Impulse Response of Elastic Half-Space in the Wave Number–Time Domain
Publication: Journal of Engineering Mechanics
Volume 130, Issue 10
Abstract
This paper presents formulas for the response functions in the mixed wave number–time domain for a homogeneous, elastic half-space subjected to impulsive, spatially harmonic sources on its surface. These functions are useful when obtaining the wave field in a half space elicited by dynamic surface sources of arbitrary spatial distribution on the surface, in either two or three dimensions. The formulas in this paper are obtained by contour integration of the Green’s functions in the frequency–wave number domain. The correctness and accuracy of these solutions is then assessed by comparison with the results of the well-known transient response functions for suddenly applied loads in both two and three dimensions.
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References
1.
Abramowitz, M., and Stegun, I.E. ( 1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables, Applied Mathematics Series 55, US Department of Commerce, National Bureau of Standards, Washington, D.C.
2.
Apsel, R. J., and Luco, J. E. (1983). “On the Green’s functions for a layered half-space. Part II.” Bull. Seismol. Soc. Am., 73, 931–951
3.
Bouchon, M., and Aki, K. (1977). “Discrete wave-number representation of seismic-source wave fields.” Bull. Seismol. Soc. Am., 67, 259–277
4.
Cagniard, L. ( 1939). “Réflexion et réfraction des ondes séismiques progressives.” Gauthier-Villars, Paris.
5.
Chao, C. (1960). “Dynamical response of an elastic half-space to tangential surface loadings.” J. Appl. Mech., 27, 559–567.
6.
De Hoop, A. T. (1960). “A modification of Cagniard’s method for solving seismic pulse problems.” Appl. Sci. Res., Sect. B, 8, 349–356.
7.
Dix, C. H. (1954). “The method of Cagniard in seismic pulse problems.” Geophysics, 19(4), 722–738.
10.
Eringen, A.C., and Suhubi, E.S. ( 1975). Elastodynamics, Vol. II, Academic, New York.
8.
Gradshteyn, I.S., and Ryzhik, I.M. ( 1980). Table of integrals, series, and products, 4th printing, Academic, New York.
9.
Johnson, L. R. (1974). “Green’s function for Lamb’s problem.” Geophys. J. R. Astron. Soc., 37, 99–131.
11.
Hisada, Y. (1994). “An efficient method for computing Green’s functions for a layered half-space with sources and receivers at close depths: Part I.” Bull. Seismol. Soc. Am., 84(5), 1456–1472.
12.
Hisada, Y. (1995). “An efficient method for computing Green’s functions for a layered half-space with sources and receivers at close depths. Part II.” Bull. Seismol. Soc. Am., 85(4), 1080–1093.
16.
Kausel, E., and Roësset, J. (1992). “Frequency domain analysis of undamped systems.” J. Eng. Mech., 118(4), 721–734.
17.
Lamb, H. (1904). “On the propagation of tremors over the surface of an elastic solid.” Philos. Trans. R. Soc. London, Ser. A, 203, 1–42.
18.
Luco, J. E., and Apsel, R. J. (1983). “On the Green’s functions for a layered half-space. Part I.” Bull. Seismol. Soc. Am., 73, 909–929.
19.
Mooney, H. M. (1974). “Some numerical solutions for Lamb’s problem.” Bull. Seismol. Soc. Am., 64(2), 473–491.
20.
Park, J. ( 2002). “Wave motion in finite and infinite media using the thin-layer method.” PhD thesis, Massachusetts Institute of Technology, Cambridge, Mass.
21.
Park, J., and Kausel, E. ( 2003). “Green’s function for a layered halfspace in the wavenumber-time domain.” 16th Engineering Mechanics Conf. ASCE, Univ. of Washington, Seattle, Wash.
22.
Pekeris, C. L. (1955). “The seismic surface pulse.” Proc., Natl. Acad. Sci. U.S.A., 41, 469–469.
23.
Spiegel, M.R. ( 1968). Mathematical handbook of formulas and tables Schaum’s outline series, McGraw–Hill, New York.
24.
Whittaker, W.L., and Christiano, P. ( 1979). “Dynamic response of flexible plates bearing on an elastic half-space.” Research Rep. No. RP-125-9-79, Dept. of Civil Engineering, Carnegie–Mellon Univ., Pittsburgh.
Information & Authors
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Published In
Journal of Engineering Mechanics
Volume 130 • Issue 10 • October 2004
Pages: 1211 - 1222
Copyright
Copyright © 2004 ASCE.
History
Published online: Oct 1, 2004
Published in print: Oct 2004
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