This paper discusses the propagation of surface waves in an elastic stratified half space, which consists of a granular elastic upper layer resting on a classic elastic substratum. We will consider both the case of an upper layer with second-gradient continuum properties, and an upper layer with Cosserat continuum properties. These enhanced continuum formulations are able to describe micro-scale effects as governed by the individual particle displacements and particle rotations. The micro-scale effects become important when the length of the surface waves is of the same order of magnitude as the particle size, which may occur in a high-speed railway track that consists of an upper layer of ballast particles. In an accompanying paper we will therefore analyze the critical structural response of the stratified half space when it is subjected to a moving, harmonically vibrating load. This model is representative for a high-speed railway system.