Asymptotic analysis for shear waves in a fluid-filled borehole

We focus on the theoretical analysis of the resonance phenomena and the geometric attenuation behaviour of critical refracted shear waves propagating along a fluid-filled borehole. Using integration by parts, we asymptotically expand the vertical branch-cut integral of shear waves in an infinite series related to each order of the derivative of the response function of the formation. It is proved that the vertical branch-cut integral of shear waves at large offsets consists mainly of the contribution of the second asymptotic series, which is related to the first derivative of the response function of the formation at the shear branch point. Using the asymptotic expansion, we develop a simplified amplitude expression for shear waves, and the resonant frequency formula. The validity of the resonance frequencies obtained by the resonant frequency formula is verified numerically by comparison with the corresponding frequencies of the numerical integral results. We also give a rational explanation for the phenomenon of two peaks appearing within each resonant peak zone: i.e. that these are the contributions of the constructive interference of the shear waves and the mode poles.