A single earthed dielectric slab cannot propagate both polarizations of surface waves with equal velocities, so a more complex structure such as a double-layer slab is needed. The nature of surface waves on 2-layer earthed-slab structures is studied quantitatively, and conditions are derived that allow for equal velocities of both polarizations; a situation that will allow a wave of arbitrary or circular polarization to propagate. The upper slab must have a greater dielectric constant, and the square of the inverse velocity ratio (of the surface wave compared with free space) is constrained to lie in a range only slightly greater than that between the two dielectric constants. Extensive numerical data for the design of these arbitrary polarization structures are included.