Expressions to Rayleigh circumferential phase velocity and dispersion relation for a cylindrical surface under mechanical pressure

  • Jean Eduardo SeboldEmail author
  • Luiz Alkimin de Lacerda


This paper describes a substantiated mathematical theory for Rayleigh waves propagated on some types of metal cylinders. More specifically, it presents not only a new way to express the dispersion relation of Rayleigh waves propagated on the cylindrical surface, but also how it can be used to construct a mathematical equation showing that the applied static mechanical pressure affects the shear modulus of the metal cylinder. All steps, required to conclude the process, consider the equation of motion as a function of radial and circumferential coordinates only, while the axial component can be overlooked without causing any problems. Some numerical experiments are done to illustrate the changes in the Rayleigh circumferential phase velocity in a metal cylindrical section due to static mechanical pressure around its external surface.


Phase velocity Dispersion relation Rayleigh waves Curved surface 

Mathematics Subject Classification

Primary 74J15 Secondary 00A69 


  1. 1.
    Hayes, M., Rivlin, R.S.: A note on the secular equation for Rayleigh waves. Z. Angew. Math. Phys. 13, 80–83 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Sharma, J.N., Pal, M.: Rayleigh–Lamb waves in magneto-thermoelastic homogeneous isotropic plate. Int. J. Eng. Sci. 42, 137–155 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kaplunov, J., Nolde, E., Prikazchikov, D.A.: A revisit to the moving load problem using an asymptotic model. Wave Motion 47, 440–451 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kaplunov, J., Prikazchikov, D.A.: Asymptotic theory for Rayleigh and Rayleigh-type waves. Adv. Appl. Mech. 50, 1–106 (2017)CrossRefGoogle Scholar
  5. 5.
    Destrad, M., Fu, Y.B.: A wave near the edge of a circular disk. Open Acoust. J. 42, 15–18 (2008)CrossRefGoogle Scholar
  6. 6.
    Guinan, M.W., Steinberg, D.J.: A simple approach to extrapolating measured polycrystalline shear moduli to very high pressure. J. Phys. Chem. Solids 36, 829–829 (1974)CrossRefGoogle Scholar
  7. 7.
    Guinan, M.W., Steinberg, D.J.: A constitutive model for metals applicable at high-strain rate. J. Appl. Phys. 51, 1498–1504 (1979)Google Scholar
  8. 8.
    Abrikosov, A.A.: Some properties of strongly compressed matter. J. Exp. Theor. Phys. 12, 1254 (1961)Google Scholar
  9. 9.
    Guinan, M.W., Steinberg, D.J.: Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements. J. Phys. Chem. Solids 41, 1501–1512 (1973)Google Scholar
  10. 10.
    Wolf, J.: Air force materials laboratory, Report AFML-TR-68-115 (1972)Google Scholar
  11. 11.
    Chandrasekharaiah, D.S.: Effects of surface stresses and voids on Rayleigh waves in an elastic solid. Int. J. Eng. Sci. 25, 205–211 (1987)CrossRefzbMATHGoogle Scholar
  12. 12.
    Stein, S., Wysession, M.: An introduction to Seismology, Earthquakes and Structure. Blackweel Publishing Ltd., Oxford (2003)Google Scholar
  13. 13.
    Duffy, D.G.: Green’s Functions With Applications. Chapman and Hall/CRC, Washington (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Timoshenko, S., Goodier, J.N.: Theory of Elasticity. The Maple Press Company, York (1951)zbMATHGoogle Scholar
  15. 15.
    Watson, G.N.: Theory of Bessel Functions. The Syndics of the Cambridge University Press, Cambriedge (1966)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jean Eduardo Sebold
    • 1
    Email author
  • Luiz Alkimin de Lacerda
    • 2
  1. 1.Federal Institute of Education, Science and Technology CatarinenseAraquariBrazil
  2. 2.Institute of Technology for DevelopmentFederal University of ParanáCuritibaBrazil

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