Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer

In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated. The layer and the half-space may be compressible or incompressible and they  are in welded contact with each other. The main aim of the paper is to derive explicit  exact secular equations of the wave for four possible combinations of a (compressible/incompressible) half-space coated by a (compressible/incompressible) layer. For the compressible/compressible case, the explicit secular equation is derived by using the effective boundary condition method. For three remaining cases, the explicit secular equations are derived from the secular equation for the compressible/compressible case by using the incompressible limit technique along with the expressions of the reduced elastic compliances in terms of the elastic stiffnesses. Based on the obtained secular equations, the effect of incompressibility on the  Raleigh wave 
propagaion  is considered numerically. It is shown that the incompressibility  strongly  affects  the Raleigh wave velocity and  it makes the Raleigh wave velocity increasing.