The dynamic effect of an out of plane distributed moving load on the surface of an elastic half-space is considered. The problem is formulated in terms of a hyperbolic-elliptic asymptotic model for a moving load where the trajectory and the distribution of the load are taken to be orthogonal. Steady-state equations are written in terms of a moving coordinate system. The near-resonant solutions are, then, obtained for sub and super-Rayleigh cases taking into account the causality principle. Numerical results of displacement components are presented for various values of the distribution parameter.