Wave mode conversion in isotropic halfspace

It is known that an incident bulk P wave propagating in a homogeneous isotropic halfspace, being reflected from the plane boundary, may exhibit a mode conversion into shear S wave without the formation of reflected P waves. The mode conversion takes place, when the incident wave hits the boundary at some critical angles, which depend upon Poisson’s ratio. Herein, it is revealed that the Jeffreys solution for the mode conversion angles needs in in corrections, mainly because of spurious roots, appeared at solving a specially constructed eighth-order polynomial for the P wave reflection coefficient. The developed approach allowed us to construct a bi-cubic polynomial and obtain analytical expressions for its roots, and to find correct values for angles of incidence, at which the mode conversion occurs.