This article presents a set of close approximations to model an elastic half-space supporting an embankment, which are evaluated in the context of phase velocity spectra, i.e. in terms of the normal wave propagation modes of the embankment-half-space system. The ultimate, intended target of these approximations is in the modeling of vertically-acting loads that may travel over the embankment with some prescribed, constant longitudinal speed. The proposed approximations are analogous to the well-known paraxial approximations that closely mimic a half-space terminating at a plane boundary, but differ from these in that the approximations herein aim at properly modeling not the waves with near normal incidence to the half-space, but waves which propagate at shallow, grazing angles along the longitudinal direction of load motion. Thus, these can be referred to as paralongitudinal approximations. The resulting expressions allow for a very effective simulation of the system at hand for loads moving with subcritical speed and solved in the context of a 2.5D solution method. Such 2.5D formulation considers a continuous model in the longitudinal direction and a discrete model in transverse planes.