A linear, viscid, continuously stratified model is used to study the response of the equatorial ocean to forcing by a wind patch moving zonally at the velocity U. For simplicity, solutions are found in an unbounded basin, and transient effects are ignored. This problem is mathematically equivalent to one in which the ocean has a uniform background current -U and the forcing is stationary, and it is possible to interpret solutions from either point of view. An important result is that resonant and near-resonant Rossby and Kelvin waves can be preferentially excited over other equatorially trapped waves forced by the wind. For two of the solutions, parameters are set as realistically as the model allows, and the solutions are compared with observations. One of them is forced by an eastward moving wind field like the westerly wind anomaly associated with the 1982--1983 EL Ni¿o event. It develops subsurface, anomalously westward flow that is strong enough to eliminate the normal Equatorial Undercurrent. The other is forced by a stationary wind in the presence of a weak, westward background current. It has a current structure in the deep ocean that resembles observations of deep equatorial jets, but the surface currents are not realistic. Near-resonant Kelvin waves contribute significantly to both solutions and are responsible for their interesting properties.