A new method for simulating the propagation of pulses radiated by a circular normal ultrasonic transducer which is directly coupled to a homogeneous and isotropic elastic half-space is proposed. Both nonuniform and uniform high-frequency asymptotics inside geometrical regions as well as boundary layers (penumbra, an axial region, and a vicinity of the critical rays) have been used to describe the transient field by means of harmonic synthesis. The nonuniform asymptotic formulas involving elementary or well-known special functions elucidate the physics of the problem and give explicit dependence of the radiated waves upon the model parameters. The formulas are applicable in the radiating near field which is the near-field with the evanescent wave zone excluded. The code based on the uniform asymptotics has been tested in all regions against an exact numerical solution. It is orders of magnitude faster, but in many realistic cases the accuracy does not suffer. The limits of applicability of the model have been established.