The structures and propagation characteristics of coastally trapped waves on a continental shelf with a barotropic coastal current are analyzed. When the current located on the shelf edge has a velocity in the direction of the phase propagation of the topographic shelf waves and when the topography stabilizes the shear instability, the linearized perturbations on the mean flow consist of propagating shelf and shear wave modes and the evanescent shelf-shelf and shelf-shear coalescence modes at complex wave numbers. At low Rossby numbers the shear waves and backward propagating shelf waves coalesce at low frequencies. As the rossby number is increased, the coalescence occurs at higher and higher frequencies and lower wave numbers. For sufficiently high Rossby numbers, the coalescence may occur directly between the forward propagating shelf wave and the shear wave, the backward propagating shelf wave being eliminated entirely. Hence the manner in which energy is transported along a continental shelf can be fundamentally altered by the presence of a mean current. ¿ American Geophysical Union 1987