Analytic resolution of time-domain half-space Green's functions for internal loads by a displacement potential-Laplace-Hankel-Cagniard transform method

A refined yet compact analytical formulation is presented for the time-domain elastodynamic response of a three-dimensional half-space subject to an arbitrary internal or surface force distribution. By integrating Laplace and Hankel transforms into a method of displacement potentials and Cagniard's inversion concept, it is shown that the solution can be derived in a straightforward manner for the generalized classical wave propagation problem. For the canonical case of a buried point load with a step time function, the response is proved to be naturally reducible with the aid of a parametrized Bessel function integral representation to six wave-group integrals on finite contours in the complex plane that stay away from all branch points and the Rayleigh pole except possibly at the starting point of the contours. On the latter occasions, the possible singularities of the integrals can be rigorously extracted by an extended method of asymptotic decomposition, rendering the residual numerical computation a simple exercise. With the new solution format, the arrival time of each wave group is derivable by simple criteria on the contour. Typical results for the time-domain response for an internal point force as well as the degenerate case of a surface point source are included for comparison and illustrations.