The polarization vector and secular equation for surface waves in an anisotropic elastic half-space

Abstract

The displacement at the free surface of an anisotropic elastic half-space x 2>0 generated by a surface wave propagating in the direction of the x 1-axis traces an elliptic path. It is represented by the polarization vector a R= e 1+ i e 2 , where e 1 , e 2 are the conjugate radii of the ellipse on the polarization plane. The displacement traces the ellipse in the direction from e 1 to e 2 . We present explicit expressions of e 1 , e 2 and the secular equation without computing the Stroh eigenvalues p and the associated eigenvectors a and b . After presenting the expressions for a general anisotropic elastic material, the special cases are studied separately. For monoclinic materials with the symmetry plane at x 3=0, the secular equation and the conjugate radii e 1 , e 2 are identical to that for orthotropic materials when s ′ 16=0 but s ′ 26 need not vanish. For monoclinic materials with the symmetry plane at x 1=0, e 1 is along the x 1-axis while e 2 is on the plane x 1=0. If the symmetry plane is at x 2=0, e 1 is on the plane x 2=0 while e 2 is along the negative x 2-axis. In both cases, e 1 , e 2 are the principal radii of the ellipse. We also present the derivative of a R with respect to x 2, the depth from the free surface, that provides information on (i) whether the conjugate radii of the ellipse increase as x 2 increases and (ii) whether the polarization plane rotates as x 2 increases. New secular equations are obtained for monoclinic materials with the symmetry plane at x 1=0 or x 2=0.