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Asymptotic Analysis of Vertical Branch-Cut Integral of Shear Waves in a Fluid-Filled Borehole Utilizing the Steepest-Descent Method |
| YAO Gui-Jin1;SONG Ruo-Long2;WANG Ke-Xie2 |
| 1College of Communication Engineering, Jilin University, Changchun 1300122School of Physics, Jilin University, Changchun 130021 |
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| Cite this article: |
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YAO Gui-Jin, SONG Ruo-Long, WANG Ke-Xie 2008 Chin. Phys. Lett. 25 578-581 |
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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.
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| Keywords:
43.30.Gp
43.20.Mv
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Received: 11 August 2007
Published: 30 January 2008
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