The paper deals with 3D dynamic response of an elastic half-space loaded by a point force moving at a constant speed along a straight line on the surface. The problem is formulated within the framework of the asymptotic hyperbolic-elliptic model developed earlier by two of the authors. The validity of the model is restricted to the range of speeds close to the Rayleigh wave speed. Steady-state near-field solutions are derived in terms of elementary functions. Transient analysis of surface motion illustrates peculiarities of the resonance associated with the Rayleigh wave. (C) 2012 Elsevier B.V. All rights reserved.