The Sommerfeld Integral in Problems of Simulating the Diffraction of Acoustic Waves Using a Triangular Lattice

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

Analytical solutions are obtained for two problems for a discrete analogue of the Helmholtz equation on a triangular lattice: (1) the problem of radiation from a point source on a plane; (2) the problem of diffraction on a half-line with Dirichlet boundary conditions. It is shown that in these problems, the complete field can be represented as an integral of an algebraic function over a family of contours located on some complex manifold. The solution to the first problem is found as an integral of some differential form over this manifold, and the asymptotics of the far field for this solution is obtained. The second problem is solved using an analogue of the Sommerfeld integral. It is checked that the obtained solution coincides with the solution of this problem by the Wiener–Hopf method.

About the authors

O I. Makarov

Moscow State University, 119991, Moscow, Russia

Email: olegmakarovlip@gmail.com
Россия, 119991, Москва, ГСП-1, Ленинские горы

A. V. Shanin

Moscow State University, 119991, Moscow, Russia

Email: olegmakarovlip@gmail.com
Россия, 119991, Москва, ГСП-1, Ленинские горы

A. I. Korolkov

Moscow State University, 119991, Moscow, Russia

Author for correspondence.
Email: olegmakarovlip@gmail.com
Россия, 119991, Москва, ГСП-1, Ленинские горы

References

  1. Зоммерфельд А. Оптика. М.: ИЛ, 1953. 486 с.
  2. Малюжинец Г.Д. Возбуждение, отражение и излучение поверхностных волн от клина с произвольными поверхностными импедансам // ДАН СССР. 1985. Т. 3. С. 752–55.
  3. Samokish B.A., Dement’ev D.B., Smyshlyaev V.P., Babich V.M. On evaluation of the diffraction coefficients for arbitrary “Nonsingular” directions of a smooth convex cone // SIAM J. Applied Mathematics. 2000. V. 60(2). P. 536–573.
  4. Бабич В.М., Лялинов М.А., Грикуров В.Э. Метод Зомерфельда–Малюжинца в теории дифракции. СПБ.: СПБГУ, 2003. 104 с.
  5. Kosevich A.M. The crystal lattice: phonons, solitons, dislocations, superlattices. John Wiley Sons, 2006.
  6. Conway J.H., Sloane N.J.A. Sphere packings, lattices and groups, volume 290. Springer Science Business Media, 2013.
  7. Yu S.-Y., Wang Q., Zheng L.-Y., He C., Liu X.-P., Lu M.-H., Chen Y.-F. Acoustic phase-reconstruction near the dirac point of a triangular phononic crystal // Applied Physics Letters. 2015. V. 106(15). 151906.
  8. Бобровницкий Ю.И., Томилина Т.М., Бахтин Б.Н., Гребенников А.С., Асфандияров Ш.А., Карпов И.А., Ким А.А. Лабораторная установка для исследования звукопоглощающих покрытий из метаматериалов при скользящем распространении звука и влияние типа источника на их эффективность // Акуст. журн. 2020. Т. 66. № 3. С. 332–341.
  9. Poblet-Puig J., Valyaev V.Yu., Shanin A.V. Boundary element method based on preliminary discretization // Mathematical models and computer simulations. 2014. V. 6(2). P. 172–182.
  10. Поблет-Пуиг Ж., Шанин А.В. О новом численном методе решения задачи излучения акустических волн // Акуст. журн. 2018. Т. 64. № 2. С. 257–265.
  11. Slepyan L.I. Models and phenomena in fracture mechanics. Springer Science Business Media, 2012.
  12. Martin P.A. Discrete scattering theory: Green’s function for a square lattice // Wave Motion. 2006. V. 43(7). P. 619–629.
  13. Berciu M. On computing the square lattice Green’s function without any integrations // J. Physics A: Mathematical and Theoretical. 2009. V. 42(39). 395207.
  14. Arnold J.M. Discrete Green’s functions and functional determinants // 2017 Int. Conf. on Electromagnetics in Advanced Applications (ICEAA), 2017.
  15. Morita T., Horiguchi T. Lattice Green’s functions for the cubic lattices in terms of the complete elliptic integral // J. Mathematical Physics. 1971. V. 12(6). P. 981–986.
  16. Cserti J. Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors // American J. Physics. 1971. V. 12(6). P. 981–986.
  17. Sharma B.L. Diffraction of waves on square lattice by semi-infinite crack // SIAM J. on Applied Mathematics. 2015. V. 75(3). P. 1171–1192.
  18. Sharma B.L. Near-tip field for diffraction on square lattice by crack // SIAM J. on Applied Mathematics. 2015. V. 75(4). P. 1915–1940.
  19. Sharma B.L. Diffraction of waves on triangular lattice by a semi-infinite rigid constraint and crack // International Journal of Solids and Structures. 2016. V. 80. P. 465–485.
  20. Shanin A.V., Korolkov A.I. Sommerfeld–type integrals for discrete diffraction problems // Wave Motion. 2020. V. 97 P. 102606.
  21. Shanin A.V., Korolkov A.I. Diffraction by a Dirichlet right angle on a discrete planar lattice // Quart. Appl. Math. 2022. V 80. P. 277–315.
  22. Шэн-шень Ч. Комплексные многообразия. М.: ИЛ, 1961. 239 с.
  23. Шабат Б.В. Введение в комплексный анализ. Часть 2. СПБ: Лань, 2004. 464 с.
  24. Тихонов А.Н., Васильева А.Б., Свешников А.Г. Дифференциальные уравнения. Курс высшей математики и математической физики. М.: Физматлит, 2002. 256 с.
  25. Steven S. Eighty years of Sommerfeld’s radiation condition // Historia mathematica. 1992. V. 19(4). P. 385–401.
  26. Свешников А.Г., Боголюбов А.Н., Кравцов В.В. Лекции по математической физике. М.: Московский Университет, 2004. 416 с.
  27. Курант Р. Теория функций. М.: Наука, 1968. 646 с.

Supplementary files


Copyright (c) 2023 О.И. Макаров, А.В. Шанин, А.И. Корольков