Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T09:38:55.705Z Has data issue: false hasContentIssue false
  • English
  • Français

On Evaluation of Lamb's Integrals for Waves in a Two-Dimension Elastic Half-Space

Published online by Cambridge University Press:  05 May 2011

Chau-Shioung Yeh ,
Tsung-Jen Teng  and
Wen-I Liao
Chau-Shioung Yeh*
Affiliation:
Department of Civil Engineering, and Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
Tsung-Jen Teng*
Affiliation:
National Center for Research on Earthquake Engineering, Taipei, Taiwan 10617, R.O.C.
Wen-I Liao*
Affiliation:
National Center for Research on Earthquake Engineering, Taipei, Taiwan 10617, R.O.C.
*
*Professor
**Research Follow
***Associate Research Fellow
Get access

Abstract

In this paper, a modified version of the method of steepest descent is proposed for the evaluation of Lamb's integrals which can be considered as basis functions dealing with the development of the transition matrix method which can be used to study the wave scattering in a two-dimensional elastic half-space. The formal solutions of the generalized Lamb's problem are studied and evaluated on the basis of the proposed method. After defining a phase function which presents in wavenumber integral, an exact mapping and an inverse mapping can be obtained according to the phase function. Thus, the original integration path can be deformed into an equivalent admissible path, namely, steepest descent path which passed through the saddle point, and then mapped onto a real axis of mapping plane, finally, resulted in an integral of Hermite type. This integral can be efficiently evaluated numerically in spite of either near- to far-field or low to high frequency. At the same time, the asymptotic value can easily be obtained by applying the proposed method. The numerical results for generalized Lamb's solutions are calculated and compared with analytic, asymptotic or other existing data, the excellent agreements are found. The properties of generalized Lamb's solutions are studied and discussed in details. Their possible applications for wave scattering in elastic half-space are also pointed out.

Type
Articles
Information
Journal of Mechanics , Volume 16 , Issue 2 , June 2000 , pp. 109 - 124
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Achenbach, J. D., Wave Propagation in Elastic Solids, North Holland, Publishing Comp., Amsterdam (1973).Google Scholar
2Apsel, R. J., “Dynamic Green's Functions for Layered Media and Applications to Boundary Value Problem,” Ph.D. Dissertation, University of California at San Diego, La Jolla, California (1979).Google Scholar
3Apsel, R. J. and Luco, J. E., “On the Green's Functions for a Layered Half Space, Part II,” Bull. Seism. Soc. Am., 73, pp. 331–351 (1983).CrossRefGoogle Scholar
4Dravinski, M. and Mossessian, T. K., “On Evaluation of Green Functions for Harmonic Loads in a Viscoelastic Half-Space,” J. Num. Mech. In Engin., 26, pp. 823–841 (1988).CrossRefGoogle Scholar
5Hudson, J. A., The Excitation and Propagation of Elastic Wave, Cambridge University Press (1980).Google Scholar
6Kundu, T., “Computation of Surface Motion in a Stratified Half Space,” Ph. D. Thesis, University of California at Los Angeles, California (1983).Google Scholar
7Kundu, T. and Mal, A. K., “Elastic Waves in Multilayered Solids due to a Dislocation Source,” Wave Motion, Vol. 7, pp. 459–471 (1985).CrossRefGoogle Scholar
8Lamb, H., “On the Propagation of Tremors Over the Surface of an Elastic Solid,” Philos. Trans. Roy. Soc., A203, pp. 1–42 (1904).Google Scholar
9Lapwood, E. R., “The Disturbance due to a Line Source in a Semi-Infinite Elastic Medium,” Proc. Roy. Soc., A242, pp. 63–100 (1949).Google Scholar
10Miller, G. F. and Pursey, H., “The Field and Radiation Impedance of Mechanical Radiators on the Free Surface on a Semi-Infinite Isotropic Solid,” Proc. Roy. Soc., A223, pp. 521–541 (1954).Google Scholar
11Pao, Y. H., “The Transition Matrix for the Scattering of Acoustic Waves and Elastic Waves,” in the Proc. of the IUTAM Symposium on Modern Problems in Elastic Wave Propagation, (ed. Miklowitz, J. and Achenbach, J.), Wiley, New York, pp. 123–144 (1978).Google Scholar
12Wong, H. L., “Dynamic Soil-Structure Interaction,” Report EER1-75-01, Earthquake Engineering Research Laboratory, California Institue of Technology, Pasadena, California (1975).Google Scholar
13Xu, P. C. and Mal, A. K., “An Adaptive Scheme for Irregularly Oscillatory Functions,” Wave Motion, Vol. 7, pp. 235–243 (1985).CrossRefGoogle Scholar
14Yeh, C. S., Teng, T. J., Liao, W. I. and Tsai, I. C., “Application of Integral Series Solution to the Diffraction of a Semi-Cylindrical Canyon,” Proc. of the 19th National Conf. on Theoretical and Applied Mechanics, Taoyuan, Taiwan, R.O.C., pp. 331–337 (1995).Google Scholar
15Yeh, C. S., Teng, T. J., Liao, W. I. and Tsai, I. C., “A Series Solution for Wave Diffraction be a Semi-Cylindrical Alluvial Valley,” Proc. of the 11th World Conf. on Earthquake Engng., Mexico, No. 157 (1996a).Google Scholar
16Yeh, C. S., Teng, T. J. and Liao, W. I., “The Dynamic Response of a Longitudinal Circular Cavity in a Half-Space Subjected an Oblique Incident Wave,” Proc. of sixth NTU-KU-Kaist Tri-Lateral Seminar/Workshop on Civil Engng., Taejon, Korea, pp. 3–8 (1996b).Google Scholar