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Edge Green’s functions on a branched surface. Asymptotics of solutions of coordinate and spectral equations

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Abstract

The problems of diffraction by a slit or a strip having ideal boundary conditions, and some other problems, can be reduced to the problem of wave propagation on a multisheet surface by applying the method of reflections. Further simplifications of the problem can be achieved by applying an embedding formula. As a result, the solution of the problem with a plane wave incidence becomes expressed in terms of the edge Green’s functions, i.e., in terms of the fields generated by dipole sources localized at branchpoints of the surface.

The present paper is devoted to finding the edge Green’s functions. For this problem, two sets of differential equations, namely, the coordinate and spectral equations, are used. The properties of solutions of these equations are studied. Bibliography: 9 titles.

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References

  1. A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann., 47, 317–374 (1896).

    Article  MathSciNet  MATH  Google Scholar 

  2. G. E. Latta, “The solution of a class of integral equations,” Jl. Rat. Mech. Anal., 5, 821–834 (1956).

    MathSciNet  Google Scholar 

  3. M. H. Williams, “Diffraction by a finite strip,” Quart. Jl. Mech. Appl. Math., 35, 103–124 (1982).

    Article  MATH  Google Scholar 

  4. A. V. Shanin, “Diffraction of a plane wave by two ideal strips,” Quart. Jl. Mech. Appl. Math., 56, 187–215 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  5. A. V. Shanin, “A generalization of the separation of variables method for some 2D diffraction problems,” Wave Motion, 37, 241–256 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. V. Shanin and E. M. Doubravsky, “Wave propagation on 2D multi-sheet surfaces. Coordinate and spectral equations,” submitted to J. Eng. Math.

  7. N. R. T. Biggs, D. Porter, and D.S.G. Stirling, “Wave diffraction through a perforated breakwater,” Quart. J. Mech. Appl. Math., 53, 375–391 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  8. R. V. Craster, A. V. Shanin, and E. M. Doubravsky, “Embedding formula in diffraction theory,” Proc. Roy. Soc. Lond. A, 459, 2475–2496 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  9. V. B. Prasolov, Problems and Theorems in Linear Algebra [in Russian], Moscow (1996).

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Authors and Affiliations

  1. Department of Physics, M. V. Lomonosov Moscow State University, Moscow, Russia

    A. V. Shanin

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  1. A. V. Shanin
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Correspondence to A. V. Shanin.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 233–256.

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Shanin, A.V. Edge Green’s functions on a branched surface. Asymptotics of solutions of coordinate and spectral equations. J Math Sci 148, 769–783 (2008). https://doi.org/10.1007/s10958-008-0024-1

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  • DOI: https://doi.org/10.1007/s10958-008-0024-1

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