## 3 Results and Discussion  
### Solution Methods  
The wave propagation process in deformable bodies and media is described by differential equations in partial derivatives of hyperbolic type. In the case of one-dimensional movements, they have the form (18) for the soil medium and the form (24) for the pipeline. Numerical methods can only solve them. The solution of these equations has not yet been obtained analytically. There are two approaches to solving them numerically:  
1. It is possible to obtain numerical solutions by the finite difference method directly. In this case, these partial differential equations are directly solved. 2. However, there is another method. Partial differential equations of hyperbolic type have real characteristics and relations on them. In this case, the characteristic relations are already ordinary differential equations. In the case of boundary conditions (30)-(33) and laws of interaction (26), the wave front lines in soil and the pipeline remain linear. The characteristic lines also remain linear on the characteristic planes \(xt\) for soil and the pipeline. The numerical finite difference method's application to these ordinary differential equations significantly increases the accuracy of the solutionsobtained. Previously the authors have used this algorithm in solving many wave problems and showed the high accuracy of the method. Examples of the application of the method are publications [35; 38; 39; 40]. This second approach was also used in this article. The linearity of characteristic lines greatly facilitates the numerical implementation of the problems under consideration. The characteristic relationships in these cases are nonlinear. They are of the following types:  
1. For underground pipeline \[\left. \begin{array}{l}\frac{d\sigma_{\xi}}{dt}-C_{c}\rho_{0c}\frac{d\sigma_{ \xi c}}{dt}\ =\ -C_{c}^{2}\rho_{0c}g(\sigma_{c},\varepsilon_{c})+\chi C_{c}\sigma_{\tau c}\\ \mbox{\it along characteristic lines}\ \frac{dx}{dt}\ =\ C_{c}\\ \frac{d\sigma_{\xi}}{dt}+C_{c}\rho_{0c}\frac{d\sigma_{\xi c}}{dt}\ =\ -C_{c}^{2}\rho_{0c}g( \sigma_{c},\varepsilon_{c})-\chi C_{c}\sigma_{\tau c}\\ \mbox{\it along characteristic lines}\ \frac{dx}{dt}\ =\ -C_{c}\\ \frac{d\sigma_{c}}{dt}-C_{c}^{2}\rho_{0c}\frac{d\varepsilon_{c}}{dt}\ =\ -C_{c}^{2}\rho_{0c}g( \sigma_{c},\varepsilon_{c})\\ \mbox{\it along lines}\ \frac{dx}{dt}\ =\ 0\end{array} \right\}\] (34) where \(g(\sigma_{c},\varepsilon_{c})\ =\ \frac{\varepsilon_{c}}{\eta_{c}}-\frac{E_{D_{c}}E_{c} \left(\varepsilon_{c}-\frac{\varepsilon_{c}}{\eta_{D_{c}}}\right)}{(E_{D_{c}} -E_{S_{c}})\eta_{c}}\), \(\eta_{c}\ =\ \frac{E_{D_{c}}E_{S_{c}}}{(E_{D_{c}}-E_{S_{c}})\mu_{c}}\). 2. For soil half-space \[\left. \begin{array}{l}\frac{d\sigma_{\xi}}{dt}-C_{g}\rho_{0g}\frac{d\sigma_{ \xi c}}{dt}\ =\ -C_{g}^{2}\rho_{0g}g(\sigma_{g},\varepsilon_{g})+\chi C_{g} \sigma_{\tau g}\\ \mbox{\it along characteristic lines}\frac{dx}{dt}\ =\ +C_{g}\\ \frac{d\sigma_{\xi}}{dt}+C_{g}\rho_{0g}\frac{d\sigma_{\xi c}}{dt}\ =\ -C_{g}^{2}\rho_{0g}g( \sigma_{g},\varepsilon_{g})-\chi C_{g}\sigma_{\tau g}\\ \mbox{\it along characteristic lines}\frac{dx}{dt}\ =\ -C_{g}\\ \frac{d\sigma_{\xi}}{dt}-C_{g}^{2}\rho_{0g}\frac{d\varepsilon_{\xi c}}{dt}\ =\ -C_{g}^{2}\rho_{0g}g( \sigma_{g},\varepsilon_{g})\\ \mbox{\it along lines}\frac{dx}{dt}\ =\ 0\end{array} \right\}\] (35) where \(g\left(\sigma_{g},\varepsilon_{g}\right)\ =\ \frac{\sigma_{\xi}}{\eta_{g}}-\frac{E_{D_{c}}E_{S_{c}} \left(\varepsilon_{g}-\frac{\varepsilon_{g}}{\eta_{g}}\right)}{\left(E_{D_{g}} -E_{S_{c}}\right)\eta_{g}}\), \(\eta_{g}\ =\ \frac{E_{D_{c}}E_{S_{c}}}{\left(E_{D_{g}}-E_{S_{c}}\right)\mu_{g}}\). To improve the accuracy of calculations on a computer, proceed to dimensionless variables (with a superscript zero) and parameters according to the formulas  
\[\left.