摘要We obtain an asymptotic solution to the vertical branch-cut integral of shear waves excited by an impulsive pressure point source in a fluid-filled borehole, by taking the effect of the infinite singularity of the Hankel functions related to shear waves in the integrand at the shear branch point into account and using the method of steepest-descent to expand the vertical branch-cut integral of shear waves. It is theoretically proven that the saddle point of the integrand is located at ks-i/z, where ks and z are the shear branch point and the offset. The continuous and smooth amplitude spectra and the resonant peaks of shear waves are numerically calculated from the asymptotic solution. These asymptotic results are generally in agreement with the numerical integral results. It is also found by the comparison and analysis of two results that the resonant factor and the effect of the normal and leaking mode poles around the shear branch point lead to the two-peak characteristics of the amplitude spectra of shear waves in the resonant peak zones from the numerical integral calculations.
Abstract：We obtain an asymptotic solution to the vertical branch-cut integral of shear waves excited by an impulsive pressure point source in a fluid-filled borehole, by taking the effect of the infinite singularity of the Hankel functions related to shear waves in the integrand at the shear branch point into account and using the method of steepest-descent to expand the vertical branch-cut integral of shear waves. It is theoretically proven that the saddle point of the integrand is located at ks-i/z, where ks and z are the shear branch point and the offset. The continuous and smooth amplitude spectra and the resonant peaks of shear waves are numerically calculated from the asymptotic solution. These asymptotic results are generally in agreement with the numerical integral results. It is also found by the comparison and analysis of two results that the resonant factor and the effect of the normal and leaking mode poles around the shear branch point lead to the two-peak characteristics of the amplitude spectra of shear waves in the resonant peak zones from the numerical integral calculations.
YAO Gui-Jin;SONG Ruo-Long;WANG Ke-Xie. Asymptotic Analysis of Vertical Branch-Cut Integral of Shear Waves in a Fluid-Filled Borehole Utilizing the Steepest-Descent Method[J]. 中国物理快报, 2008, 25(2): 578-581.
YAO Gui-Jin, SONG Ruo-Long, WANG Ke-Xie. Asymptotic Analysis of Vertical Branch-Cut Integral of Shear Waves in a Fluid-Filled Borehole Utilizing the Steepest-Descent Method. Chin. Phys. Lett., 2008, 25(2): 578-581.
[1]Biot M A 1952 J. Appl. Phys. 23 997 [2] White J E 1962 Geophysics 27 327 [3] Crawford G E, Hoyer W A and Span M M 1973 The Log.Analyst. 14 3 [4] Peterson E W 1974 J. Appl. Phys. 45 3340 [5] Paillet F L, and Cheng C H 1986 Geophysics 511438 [6] Winbow G A 1988 Geophysics 53 1334 [7] Zhang H L et al 1995 Sci. Chin. A 7 742 (inChinese) [8] Zhang H L, Wang X M and Zhang B X 2004 Acoustic Fieldand Waves of Borehole (Beijing: Science Press) chap 4 p 39 (inChinese) [9] Kurkjian A L 1985 Geophysics 50 852 [10] Roever W L et al 1974 J. Acoust. Soc. Am. 551144 [11] Tsang L and Kong J A 1979 J. Acoust. Soc. Am. 65 647 [12] Dong Q D et al 1991 Chin. J. Geophys. 34 228(in Chinese) [13] Yao G J et al 2007 Geophys. Prosp. 55 561 [14]Winbow G A 1985 Geophysics 50 1119 [15] Brekhovskih L M 1960 Waves in Layered Media(Beijing: Science Press) chap 4 p 157 (in Chinese)
2008, Vol. 25 
