Skip to main content
SpringerLink
Log in

Pochhammer–Chree waves: polarization of the axially symmetric modes

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Pochhammer, L.: Ueber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiscylinder. J. Reine Angew. Math. 81, 324–336 (1876)

    MathSciNet  MATH  Google Scholar 

  2. Chree, C.: Longitudinal vibrations of a circular bar. Q. J. Pure Appl. Math. 21, 287–298 (1886)

    MATH  Google Scholar 

  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. Field, G.S.: Velocity of sound in cylindrical rods. Can. J. Res. 5, 619–624 (1931)

    Article  Google Scholar 

  5. Field, G.S.: Longitudinal waves in cylinders of liquid, in hollow tubes and in solid rods. Can. J. Res. 11, 254–263 (1934)

    Article  MATH  Google Scholar 

  6. Field, G.S.: Dispersion of supersonic waves in cylindrical rods. Phys. Rev. 57, 1188 (1940)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  8. Bancroft, D.: The velocity of longitudinal waves in cylindrical bars. Phys. Rev. 59, 588–593 (1941)

    Article  Google Scholar 

  9. Hudson, G.E.: Dispersion of elastic waves in solid circular cylinders. Phys. Rev. 63, 46–51 (1943)

    Article  Google Scholar 

  10. Holden, A.H.: Longitudinal modes of elastic waves in isotropic cylinders and slabs. Bell Syst. Tech. J. 30, 956–969 (1951)

    Article  MathSciNet  Google Scholar 

  11. Adem, J.: On the axially-symmetric steady wave propagation in elastic circular rods. Q. Appl. Math. 12, 261–275 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  12. Redwood, M., Lamb, J.: On propagation of high frequency compressional waves in isotropic cylinders. Proc. Phys. Soc. Lond. 70, 136–143 (1957)

    Article  MATH  Google Scholar 

  13. Mindlin, R.D., McNiven, H.D.: Axially symmetric waves in elastic rods. Trans. ASME. J. Appl. Mech. 27, 145–151 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  14. McNiven, H.D., Perry, D.C.: Axially symmetric waves in infinite, elastic rods. J. Acoust. Soc. Am. 34, 433–437 (1962)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  17. Kolsky, H.: Stress waves in solids. J. Sound Vib. 1, 88–110 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hutchinson, J.R., Percival, C.M.: Higher modes of longitudinal wave propagation in thin rod. J. Acoust. Soc. Am. 44, 1204–1210 (1968)

    Article  Google Scholar 

  19. Zemanek, J.: An experimental and theoretical investigation of elastic wave propagation in a cylinder. J. Acoust. Soc. Am. 51, 265–283 (1972)

    Article  MATH  Google Scholar 

  20. Thurston, R.N.: Elastic waves in rods and clad rods. J. Acoust. Soc. Am. 64, 1–37 (1978)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Kuznetsov, S.V.: Lamb waves in anisotropic plates. Acoust. Phys. 60, 95–103 (2014). (Review)

    Article  Google Scholar 

  24. Kuznetsov, S.V.: Love waves in nondestructive diagnostics of layered composites. Surv. Acoust. Phys. 56, 877–892 (2010)

    Article  Google Scholar 

  25. Tyutekin, V.V., Boiko, A.I.: Helical normal waves near a cylindrical cavity in an elastic medium. Acoust. Phys. 56, 141–144 (2010)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  28. Seemann, W.: Wellenausbreitung in rotierenden und statisch konservativ vorbelasteten Zylindern, Ph.D. Thesis. Universität Karlsruhe, Fakultät für Maschinenbau, Diss. v. (1991)

Download references

Author information

Authors and Affiliations

  1. Moscow State University of Civil Engineering, Moscow, Russia

    Alla V. Ilyashenko

  2. Institute for Problems in Mechanics, Moscow, Russia

    Sergey V. Kuznetsov

Authors
  1. Alla V. Ilyashenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey V. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey V. Kuznetsov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ilyashenko, A.V., Kuznetsov, S.V. Pochhammer–Chree waves: polarization of the axially symmetric modes. Arch Appl Mech 88, 1385–1394 (2018). https://doi.org/10.1007/s00419-018-1377-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-018-1377-7

Keywords

Search

Navigation