  We present a comparison of the dispersion relations derived for anti-plane
surface waves using the two distinct approaches of the surface elasticity
vis-a-vis the lattice dynamics. We consider an elastic half-space with surface
stresses described within the Gurtin-Murdoch model, and present a formulation
of its discrete counterpart that is a square lattice half-plane with surface
row of particles having mass and elastic bonds different from the ones in the
bulk. As both models possess anti-plane surface waves we discuss similarities
between the continuum and discrete viewpoint. In particular, in the context of
the behaviour of phase velocity, we discuss the possible characterization of
the surface shear modulus through the parameters involved in lattice
formulation.
