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Fourier Expansion to Elastic Vector Wave Functions and Applications of Wave Bases to Scattering in Half-Space

Published online by Cambridge University Press:  22 March 2012

P.-J. Shih ,
T.-J. Teng  and
C.-S. Yeh
P.-J. Shih*
Affiliation:
Department of Civil and Environmental Engineering, National University of Kaohsiung, Kaohsiung, Taiwan 81148, R.O.C.
T.-J. Teng
Affiliation:
National Center for Research on Earthquake Engineering, Taipei, Taiwan 10671, R.O.C.
C.-S. Yeh
Affiliation:
Department of Civil Engineering and Institute of Applied Mechanics, National Taiwan University Taipei, Taiwan 10617, R.O.C.
*
*Corresponding author ([email protected])
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Abstract

This paper proposes a complete basis set for analyzing elastic wave scattering in half-space. The half-space is an isotropic, linear, and homogeneous medium except for a finite inhomogeneity. The wave bases are obtained by combining buried source functions and their reflected counter-waves generated from the infinite-plane boundary. The source functions are the vector wave functions of infinite-space. Based on the source functions expressed in the Fourier expansion form, the reflected counter-waves are easily obtained by solving the infinite-plane boundary conditions. Few representations adopt Wely's integration, but the Fourier expansion is developed from it and applied to decouple the angular-differential terms of the vector wave functions. In addition to the scattering of the finite inhomogeneity, the transition matrix method is extended to express the surface boundary conditions. For the numerical application in this paper, the P- and the SV- waves are assumed as the incoming fields. As an example, this paper computes stress concentrations around a cavity. The steepest-descent path method yielding the optimum integral paths is used to ensure the numerical convergence of the wave bases in the Fourier expansion. The resultant patterns from these approaches are compared with those obtained from numerical simulations.

Keywords

Fourier expressionElastic half-spaceScattering
Type
Articles
Information
Journal of Mechanics , Volume 28 , Issue 1 , March 2012 , pp. 19 - 39
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2012

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References

REFERENCES

1. Whittaker, E. T., “On the Partial Differential Equations of Mathematical Physics,” Mathematische Annalen, 57, pp. 333–355 (1903).CrossRefGoogle Scholar
2. Sommerfeld, A., “Über Die Ausbreitung Der Wellen in Der Drahtlosen Telegraphie,” Annalen der Physik, 28, pp. 665–736 (1909).CrossRefGoogle Scholar
3. Kristensson, G. and Ström, S., “Scattering from Buried Inhomogeneities—A General Three-Dimensional Formalism,” Journal of the Acoustical Society of America, 64, pp. 917–936 (1978).CrossRefGoogle Scholar
4. Boström, A. and Kristensson, G., “Elastic Wave Scattering by a Three-Dimensional Inhomogeneity in an Elastic Half Space,” Wave Motion, 2, pp. 335–353 (1980).CrossRefGoogle Scholar
5. Boström, A. and Kristensson, G.., “Scattering of a Pulsed Rayleigh Wave by a Spherical Cavity in an Elastic Half Space,” Wave Motion, 5, pp.137–143 (1983).CrossRefGoogle Scholar
6. Kristensson, G., “Electromagnetic Scattering from Buried Inhomogeneities—A General Three Dimensional Formalism,” Journal of Applied Physics, 51, pp. 3486–3500 (1980).CrossRefGoogle Scholar
7. Boström, A. and Karlsson, A., “Exact Synthetic Seismograms for an Inhomogeneity in a Layered Elastic Half-Space,” Geophysical Journal of the Royal Astronomical Society, 79, pp. 835–862 (1984).CrossRefGoogle Scholar
8. Hackman, R. H. and Sammelmann, G. S., “Multiple-Scattering Analysis for a Target in an Oceanic Wave-Guide,” Journal of the Acoustical Society of merica, 84, pp. 1813–1825 (1988).CrossRefGoogle Scholar
9. Danos, M. and Maximon, L. C., “Multipole Matrix Elements of the Translation Operator,” Journal of Mathematical Physics, 6, pp. 766–778 (1965).CrossRefGoogle Scholar
10. Devaney, A. J. and Wolf, E., “Multipole Expansions and Plane Wave Representations of the Electromagnetic Field,” Journal of Mathematical Physics, 15, pp. 234–244 (1974).CrossRefGoogle Scholar
11. Ben-Menahem, A. and Cisternas, A., “The Dynamic Response of an Elastic Half-Space to an Explosion in a Spherical Cavity,” Journal of Mathematical Physics, 42, pp. 112–125 (1963).CrossRefGoogle Scholar
12. Ben-Menahem, A. and Singh, S. J., “Multiploar Elastic Fields in a Layered Half-Space,” Bulletin of the Seismological Society of America, 58, pp. 1519–1572 (1968).CrossRefGoogle Scholar
13. Chang, S. K. and Mei, K. K., “Multipole Expansion Technique for Electromagnetic Scattering by Buried Objects,” Electromagnetics, 1, pp. 73–89 (1981).CrossRefGoogle Scholar
14. Ben-Menahem, A. and Singh, S. J., Seismic Wave and Sources, Dover Publications, New York (2000).Google Scholar
15. Weyl, H., “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Annalen der Physik, 60, 481–500 (1919).CrossRefGoogle Scholar
16. Erdélyi, A., Asymptotic Expansions, Dover Publishing Company, NY (1955).CrossRefGoogle Scholar
17. Yeh, C. S., Teng, T. J. and Liao, W. I., “On Evaluation of Lamb's Integrals for Waves in a Two-Dimension Elastic Half-Space,” Journal of Mechanics, 16, pp.109–124 (2000).CrossRefGoogle Scholar
18. Morse, P. M. and Feshbach, H., Method of Theoretical Physics, McGraw-Hill, New York (1953).Google Scholar
19. Boström, A. and Karlsson, A., “Broad-Band Synthetic Seismograms for a Spherical Inhomogeneity in a Many-Layered Elastic Half-Space,” Geophysical Journal of the Royal Astronomical Society, 89, pp. 527–547 (1987).CrossRefGoogle Scholar
20. Kennett, B. L. N., “Reflection Operator Methods for Elastic Waves I—Irregular Interfaces and Regions,” Wave Motion, 9, pp. 407–418 (1984).CrossRefGoogle Scholar
21. Kennett, B. L. N., “Reflection Operator Methods for Elastic Waves II—Composite Regions and Source Problems,” Wave Motion, 9, pp. 419–429 (1984).CrossRefGoogle Scholar
22. Waterman, P. C., “Matrix Theory of Elastic Wave Scattering,” Journal of the Acoustical Society of America, 60, pp. 567–587 (1976).CrossRefGoogle Scholar
23. Waterman, P. C. “New Formulation of Acoustic Scattering,” Journal of the Acoustical Society of America, 45, pp. 1417–1429 (1969).CrossRefGoogle Scholar
24. Chen, J. T. and Chen, P. Y., “A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes Under Bending,” Journal of Mechanics, 23, pp. 211–221 (2007).CrossRefGoogle Scholar
25. Chen, J. T., Liao, H. Z. and Lee, W. M., “An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains,” Journal of Mechanics, 25, pp. 59–74 (2009).CrossRefGoogle Scholar
26. Chen, J. T, Lee, Y. T. and Chou, K. S., “Revisit of Two Classical Elasticity Problems by Using the Null-Field Integral Equations,” Journal of Mechanics, 26, pp. 393–401 (2010).CrossRefGoogle Scholar
27. Liao, W. I., “Dynamic Response of a Half-Space with Defeats Subjected to an Elastic Wave,” Ph.D. thesis, National Taiwan University (1997).Google Scholar
28. Liao, W. I., Teng, T. J. and Yeh, C. S., “A Series Solution and Numerical Technique for Wave Diffraction by a Three-Dimensional Canyon,” Wave Motion, 39, pp. 129–142 (2004).CrossRefGoogle Scholar
29. Liao, W. I., “Application of Transition Matrix to Scattering of Elastic Waves in Half-Space by a Surface Scatterer,” Wave Motion, in press (2010).Google Scholar
30. Pao, Y. H., “Betti's Identity and Transition Matrix for Elastic Waves,” Journal of the Acoustical Society of America, 64, pp. 302–310 (1978).CrossRefGoogle Scholar
31. Shyu, W. S., “Scattering Behavior from an Inclusion in Elastic Half-plane,” Ph.D. thesis, National Taiwan University (2002).Google Scholar
32. Chang, S. K. and Mei, K. K., “Generalized Sommerfeld Integrals and Field Expansions in Two Medium Half-Space,” IEEE Transactions on Antennas and Propagation, 28, pp. 504–512 (1980).CrossRefGoogle Scholar
33. Greengard, L., Huang, J.Rokhlin, V. and Wandzura, S., “Accelerating Fast Multipole Methods for the Helmholtz Equation at Low Frequencies,” IEEE Computational Science and Engineering, 5, pp. 32–38 (1998).CrossRefGoogle Scholar
34. Achenbach, J. D., Wave Propagation in Elastic Solids, North-Holland Publishing Company (1973).Google Scholar
35. Goodier, J. N. and Bishop, R. E. D., “A Note on Critical Reflections of Elastic Waves at Free Surface,” Journal of Applied Physics, 23, pp. 124–126 (1952).CrossRefGoogle Scholar
36. Stratton, J. A., Electromagnetic Theory, McGraw-Hall Book Company (1941).Google Scholar
37. Chew, W. C., “Recurrence Relations for Three Dimesnional Scalar Addition Theorem,” Journal of Electromagnetic Waves and Applications, 6, pp. 133–142 (1992).CrossRefGoogle Scholar
38. Chew, W. C. and Wang, Y. M., “Efficient Ways to Compute the Vector Addition Theorem,” Journal of Electromagnetic Waves and Applications, 7, pp. 651–665 (1993).CrossRefGoogle Scholar