Acta Mechanica

, Volume 137, Issue 1–2, pp 65–81 | Cite as

Wave propagation modeling in human long bones

  • D. I. Fotiadis
  • G. Foutsitzi
  • C. V. Massalas
Original Papers


The dynamic behavior of a dry long bone that has been modeled as a piezoelectric hollow cylinder of crystal class 6 is investigated. The solution for the wave propagation problem is expressed in terms of a potential function which satisfies an eighth-order partial differential equation, whose solutions lead to the derivation of the explicit solution of the wave equation. The mechanical boundary conditions correspond to those of stress free lateral surfaces, while the electrical boundary conditions correspond to those of short circuit. The satisfaction of the boundary conditions leads to the dispersion relation which is solved numerically. The frequencies obtained are presented as functions of various parameters and they compare well with other researchers' theoretical results.


Boundary Condition Partial Differential Equation Fluid Dynamics Wave Propagation Wave Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Wilson, L. A., Morrison, J. A.: Wave propagation in piezoelectric rods of hexagonal crystal symmetry. Q. J. Appl. Math.30, 387–395 (1977).Google Scholar
  2. [2]
    Mirsky, I.: Wave propagation in transversely isotropic circular cylinders, Part I: Theory. J. Acoust. Soc. Am.37, 1016–1021 (1965).Google Scholar
  3. [3]
    Ambardar, A., Ferris, C. D.: Wave propagation in a piezoelectric two-layered cylindrical shell with hexagonal symmetry: Some implications for long bone. J. Acoust. Soc. Am.63, 781–792 (1978).Google Scholar
  4. [4]
    Fukada, E., Yasuda, I.: Piezoelectric effects in collagen. Jpn. J. Appl. Phys.3, 117–121 (1964).Google Scholar
  5. [5]
    Güzelsu, N., Saha, S.: Electro-mechanical wave propagation in long bones. J. Biomech.14, 19–33 (1981).Google Scholar
  6. [6]
    Paul, H. S., Venkatesan, M.: Wave propagation in a piezoelectric human bone of arbitrary cross section with a circular cylindrical cavity. J. Acoust. Soc. Am.89, 196–199 (1991).Google Scholar
  7. [7]
    Paul, H. S., Venkatesan, M.: Wave propagation in a piezoelectrical bone with a cylindrical cavity of arbitrary shape. Int. J. Eng. Sci.29, 1601–1607 (1991).Google Scholar
  8. [8]
    Ding, H., Chenbuo, L.: General solutions for coupled equations for piezoelectric media. Int. J. Solids Struct.16, 2283–2298 (1996).Google Scholar
  9. [9]
    Lang, S. B.: Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dry bovine bones. IEEE Trans. Biomech. Engng.17, 101–105 (1970).Google Scholar
  10. [10]
    Reinish, G. B.: Dielectric and piezoelectric properties of bone as function of moisture content. Ph. D. Thesis, Columbia University (1974).Google Scholar
  11. [11]
    Charalambopoulos, A., Fotiadis, D. I., Massalas, C. V.: Free vibrations of a double layered elastic isotropic cylindrical rod. Int. J. Eng. Sci.36, 711–731 (1998).Google Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • D. I. Fotiadis
    • 1
  • G. Foutsitzi
    • 2
  • C. V. Massalas
    • 2
  1. 1.Department of Computer ScienceUniversity of IoanninaIoanninaGreece
  2. 2.Department of MathematicsUniversity of IoanninaIoanninaGreece

Personalised recommendations