On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

The  secular equation  of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Mechanics of Materials 31 (1999), 215-233] and  Sotiroplous $\&$ Tougelidis [Ultrasonics 36 (1998), 371-374]. However, it  is not totally explicit and some misprints have been occurred in this secular equation in both papers.  This secular equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through  the interface between the layer and the half-space.  Since the expansion of this six-order determinant was not shown in both two papers, it has been  difficult to readers to recognize these misprints.  This paper presents a technique  that provides a
totally explicit secular equation of the wave. The technique  makes  clear  the way from the traction-free and  continuity conditions to the  secular equation   and  enables  us to recognize   the misprints  appearing  in the reported secular equation.  The technique can be  employed to obtain explicit secular equations of Rayleigh waves for many other  cases.  Moreover, the paper introduces  a transfer matrix in explicit form for  an orthotropic layer that is much simpler in form than the one obtained previously.