Geodesic
Sommerfeld’s method for solving the dynamic rigid stamp indentation problem
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 18 (2025) no. 3, pp. 300-308

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The work is based on Sommerfeld's ideas in 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 a functional. Sommerfeld's method is used to select the only physically correct solution. Namely, the expressions in the minimized functional are reduced to dimensionless form. This allowed us to create a method for calculating wave acoustic fields for arbitrary radius of a rigid stamp. Applied to vibration problems, the solution for a small rigid stamp is obtained in explicit form. This allows stable calculation of vibrating wave fields for teleseismic distances. The program created on this basis allows carrying out calculations even on personal computers with OpenMP parallelization. A result of analytical calculations the distinction of wave fields for a stamp and a distributed source of small dimensions are shown.
Keywords: Sommerfeld method, mixed problem, hard stamp, functional minimization, dimensionality equalization, acoustic waves. 
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     title = {Sommerfeld{\textquoteright}s method for solving the dynamic rigid stamp indentation problem},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {300--308},
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     number = {3},
     year = {2025},
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     url = {http://geodesic.mathdoc.fr/item/JSFU_2025_18_3_a1/}
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Alexey G. Fatyanov. Sommerfeld’s method for solving the dynamic rigid stamp indentation problem. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 18 (2025) no. 3, pp. 300-308. http://geodesic.mathdoc.fr/item/JSFU_2025_18_3_a1/