This paper presents wave propagation over a finitely conducting half-space whose surface is bounded by a rough surface of small RMS height. An electric or magnetic current source is used to excite the rough surface, and the Green's function for transverse electric (TE) and transverse magnetic wave propagation is obtained. The appearance of roughness at the boundary produces both a coherent (mean) and incoherent (fluctuating) field distribution, which is obtained from Dyson's equation and Bethe-Salpeter's equation, respectively. The coherent Green's function for vertical polarization exhibits similar characteristics to the Sommerfeld dipole problem where the Zenneck wave pole is modified by roughness. The incoherent field generated by rough surfaces is obtained for both vertical and horizontal polarization, and the conventional cross section per unit length of the rough surface is modified to include the effects of surface roughness. For angles near grazing, a low-grazing-angle cross section is obtained by evaluating the Bethe-Salpeter's equation with the Sommerfeld solution. Finally, the coherent and incoherent intensity for the TE rough surface Green's function is obtained and compared to Monte Carlo simulations.