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

## Abstract

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.

## Keywords

Scattering Surface wave Cavity Equivalent body forces Reciprocity theorem## Notes

### Acknowledgements

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.

## References

- 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.Heslehurst, R.B.: Defects and Damage in Composite Materials and Structures. CRC Press, Boca Raton, Florida (2014)CrossRefGoogle Scholar
- 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.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.Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland Publishing Company (1973). Paperback edition published by Elsevier (2012)Google Scholar
- 6.Graff, K.F.: Wave Motion in Elastic Solids. Courier Corporation, North Chelmsford, Massachusetts (2012)zbMATHGoogle Scholar
- 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.Aki, K., Richards, P.G.: Quantitative Seismology: Theory and Methods. W. H. Freeman and Company, San Francisco, California (1980)Google Scholar
- 9.Rice, J.R.: Elastic wave emission from damage processes. J. Nondestr. Eval.
**1**(4), 215–224 (1980)CrossRefGoogle Scholar - 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.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.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.Betti, E.: Teoria della elasticita’. Il Nuovo Cimento (1869–1876), 7(1), 69–97 (1872)CrossRefGoogle Scholar
- 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.Achenbach, J.D.: Reciprocity in Elastodynamics. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
- 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.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.Achenbach, J.D.: Reciprocity and related topics in elastodynamics. Appl. Mech. Rev.
**59**(1), 13–32 (2006)CrossRefGoogle Scholar - 19.Achenbach, J.D.: A new use of the elastodynamic reciprocity theorem. Math. Mech. Solids
**19**(1), 5–18 (2014)MathSciNetCrossRefGoogle Scholar - 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.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