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Application of topological methods to estimate the number of longitudinal elastic waves in crystals

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Abstract

In the problem of estimating the number of longitudinal normals of elastic waves in crystals, several approaches based on the methods of modern topology and associated with the use of topological invariants are proposed. It is shown that for any symmetry class of the crystals there exists not less than three directions along which purely longitudinal waves can propagate. Under the Morse condition on the fundamental function, connections are obtained between the numbers of directions of longitudinal normals corresponding to different types of critical points of this function. Further prospects are discussed.

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References

  1. I. Fedorov,The Theory of Elastic Waves in Crystals [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  2. Yu. I. Sirotin and M. P. Shaskol'skaya,Fundamentals of Crystallography [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  3. J. W. S. Rayleigh,The Theory of Sound, Macmillan, London (1894).

    Google Scholar 

  4. H. Kolsky,Stress Waves in Solids, Oxford (1953).

  5. M. Musgrave, in:Progress in Solid Mechanics, Vol. 2 (eds. Sheddon and Hill), North-Holland, Amsterdam (1961).

    Google Scholar 

  6. P. S. Aleksandrov,Combinatorial Topology [in Russian], Gostekhizdat, Moscow (1947).

    Google Scholar 

  7. Yu. G. Borisovich, N. M. Bliznyakov, Ya. I. Izrailevich, and T. N. Fomenko,Introduction to Topology [in Russian], Vyssh. Shkola, Moscow (1980).

    Google Scholar 

  8. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko,Modern Geometry. Methods of Cohomology Theory [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  9. M. A. Krasnosel'skii,Topological Methods in the Theory of Nonlinear and Integral Equations [in Russian], Gostekhizdat, Moscow (1956).

    Google Scholar 

  10. M. A. Krasnosel'skii and P. P. Zabreiko,Geometrical Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  11. Yu. G. Borisovich,Dokl. Akad. Nauk SSSR 101, 205 (1955).

    Google Scholar 

  12. Yu. G. Borisovich,Dokl. Akad. Nauk SSSR,104, 165 (1955).

    Google Scholar 

  13. A. S. Shvarts,Tr. Mosk. Mat. Ob-va, No. 10, 217 (1961).

    Google Scholar 

  14. A. S. Shvarts,Tr. Mosk. Mat. Ob-va, No. 11, 99 (1962).

    Google Scholar 

  15. M. W. Hirsch,Differential Topology, New York (1976).

  16. W. Marzantouriez,Zesz. Nauk. Wydz. Mat. Fiz. Chem. Mat. UG, No. 6, 55 (1987).

    Google Scholar 

  17. V. I. Arnol'd,Usp. Mat. Nauk,34, 3 (1979).

    Google Scholar 

  18. M. A. Krasnosel'skii,Vector Fields on the Plane [in Russian], Fizmatgiz, Moscow (1963).

    Google Scholar 

  19. O. V. Kunakovskaya, “Some comments on the indices of singularities of 1-forms,” in:Topological and Geometrical Methods in Mathematical Physics [in Russian], published by Voronezh University, Voronezh (1983), p. 118.

    Google Scholar 

  20. O. V. Kunakovskaya, “Boundary singularities of a pair of sections of ann-dimensional vector bundle over ann-dimensional manifold with boundary,” [in Russian], State Pedagogical Institute, Tambov (1986), 23 p., deposited at VINITI August 29, 1986. No. 6317-V86.

    Google Scholar 

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Authors
  1. Yu. G. Borisovich
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  2. B. M. Darinskii
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  3. O. V. Kunakovskaya
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Additional information

Voronezh Polytechnic Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 146–152. January, 1993.

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Borisovich, Y.G., Darinskii, B.M. & Kunakovskaya, O.V. Application of topological methods to estimate the number of longitudinal elastic waves in crystals. Theor Math Phys 94, 104–108 (1993). https://doi.org/10.1007/BF01017000

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  • DOI: https://doi.org/10.1007/BF01017000

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