An explicit secular equation for surface waves in an elastic material of general anisotropy

Abstract

An explicit secular equation for surface waves in an elastic half-space was obtained by Rayleigh in 1885 (Proc. London Math. Soc. 17) for isotropic materials and by Stoneley in 1963 (Geophys. J. Astron. Soc. 8) for orthotropic materials. Since then no explicit secular equation was found for a material more general than orthotropic materials. Recently, Destrade (J. Acoust. Soc. Amer. 109, 2001) employed the first integrals (originally proposed by Mozhaev) to obtain an explicit secular equation for monoclinic materials with the symmetry plane at x3 = 0, which is the plane of motion. The first integral approach does not work for monoclinic materials with the symmetry plane at x1 = 0 or x2 = 0. Ting (Proc. R. Soc. 32, 2002) proposed a new approach that not only recovers Destrade's secular equation easily, but also provides an explicit expression of the Stroh eigenvalues p and the eigenvectors a, b without solving the quartic equation for p. These eigenvalues and eigenvectors are needed in the surface wave solution. Employing this new approach, Ting also obtained explicit secular equations for monoclinic materials with the symmetry plane at x1 = 0 or x2 = 0. This new approach is extended here to derive an explicit secular equation for surface waves in a general anisotropic elastic material. The secular equation for a general anisotropic material is then specialized to monoclinic materials with the symmetry plane at x1 = 0, x2 = 0 or x3 = 0. It does recover the secular equations of Destrade, Stoneley and Rayleigh.