Advertisement

Pochhammer–Chree waves in rods: degeneracy at the bulk wave velocities

  • Sergey V. Kuznetsov
Article
  • 23 Downloads

Abstract

Degeneracy of the linear Pochhammer–Chree wave equation at phase velocities coinciding with the bulk wave velocities is observed and analyzed. Spectral analysis of Pochhammer–Chree equation is performed. The corrected analytical solutions for components of the displacement fields are constructed, accounting degeneracy of the secular equations and the corresponding solutions.

Keywords

Pochhammer–Chree wave Spectral analysis Degenerate solution Dispersion 

Mathematics Subject Classification

74J15 

References

  1. 1.
    Pochhammer, L.: Ueber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiscylinder. J. Reine Angew. Math. 81, 324–336 (1876)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chree, C.: Longitudinal vibrations of a circular bar. Q. J. Pure Appl. Math. 21, 287–298 (1886)zbMATHGoogle Scholar
  3. 3.
    Chree, C.: The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and applications. Trans. Camb. Philos. Soc. 14, 250–309 (1889)Google Scholar
  4. 4.
    Mindlin, R.D., McNiven, H.D.: Axially symmetric waves in elastic rods. Trans. ASME. J. Appl. Mech. 27, 145–151 (1960)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Meeker, T.R., Meitzler, A.H.: Guided wave propagation in elongated cylinders and plates. In: Mason, W.P. (ed.) Physical Acoustics: Principles and Methods, vol. 1A, pp. 111–167. Academic Press, New York (1964)CrossRefGoogle Scholar
  6. 6.
    Kolsky, H.: Stress waves in solids. J. Sound Vib. 1, 88–110 (1964)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zemanek, J.: An experimental and theoretical investigation of elastic wave propagation in a cylinder. J. Acoust. Soc. Am. 51, 265–283 (1972)CrossRefGoogle Scholar
  8. 8.
    Shawagfeh, N., Kaya, D.: Series solution to the Pochhammer–Chree equation and comparison with exact solutions. Comput. Math. Appl. 47, 1915–1920 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Valsamos, G., Casadei, F., Solomos, G.: A numerical study of wave dispersion curves in cylindrical rods with circular cross-section. Appl. Comput. Mech. 7, 99–114 (2013)Google Scholar
  10. 10.
    Ilyashenko, A.V., Kuznetsov, S.V.: Abnormality of the longitudinal Pochhammer–Chree waves in the vicinity of \(\text{c}_{2}\) phase speed. Arch. Appl. Mech. (2018)  https://doi.org/10.1007/s00419-018-1377-7 CrossRefGoogle Scholar
  11. 11.
    Kuznetsov, S.V.: Lamb waves in anisotropic plates (review). Acoust. Phys. 60, 95–103 (2014)CrossRefGoogle Scholar
  12. 12.
    Djeran-Maigre, I., Kuznetsov, S.V.: Solitary SH waves in two-layered traction-free plates. C. R. Méc. 336, 102–107 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute for Problems in MechanicsMoscowRussia

Personalised recommendations