## 2 Materials and Methods
### Interaction Laws
This factor is taken into account in law (4) by introducing the parameter \(I_{S}\)  
\[\tau\ =\ K_{x}(\sigma_{N},I_{S})u \tag{14}\]  
where \(I_{S}\ =\ abs\Big{(}\frac{u}{u_{s}}\Big{)}\) is the parameter characterizing the degree of damage of the soil contact layer structure. The range in parameter change is \(0\leq I_{S}\leq 1\). At \(I_{S}\ =\ 0\), the soil contact layer is not destroyed, and at \(I_{S}\ =\ 1\) it is destroyed completely. The change in \(I_{S}\) is irreversible. With these circumstances in view, a function was proposed in [14; 42], which takes into account the changes in \(K_{x}(\sigma_{N},I_{S})\) in the following form  
\[K_{x}(\sigma_{N},I_{S})\ =\ K_{x}^{*}(\sigma_{N})exp(\beta(1-I_{S})) \tag{15}\]  
where \(K_{x}^{*}\) is the function of rigidity (interaction) of the pipe with structurally disturbed soil, \(\beta\) is the dimensionless coefficient characterizing the range of variation in \(K_{x}\). At \(I_{S}\ =\ 0\) from (15) we obtain  
\[K_{x}^{N}(\sigma_{N})\ =\ K_{x}^{*}(\sigma_{N})e^{\beta} \tag{16}\]  
In [14; 42], based on the experimental results, the relation for the function \(K_{x}^{*}(\sigma)\) is proposed  
\[K_{x}^{*}(\sigma_{N})\ =\ K_{N}\sigma_{N} \tag{17}\]  
where \(K_{N}\) is the coefficient of rigidity of the connection of the underground pipeline with the soil contact layer (the coefficient of shear rigidity for the soil contact layer). The value of relative displacement \(u\ =\ u_{g}-u_{c}\) is determined by solving seismic wave propagation problems in the pipeline and soil. This circumstance requires joint consideration of the process of deformation of the underground pipeline and the surrounding soil. In the general case, such a problem is a three-dimensional one. In this statement, the problem of the underground pipeline interaction with soil is rather complicated. In a two-dimensional statement, this problem was considered in [27]. In non-one-dimensional cases, as shown in [38], at the adjacent boundary of soil contact with the underground pipeline, the condition of complete cohesion \(u\ =\ 0\) could be accepted if the soil deformation laws take into account its structural destruction. In the cases where the structural destruction of soil is not considered, and in the cases of one-dimensional statement, it is necessary to use the condition in the form (15) at the underground pipeline-soil contact boundary. In a one-dimensional case, the soil environment around the underground pipeline is considered as a half-space. The spatial coordinate \(x\) coincides with the pipeline axis. The initial section of the half-space (\(x\ =\ 0\)) also coincides with the initial section of the underground pipeline. A seismic plane wave propagating along the soil half-space (in the direction of the pipeline axis) is considered without taking into account the pipeline itself. As a result of solving such a wave problem, all wave parameters for the one-dimensional case are determined along the pipeline axis at each fixed point, i.e., \(\sigma_{g},v_{g},u_{g},\epsilon_{g}\), longitudinal seismic stress, soil particle velocity, soil particle displacement, and soil strain, respectively. They allow determining the forces of interaction of the pipeline with soil at each point along the pipeline axis. At the same time, the problem of the same plane seismic wave propagation in an underground pipeline is solved, taking into account the interaction force (friction) on its outer surface in contact with the soil medium. This statement of the problem is the closest to the real situation. In contrast to the dynamic theory of seismic resistance of underground pipelines, in this case, the values of \(\sigma_{N},v_{g},u_{g},\epsilon_{g}\), and hence \(I_{S}\) are determined from the solution of the wave problem for soil in the pipelines axial direction at each point. Therefore, the theory proposed here is called the wave theory of seismic resistance of underground pipelines. The wave theory is the next stage in developing the theory of seismic resistance of underground pipelines.