Modified Steepest-Descent Path Method in Solving Weyl Integration Representation of Vector Wave Bases

Abstract

Weyl integration representation is ever regarded as a wave source. Weyl integration has feature of double Fourier integral formulas, and the traditional steepest-descent path method has been dealt with convergence of oscillatory terms in the integrands of wave source. Unfortunately, to solve the reflective or scattering waves, the equations contain singular poles and branch cuts on the complex plane, because variables of the integrands are shown in denominators and in square-root terms. Singular poles represent the Rayleigh waves, and they can be solved by residue values. However, the branch cuts on the complex plane represent the head waves, and integral paths are not allowed to pass across the branch cuts. They need to solve through applying numerical integration. This paper provides a deformed integral path from the traditional integral path to the path in which the exponential terms could decay rapidly, the singular poles are considered, and the branch cut paths are counted. This demonstrates benefits of the modified steepest-descent path method in solving the vector wave bases formed in Weyl integration for elastodynamic, poroelastodynamic, and electromagnetic waves.