Generalized characteristic method of elastodynamics

Abstract A generalized characteristic method to solve two-dimensional time domain elastodynamics problems is developed. The (real) characteristic surfaces in x, y, t -space for the Navier's equations are continued to the complex domain, in which they are called generalized characteristics. The first order ordinary differential equations along the generalized characteristics which are equivalent to the Navier's equations are used to simplify an initial-boundary value problem of elastodynamics to a set of algebraic equations whose unknowns are the first order derivatives of the displacement components. The reflection coefficients of longitudinal and transverse waves, whose fronts can be of arbitrary convex shape, from a traction free boundary are also found to be algebraic expressions. The method makes the inversion of Laplace transform required by Cagniard's technique unnecessary and can be used directly to analyse the head wave and the regions of influence of the reflected waves. Closed form expressions for the time domain Green's function in the whole plane and the solution of Lamb's problem for a buried concentrated force are given as applications of the method. Formulae for the surface displacement take on a simpler form than those reported previously. Finally, some numerical results for the displacement of receivers on and below the surface are given. The displacements of reflected waves approach infinity when they are treated separately but superposing the displacements due to the reflected p - and s -waves shows that the transient Rayleigh pulse is finite.