Radiation from Equivalent Body Forces for Scattering of Surface Waves by a Near-Surface Cylindrical Cavity

  • Chao Yang
  • Jan D. AchenbachEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 90)


The scattering of incident surface waves by a cylindrical cavity of arbitrary shape near the free surface of an elastic half-space is considered in this paper. The scattered field is represented by the radiation from equivalent body forces. The equivalent body forces due to the horizontal and vertical displacement components of the incident surface wave are determined separately. It is found that the equivalent body forces are double forces parallel and normal to the free surface of the half-space. By the use of the elastodynamic reciprocity theorem, the surface waves generated by the equivalent double forces are obtained in terms of properties of the incident wave, the cross-sectional area of the cavity and the elastic constants of the elastic half-space. The superposition of the surface waves generated by the equivalent body forces represents the scattered field of surface waves.


Scattering Surface wave Cavity Equivalent body forces Reciprocity theorem 



This work was supported by the National Natural Science Foundation of China (No. 51335001). We thank the China Scholarship Council for the funding to support Chao Yang’s study at Northwestern University.


  1. 1.
    Lange, F.F., Davis, B.I., Wright, E.: Processing-related fracture origins: IV, elimination of voids produced by organic inclusions. J. Am. Ceram. Soc. 69(1), 66–69 (1986)CrossRefGoogle Scholar
  2. 2.
    Heslehurst, R.B.: Defects and Damage in Composite Materials and Structures. CRC Press, Boca Raton, Florida (2014)CrossRefGoogle Scholar
  3. 3.
    Rice, J.R., Tracey, D.M.: On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17(3), 201–217 (1969)CrossRefGoogle Scholar
  4. 4.
    Tvergaard, V., Hutchinson, J.W.: Two mechanisms of ductile fracture: void by void growth versus multiple void interaction. Int. J. Solids Struct. 39(13), 3581–3597 (2002)CrossRefGoogle Scholar
  5. 5.
    Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland Publishing Company (1973). Paperback edition published by Elsevier (2012)Google Scholar
  6. 6.
    Graff, K.F.: Wave Motion in Elastic Solids. Courier Corporation, North Chelmsford, Massachusetts (2012)zbMATHGoogle Scholar
  7. 7.
    Rayleigh, L.: On waves propagated along the plane surface of an elastic solid. Proc. Lond. Math. Soc. 1(1), 4–11 (1885)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Aki, K., Richards, P.G.: Quantitative Seismology: Theory and Methods. W. H. Freeman and Company, San Francisco, California (1980)Google Scholar
  9. 9.
    Rice, J.R.: Elastic wave emission from damage processes. J. Nondestr. Eval. 1(4), 215–224 (1980)CrossRefGoogle Scholar
  10. 10.
    Zhang, H., Achenbach, J.D.: Use of equivalent body forces for acoustic emission from a crack in a plate. Mech. Res. Commun. 68, 105–108 (2015)CrossRefGoogle Scholar
  11. 11.
    Yang, C., Achenbach, J.D.: Time domain scattering of elastic waves by a cavity, represented by radiation from equivalent body forces. Int. J. Eng. Sci. 115, 43–50 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lamb, H.: On the propagation of tremors over the surface of an elastic solid. Philos. Trans. R. Soc. Lond. Ser. A 203, 1–42 (1904)CrossRefGoogle Scholar
  13. 13.
    Betti, E.: Teoria della elasticita’. Il Nuovo Cimento (1869–1876), 7(1), 69–97 (1872)CrossRefGoogle Scholar
  14. 14.
    Achenbach, J.D., Xu, Y.: Wave motion in an isotropic elastic layer generated by a time-harmonic point load of arbitrary direction. J. Acoust. Soc. Am. 106(1), 83–90 (1999)CrossRefGoogle Scholar
  15. 15.
    Achenbach, J.D.: Reciprocity in Elastodynamics. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  16. 16.
    Phan, H., Cho, Y., Achenbach, J.D.: Application of the reciprocity theorem to scattering of surface waves by a cavity. Int. J. Solids Struct. 50(24), 4080–4088 (2013)CrossRefGoogle Scholar
  17. 17.
    Phan, H., Cho, Y., Achenbach, J.D.: Validity of the reciprocity approach for determination of surface wave motion. Ultrasonics 53(3), 665–671 (2013)CrossRefGoogle Scholar
  18. 18.
    Achenbach, J.D.: Reciprocity and related topics in elastodynamics. Appl. Mech. Rev. 59(1), 13–32 (2006)CrossRefGoogle Scholar
  19. 19.
    Achenbach, J.D.: A new use of the elastodynamic reciprocity theorem. Math. Mech. Solids 19(1), 5–18 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sorokin, S.V.: On the bi-orthogonality conditions for multi-modal elastic waveguides. J. Sound Vib. 332(21), 5606–5617 (2013)CrossRefGoogle Scholar
  21. 21.
    Shi, F., Lowe, M.J.S., Xi, X., Craster, R.V.: Diffuse scattered field of elastic waves from randomly rough surfaces using an analytical Kirchhoff theory. J. Mech. Phys. Solids 92, 260–277 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mechanical EngineeringBeijing Institute of TechnologyBeijingChina
  2. 2.McCormick School of Engineering and Applied ScienceNorthwestern UniversityEvanstonUSA

Personalised recommendations