}\overline{\tau}_{0}H_{0}^{(1)} \Big{(}\overline{r}^{\prime}\Big{)}|d\overline{\tau}_{0}d\theta_{0}\\ &+\int_{0}^{\overline{D}}\int_{0}^{2\pi}\big{[}(0)\Big{(}d_{4}e^{-\overline{ \
u}_{0}^{2}}+d_{8}e^{-\overline{\sigma}_{0}^{2}}\Big{)}\overline{R}_{0}\cos \theta_{0}e^{\overline{R}_{0}\cos\theta_{0}}H_{1}^{(1)}(\overline{R}^{\prime}) \cos\Phi^{\prime}\\ &+U(0)d_{10}e^{-\overline{\
u}_{0}^{2}}\overline{R}_{0}e^{\overline{R}_{0}\cos \theta_{0}}\sin\theta_{0}H_{1}^{(1)}(\overline{R}^{\prime})\sin\Phi^{\prime}\\ &iW(0)\Big{(}d_{6}e^{-\overline{\
u}_{0}^{2}}+d_{7}e^{-\overline{\sigma}_{0}^{ 2}}\Big{)}\overline{R}_{0}\cos\theta_{0}e^{\overline{R}_{0}\cos\theta_{0}}H_{0 }^{(1)}(\overline{R}^{\prime})|d\overline{z}_{0}d\theta_{0}\Big{\}},\end{array} \tag{17}\]  
where  
\[\overline{R}^{\prime}=\sqrt{\overline{R}_{0}^{2}+\overline{r}^{2}-2\overline{ \tau}\overline{R}_{0}\cos\theta_{0}}, \tag{18}\]  
\[\cos\Phi\prime=\frac{\overline{\tau}-\overline{R}_{0}\cos\theta_{0}}{\sqrt{ \overline{R}_{0}^{2}+\overline{r}^{2}-2\overline{R}_{0}\cos\theta_{0}}},\ \ \sin\Phi\prime=\frac{\overline{R}_{0}\sin\theta_{0}  
\[\begin{array}{ll}u_{z}=\frac{A_{\alpha\mu}}{4E}W(z)&\left\{\int_{-\overline{R}_{0}}^ {\overline{R}_{0}}\int_{-\overline{R}_{0}}^{\overline{R}_{0}}[iI(0)\big{(}d_{6}e ^{-p\overline{D}}+d_{7}e^{-q\overline{D}}\big{)}e^{i\overline{x}_{0}}H_{1}^{(1) }\Big{(}\overline{r}\Big{)}\cos\theta^{\prime}\right.\\ &\left.-W(0)\big{(}d_{8}e^{-p\overline{D}}+d_{9}e^{-q\overline{D}}\big{)}e^{i \overline{x}_{0}}H_{1}^{(1)}\Big{(}\overline{r}^{\prime}\Big{)}\big{|}d \overline{x}_{0}d\overline{y}_{0}\right.\\ &\left.+\int_{0}^{\overline{D}}\int_{-\overline{R}_{0}}^{\overline{R}_{0}}|2 iU (0)\sin\overline{R}_{0}\Big{(}d_{4}e^{-p\overline{x}_{0}}+d_{5}e^{-q\overline {x}_{0}}\Big{)}H_{1}^{(1)}\Big{(}\overline{r}^{\prime}\Big{)}\cos\theta^{ \prime}\right.\\ &\left.+2 iU(0)\sin\overline{R}_{0}d_{10}e^{-p\overline{x}_{0}}H_{1}^{(1)} \Big{(}\overline{r}^{\prime}\Big{)}\cos\theta^{\prime}\right.\\ &\left.+2W(0)\sin\overline{R}_{0}\Big{(}d_{6}e^{-p\overline{x}_{0}}+d_{7}e^{- q\overline{x}_{0}}\Big{)}H_{0}^{(1)}\Big{(}\overline{r}^{\prime}\Big{)}\big{|}d \overline{y}_{0}d\overline{z}_{0}\right\},\end{array} \tag{20}\]  
where \(\overline{x}_{0}=kx_{0},\ \overline{y}_{0}=ky_{0},\ \overline{z}_{0}=kz_{0}\).