Archive of Applied Mechanics

, Volume 88, Issue 8, pp 1385–1394 | Cite as

Pochhammer–Chree waves: polarization of the axially symmetric modes

  • Alla V. Ilyashenko
  • Sergey V. Kuznetsov


The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of wave polarization on the free surface due to variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed (\(c_2\)) all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree wave can propagate at \(c_2\) phase speed.


Pochhammer–Chree waves Polarization Dispersion Spectral analysis 


  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.
    Field, G.S.: Velocity of sound in cylindrical rods. Can. J. Res. 5, 619–624 (1931)CrossRefGoogle Scholar
  5. 5.
    Field, G.S.: Longitudinal waves in cylinders of liquid, in hollow tubes and in solid rods. Can. J. Res. 11, 254–263 (1934)CrossRefzbMATHGoogle Scholar
  6. 6.
    Field, G.S.: Dispersion of supersonic waves in cylindrical rods. Phys. Rev. 57, 1188 (1940)CrossRefGoogle Scholar
  7. 7.
    Shear, S.K., Focke, A.B.: The dispersion of supersonic waves in cylindrical rods of polycrystalline silver, nickel, and magnesium. Phys. Rev. 57, 532–537 (1940)CrossRefGoogle Scholar
  8. 8.
    Bancroft, D.: The velocity of longitudinal waves in cylindrical bars. Phys. Rev. 59, 588–593 (1941)CrossRefGoogle Scholar
  9. 9.
    Hudson, G.E.: Dispersion of elastic waves in solid circular cylinders. Phys. Rev. 63, 46–51 (1943)CrossRefGoogle Scholar
  10. 10.
    Holden, A.H.: Longitudinal modes of elastic waves in isotropic cylinders and slabs. Bell Syst. Tech. J. 30, 956–969 (1951)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Adem, J.: On the axially-symmetric steady wave propagation in elastic circular rods. Q. Appl. Math. 12, 261–275 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Redwood, M., Lamb, J.: On propagation of high frequency compressional waves in isotropic cylinders. Proc. Phys. Soc. Lond. 70, 136–143 (1957)CrossRefzbMATHGoogle Scholar
  13. 13.
    Mindlin, R.D., McNiven, H.D.: Axially symmetric waves in elastic rods. Trans. ASME. J. Appl. Mech. 27, 145–151 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McNiven, H.D., Perry, D.C.: Axially symmetric waves in infinite, elastic rods. J. Acoust. Soc. Am. 34, 433–437 (1962)CrossRefGoogle Scholar
  15. 15.
    Onoe, M., McNiven, H.D., Mindlin, R.D.: Dispersion of axially symmetric waves in elastic rods. Trans. ASME. J. Appl. Mech. 29, 729–734 (1962)CrossRefzbMATHGoogle Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    Kolsky, H.: Stress waves in solids. J. Sound Vib. 1, 88–110 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hutchinson, J.R., Percival, C.M.: Higher modes of longitudinal wave propagation in thin rod. J. Acoust. Soc. Am. 44, 1204–1210 (1968)CrossRefGoogle Scholar
  19. 19.
    Zemanek, J.: An experimental and theoretical investigation of elastic wave propagation in a cylinder. J. Acoust. Soc. Am. 51, 265–283 (1972)CrossRefzbMATHGoogle Scholar
  20. 20.
    Thurston, R.N.: Elastic waves in rods and clad rods. J. Acoust. Soc. Am. 64, 1–37 (1978)CrossRefzbMATHGoogle Scholar
  21. 21.
    Pao, Y.-H., Mindlin, R.D.: Dispersion of flexural waves in an elastic, circular cylinder. Trans. ASME. J. Appl. Mech. 27, 513–520 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    Kuznetsov, S.V.: Lamb waves in anisotropic plates. Acoust. Phys. 60, 95–103 (2014). (Review) CrossRefGoogle Scholar
  24. 24.
    Kuznetsov, S.V.: Love waves in nondestructive diagnostics of layered composites. Surv. Acoust. Phys. 56, 877–892 (2010)CrossRefGoogle Scholar
  25. 25.
    Tyutekin, V.V., Boiko, A.I.: Helical normal waves near a cylindrical cavity in an elastic medium. Acoust. Phys. 56, 141–144 (2010)CrossRefGoogle Scholar
  26. 26.
    Pavić, G., Chevillotte, F., Heraud, J.: Dynamics of large-diameter water pipes in hydroelectric power plants. J. Phys. Conf. Ser. 813, 1–5 (2017)Google Scholar
  27. 27.
    Sharma, G.S., Skvortsov, A., MacGillivray, I., Kessissoglou, N.: Acoustic performance of gratings of cylindrical voids in a soft elastic medium with a steel backing. J. Acoust. Soc. Am. 141, 4694–4704 (2017)CrossRefGoogle Scholar
  28. 28.
    Seemann, W.: Wellenausbreitung in rotierenden und statisch konservativ vorbelasteten Zylindern, Ph.D. Thesis. Universität Karlsruhe, Fakultät für Maschinenbau, Diss. v. (1991)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Moscow State University of Civil EngineeringMoscowRussia
  2. 2.Institute for Problems in MechanicsMoscowRussia

Personalised recommendations