## 4 Conclusions  
The critical analysis of the existing theories considers the advantages and disadvantages of seismic resistance of underground pipelines. The results obtained led to the following conclusions.  
1. The main disadvantage of the dynamic theory of seismic resistance of underground pipelines is the neglect of dynamic stress state in soil under seismic wave propagation. The next drawback of the dynamic theory is an inaccurate, approximate accounting for the displacement of the soil medium to which the underground pipeline is embedded.
2. The complete interaction process includes the stages of nonlinear changes in the interaction force (the friction force) by manifesting its peak value and the Coulomb friction. The contact layer of soil undergoes shear deformations until complete structural destruction of the soil contact layer. The interaction force is the friction force, and its peak value does not appear.
3. The problems of seismic resistance of underground pipelines should be considered based on the theory of propagating seismic waves in a soil medium and the interaction of seismic waves with underground pipelines, i.e., based on the wave theory of seismic resistance of underground pipelines.
4. A one-dimensional coupled problem of seismic resistance of underground pipelines under seismic impacts was posed based on the wave theory. An algorithm and a program for the numerical solution of the stated wave problems were developed using the method of characteristics and the method of finite differences.  
Figure 16: Relative displacement depending on time for \(x=5\) m, 10 m, and 15 m (curves 1, 2, and 3, respectively).  
5. An analysis of the laws of interaction of underground pipelines with soil under seismic influences shows that it is necessary to use in the calculations the laws of interaction that account for the complete interaction processes observed in experiments.
6. The analysis of the obtained numerical solutions and the posed coupled problems of the wave theory of seismic resistance of underground pipelines show the occurrence mechanisms of longitudinal stresses in underground pipelines under seismic influences.
7. The results of calculations stated that an account for the dynamic stress normal to the underground pipeline's outer surface leads to multiple increases in longitudinal stress in the underground pipeline. This multiple increase is due to the transformation of the interaction force into an active frictional force, resulting from a greater strain in soil than the one in the underground pipeline.
8. The wave theory's efficiency and reliability are shown in comparison with the dynamic theory of seismic resistance of underground pipelines.  
Conceptualization, K.S.S. and N.I.V.; methodology, K.S.S.; software, K.S.S.; validation, K.S.S.; formal analysis, K.S.S. and N.I.V.; investigation, K.S.S.; resources, K.S.S.; data curation, K.S.S.; writing--original draft preparation, K.S.S.; writing--review and editing, N.I.V.; visualization, K.S.S.; supervision, K.S.S.; project administration, K.S.S.; funding acquisition, N.I.V. All authors have read and agreed to the published version of the manuscript.  
The research is funded by the Ministry of Science and Higher Education of the Russian Federation as part of World-class Research Center program: Advanced Digital Technologies (contract No. 075-15-2020-934 dated 17 November 2020).  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.