Diffraction by a Dirichlet right angle on a discrete planar lattice
Authors:
A. V. Shanin and A. I. Korolkov
Journal:
Quart. Appl. Math. 80 (2022), 277-315
MSC (2020):
Primary 12F05, 57Z05, 12E05
DOI:
https://doi.org/10.1090/qam/1612
Published electronically:
February 23, 2022
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Additional Information
Abstract: A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of the work by A. V. Shanin and A. I. Korolkov, Sommerfeld-type integrals for discrete diffraction problems, Wave Motion 97 (2020).
References
- A. V. Shanin and A. I. Korolkov, Sommerfeld-type integrals for discrete diffraction problems, Wave Motion 97 (2020), 102606, 24. MR 4107117, DOI 10.1016/j.wavemoti.2020.102606
- È. I. Zverovič, The Behnke-Stein kernel and the solution in closed form of the Riemann boundary value problem on the torus, Dokl. Akad. Nauk SSSR 188 (1969), 27–30 (Russian). MR 0262519
- È. I. Zverovič, Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces, Uspehi Mat. Nauk 26 (1971), no. 1(157), 113–179 (Russian). MR 0409841
- Yu. A. Antipov and N. G. Moiseev, Exact solution of the two-dimensional problem for a composite plane with a cut that crosses the interface line, Prikl. Mat. Mekh. 55 (1991), no. 4, 662–671 (Russian); English transl., J. Appl. Math. Mech. 55 (1991), no. 4, 531–539 (1992). MR 1141586, DOI 10.1016/0021-8928(91)90018-P
- Y. A. Antipov and V. V. Silvestrov, Factorization on a Riemann surface in scattering theory, Quart. J. Mech. Appl. Math. 55 (2002), no. 4, 607–654. MR 1936120, DOI 10.1093/qjmam/55.4.607
- Askold Khovanskii, Topological Galois Theory, Springer, Berlin, Heidelberg, 2014.
- Simon Donaldson, Riemann surfaces, Oxford Graduate Texts in Mathematics, vol. 22, Oxford University Press, Oxford, 2011. MR 2856237, DOI 10.1093/acprof:oso/9780198526391.001.0001
- B. V. Shabat, Introduction to complex analysis, American Mathematical Society, Providence, RI, 1992.
- Helmut Koch, Introduction to classical mathematics. I, Mathematics and its Applications, vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. From the quadratic reciprocity law to the uniformization theorem; Translated and revised from the 1986 German original; Translated by John Stillwell. MR 1149600, DOI 10.1007/978-94-011-3218-3
- Emil Artin, Galois Theory, 2nd ed., Notre Dame Mathematical Lectures, no. 2, University of Notre Dame, Notre Dame, Ind., 1944. MR 0009934
- George Springer, Introduction to Riemann surfaces, American Mathematical Society, Providence, RI, 2001.
- N. R. T. Biggs, A new family of embedding formulae for diffraction by wedges and polygons, Wave Motion 43 (2006), no. 7, 517–528. MR 2252752, DOI 10.1016/j.wavemoti.2006.04.002
- E. A. Skelton, R. V. Craster, and A. V. Shanin, Embedding formulae for diffraction by non-parallel slits, Quart. J. Mech. Appl. Math. 61 (2008), no. 1, 93–116. MR 2382766, DOI 10.1093/qjmam/hbm026
References
- A. V. Shanin and A. I. Korolkov, Sommerfeld-type integrals for discrete diffraction problems, Wave Motion 97 (2020), 102606, 24. MR 4107117, DOI 10.1016/j.wavemoti.2020.102606
- È. I. Zverovič, The Behnke-Stein kernel and the solution in closed form of the Riemann boundary value problem on the torus, Dokl. Akad. Nauk SSSR 188 (1969), 27–30 (Russian). MR 0262519
- È. I. Zverovič, Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces, Uspehi Mat. Nauk 26 (1971), no. 1(157), 113–179 (Russian). MR 0409841
- Yu. A. Antipov and N. G. Moiseev, Exact solution of the two-dimensional problem for a composite plane with a cut that crosses the interface line, Prikl. Mat. Mekh. 55 (1991), no. 4, 662–671 (Russian); English transl., J. Appl. Math. Mech. 55 (1991), no. 4, 531–539 (1992). MR 1141586, DOI 10.1016/0021-8928(91)90018-P
- Y. A. Antipov and V. V. Silvestrov, Factorization on a Riemann surface in scattering theory, Quart. J. Mech. Appl. Math. 55 (2002), no. 4, 607–654. MR 1936120, DOI 10.1093/qjmam/55.4.607
- Askold Khovanskii, Topological Galois Theory, Springer, Berlin, Heidelberg, 2014.
- Simon Donaldson, Riemann surfaces, Oxford Graduate Texts in Mathematics, vol. 22, Oxford University Press, Oxford, 2011. MR 2856237, DOI 10.1093/acprof:oso/9780198526391.001.0001
- B. V. Shabat, Introduction to complex analysis, American Mathematical Society, Providence, RI, 1992.
- Helmut Koch, Introduction to classical mathematics. I, Mathematics and its Applications, vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. From the quadratic reciprocity law to the uniformization theorem; Translated and revised from the 1986 German original; Translated by John Stillwell. MR 1149600, DOI 10.1007/978-94-011-3218-3
- Emil Artin, Galois Theory, 2nd ed., Notre Dame Mathematical Lectures, no. 2, University of Notre Dame, Notre Dame, Ind., 1944. MR 0009934
- George Springer, Introduction to Riemann surfaces, American Mathematical Society, Providence, RI, 2001.
- N. R. T. Biggs, A new family of embedding formulae for diffraction by wedges and polygons, Wave Motion 43 (2006), no. 7, 517–528. MR 2252752, DOI 10.1016/j.wavemoti.2006.04.002
- E. A. Skelton, R. V. Craster, and A. V. Shanin, Embedding formulae for diffraction by non-parallel slits, Quart. J. Mech. Appl. Math. 61 (2008), no. 1, 93–116. MR 2382766, DOI 10.1093/qjmam/hbm026
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Additional Information
A. V. Shanin
Affiliation:
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
MR Author ID:
354791
Email:
andrey_shanin@mail.ru
A. I. Korolkov
Affiliation:
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
MR Author ID:
1124287
Email:
korolkov@physics.msu.ru
Received by editor(s):
November 29, 2021
Received by editor(s) in revised form:
December 17, 2021
Published electronically:
February 23, 2022
Additional Notes:
The work was supported by the RFBR grant 19-29-06048
Article copyright:
© Copyright 2022
Brown University
