Abstract: The main problem considered here is the transient response of an elastic half space under a dynamic impulse loading of static rigid distribution. By Laplace-Hankel transform to the governing equations and boundary conditions, the solution of the basic equations is obtained in the integral transformation domain. Then the exact analytical solution for the vertical surface displacement of the ground is acquired applying inverse Laplace transform and Cagniard-De hoop method. The solution is composed of a number of different terms which represent P-wave, S-wave and R-wave contributions to the displacement. The paper has given the displacement-time curves of the location which is farther than the end of the radius of the area that the load acts on. The curves indicate that each stress wave arrives at the location two times successively with the process of time, and the location tends towards stillness then. The answer of the problem is at first in the history of elastodynamics and can be used to study the dynamic interaction and contact problems between soil and structures.