Effects of the Rayleigh Secular Function on Time-Harmonic Asymptotic Solutions Due to Horizontal Vibration Sources

The time-harmonic asymptotic solutions due to the surface horizontal vibration sources provide the theoretical basis in the applications of buried object detection. In the integral transformation method, the Rayleigh secular function appears in the denominator of the integrand of the inverse transformation. This leads to the multi-leaf characteristics of the integrand and the asymptotic solution is affected by the Rayleigh poles, resulting in a mismatch between the asymptotic time-harmonic solution and the finite element results. In this paper, an integral expression for the time-harmonic solution of the surface horizontal vibration source is derived using the integral transformation method. The asymptotic results using the steepest descent method are decomposed into the analytical component, the modified component of the poles and the residual component of the poles. Expressions for each component are given, with particular emphasis on the effect of the Rayleigh secular function on the asymptotic solution. It is found that for the multi-leaf problem, the asymptotic expressions related to shear waves should use the results of the γ + + leaf, while the asymptotic expressions related to compressional waves should use the results of the γ + − leaf when > arcsin h / k . Comparison of the numerical and semi-analytical solutions is made to verify the expressions for the analytical components, along with the selection of the Riemann surface.