The group velocity of a wave which propagates over an impedance plane (ground wave) is frequency dependent. Therefore when a narrow pulse (wideband signal) propagates over such a surface, it exhibits delay dispersion. Previous efforts to determine the delay dispersion of wideband ground wave signals have been limited to propagation over a smooth spherical earth. The analysis presented in this paper includes the effects of surface roughness and is applicable to the case of a rough ocean surface. The results are presented as a function of distance and sea state. It is shown that at short distances surface roughness tends to increase the amount of delay dispersion. Beyond a certain distance, delay dispersion depends on roughness: a small amount of roughness surprisingly causes a reduction of the delay dispersion, a large amount of roughness causes increased delay dispersion as expected.