## 3 Results and Discussion
### Numerical Calculations
#### 3.2.3 Influence of Shear Stresses on Longitudinal Stresses in Pipeline Cross-Sections  
Consider dependence the shear stresses \(\tau(u)\) in the pipeline sections on longitudinal stresses' values. The laws (26) at \(\sigma_{N}~{}=~{}\sigma_{N}^{5}~{}=~{}const\) were fulfilled on the contact surface of the pipeline with soil (the dynamic component of the normal stress on the pipeline was not taken into account). Therefore, the law (26) was equivalent to the condition of the \"elastic-plastic\" law of interaction involving the Coulomb friction law.  
In contrast to the dependencies \(\tau(u)\) in Figure 3, the interaction patterns had the form shown in Figure 7. Changes in shear stress on the contact surface of the pipeline with soil, in this case, occurred by analogy with \"loading-unloading\" (Figure 7). Curves 1, 2, and 3 in Figure 7 refer to the same sections of the pipeline \(x\)--5.10 and 15 m. Here, the interaction process did not reach the Coulomb law since \(u<u_{*}\).  
The corresponding stress changes in these pipe sections are shown in Figure 8 (curves 0-3). As seen from the comparison of the dependencies \(\sigma(t)\) in Figures 2 and 8, the changes in the law of the pipeline-soil interaction \(\tau(u)\) under considered initial data of the problem have practically no effect on the dependence \(\sigma(t)\). A similar pattern is observed in the dependences \(v_{c}(t)\) and \(u(t)\).  
Figure 8: Change in stresses in pipeline cross-sections for \(x\) = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively).  
Figure 7: Dependence of shear stress on relative displacement without considering \(\sigma_{N}^{D}\) for x = 5 m, 10 m, and 15 m (curves 1, 2, and 3, respectively).  
It should be noted here that in the calculation options considered above, \(\sigma_{N}~{}=~{}const\) is taken in the laws of interaction (26). In these cases, only the changes in the absolute displacement of soil affected the wave parameters (stress, strain, and velocity of pipeline sections) in the pipeline. This led to the formation of an active force of interaction on the pipeline's contact surface with soil, as noted above, so the stresses in the pipeline sections increased. At the same time, the wave parameters in the sections of soil remained unchanged.  
Figure 9 shows the change in stresses in the sections of soil. Curves 0-3 refer, in this case, to the same sections of soil at \(x~{}=~{}0\) m, \(5\) m, \(10\) m, and \(15\) m. As seen from Figure 9, the amplitudes of stress waves did not change with distance in soil. Changes in the interaction force did not affect the values of stress amplitude in soil due to relation (19). Obviously, in the case of \(\sigma_{N}~{}=~{}\sigma_{N}^{S}~{}=~{}const\), the results of calculations, according to the dynamic theory and the wave theories, coincided. In the calculations, according to the dynamic theory, the basic question was how accurately the change in absolute displacements of soil \(u_{g}(t)\) could be set at each point of the contact surface of the underground pipeline with soil. As the answer to this question, the function of change in absolute displacement of soil \(u_{g}(t)\) was determined from the soil motion equation's solution. This naturally increased the accuracy of the results obtained and their reliability.