Behavior of surface waves at transition through a junction line on the boundary of an elastic homogenous isotropic body

Abstract

Consider a convex homogeneous isotropic elastic body ~ C R~_ = {(z, y, z)lz >_ 0} bounded by a stress-free surface Of}. The surface 0f~ consists of a smooth cylindrical surface 0f/-(with z _> 0, x < 0) with generator parallel to the y-axis and of the half-plane 0~ + (z = 0, x >_ 0). The junction line 10 = {(x, y, z)lx = z = 0} coincides with the y-axis. The plane tangent to the surface Oft is continuous on the junction line, but the curvature of the normal section of the surface 012 has a discontinuity of the first kind at each point of the y-axis (see Fig. 1). The value of the discontinuity depends on the y-coordinate.Let L be a ray lying in a plane coplanar to the x, z plane and belonging to the surface 012. By definition, rays are extremals of the Fermat functional f ds T' where ds is an element of the arc length of the curve along which the integration is carried out, and b is the velocity of propagation of the transversal elastic waves in ft.In this paper, we consider high-frequency asymptotic expansions of the solutions of the equations of elasticity theory concentrated in the vicinity of the boundary Oft-. These solutions determine waves which propagate along the ray L with a velocity close to b and overrun on the line I0 of junction of the surfaces Of/-and 0f~ +.