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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2024 MCQ for St. Petersburg Mathematical Journal is 0.68.

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Scattering of a surface wave in a polygonal domain with impedance boundary
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by M. A. Lyalinov and N. Y. Zhu
St. Petersburg Math. J. 33 (2022), 255-282
DOI: https://doi.org/10.1090/spmj/1700
Published electronically: March 4, 2022
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Abstract:

The two-dimensional (2D) domain under study is bounded from below by two semi-infinite and, between them, two finite straight lines; on each of the straight lines (segments), a usually individual impedance boundary condition is imposed. An incident surface wave, propagating from infinity along one semi-infinite segment of the polygonal domain, excites outgoing surface waves both on the same segment (a reflected wave) and on the second semi-infinite segment (a transmitted wave); in addition, a circular (cylindrical) outgoing wave will be generated in the far field. The scattered wave field satisfies the Helmholtz equation and the Robin (in other words, impedance) boundary conditions as well as some special integral form of the Sommerfeld radiation conditions. It is shown that a classical solution of the problem is unique. By the use of some known extension of the Sommerfeld–Malyuzhinets technique, the problem is reduced to functional Malyuzhinets equations and then to a system of integral equations of the second kind with integral operator depending on a characteristic parameter. The Fredholm property of the equations is established, which also leads to the existence of the solution for noncharacteristic values of the parameter. From the Sommerfeld integral representation of the solution, the far-field asymptotics is developed. Numerical results for the scattering diagram are also presented.
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Bibliographic Information
  • M. A. Lyalinov
  • Affiliation: St. Petersburg University, 7/9 Universitetskaya nab., 199034, St.Petersburg, Russia
  • Email: lyalinov@yandex.ru, m.lyalinov@spbu.ru
  • N. Y. Zhu
  • Affiliation: Institute of Radio Frequency Technology, University of Stuttgart, Pfaffenwaldring 47, D–70550, Stuttgart, Germany
  • Email: zhu@ihf.uni-stuttgart.de
  • Received by editor(s): September 4, 2020
  • Published electronically: March 4, 2022
  • Additional Notes: The work of the first author was supported by the grant no. 20-01-00451 of the Russian Foundation for Basic Research

    • Dedicated: Dedicated to Vasiliĭ Mikhaĭlovich Babich
  • © Copyright 2022 American Mathematical Society
  • Journal: St. Petersburg Math. J. 33 (2022), 255-282
  • MSC (2020): Primary 74J20
  • DOI: https://doi.org/10.1090/spmj/1700
  • MathSciNet review: 4445759