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<p>In this paper, a&#xa0;mixed dynamic problem for a&#xa0;rigid stamp in an elastic homogeneous half-space in a&#xa0;cylindrical coordinate system is considered. The work is based on Sommerfeld’s ideas for solving the diffraction problem on a&#xa0;mirror segment. On this basis, a&#xa0;new method for solving the dynamic problem for a&#xa0;vibrating rigid stamp is developed. The solution is sought by minimizing the functional. To select a&#xa0;unique solution, existing methods for solving this problem currently use the behavior of the solution in the vicinity of the discontinuity point of the boundary conditions (the condition on the rib). In this paper, following Sommerfeld, to select a&#xa0;unique physically correct solution in the minimized functional, the expressions are reduced to dimensionless form. As a&#xa0;result, the problem is reduced to solving the system of linear algebraic equations with a&#xa0;Hermitian matrix. Software developed on this basis with OpenMP parallelization allows to calculate the seismic wave fields for an arbitrary radius of a&#xa0;rigid stamp. The following result was obtained as a&#xa0;result of the modeling. As the stamp radius increases, the number of waves increases. It should be noted that in Babeshko’s work et&#xa0;al. (1989) mathematically demonstrated the occurrence of multiple Rayleigh surface waves. This confirms the modeling conclusions obtained in this work.</p>

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Seismic wave fields for the rigid stamp in elastic medium. Sommerfeld method

  • 期刊论文

  • 发表日期：

    2025年12月8日

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  • Alexey G. Fatyanov

摘要

In this paper, a mixed dynamic problem for a rigid stamp in an elastic homogeneous half-space in a cylindrical coordinate system is considered. The work is based on Sommerfeld’s ideas for solving the diffraction problem on a mirror segment. On this basis, a new method for solving the dynamic problem for a vibrating rigid stamp is developed. The solution is sought by minimizing the functional. To select a unique solution, existing methods for solving this problem currently use the behavior of the solution in the vicinity of the discontinuity point of the boundary conditions (the condition on the rib). In this paper, following Sommerfeld, to select a unique physically correct solution in the minimized functional, the expressions are reduced to dimensionless form. As a result, the problem is reduced to solving the system of linear algebraic equations with a Hermitian matrix. Software developed on this basis with OpenMP parallelization allows to calculate the seismic wave fields for an arbitrary radius of a rigid stamp. The following result was obtained as a result of the modeling. As the stamp radius increases, the number of waves increases. It should be noted that in Babeshko’s work et al. (1989) mathematically demonstrated the occurrence of multiple Rayleigh surface waves. This confirms the modeling conclusions obtained in this work.