A linear stratified ocean model is used to study the wind-driven response of the equatorial ocean. The model is an extension of the Lighthill (1969) model that allows the diffusion of heat and momentum into the deeper ocean, and so can develop non-trivial steady solutions. To retain the ability to expand solutions into sums of vertical normal modes, mixing coefficients must be inversely proportional to the square of the background Vaisala frequency. The model is also similar to the earlier homogeneous ocean model of Stommel (1960). He extended Ekman dynamics to the equator by allowing his model to generate a barotropic pressure field. The present model differs in that the presence of stratification allows the generation of a baroclinic pressure field as well. The most important result of this paper is that linear theory can produce a realistic equatorial current structure. The model Undercurrent has a reasonable width and depth scale. There is westward flow both above and below the Undercurrent. The meridional circulation conforms to the 'classical' picture suggested by Cromwell (1953). Unlike the Stommel solution, the response here is less sensitive to variations of parameters. Ocean boundaries are not necessary for the existence of the Undercurrent but are necessary for the existence of the deeper Equatorial Intermediate Current. The radiation of equatorially trapped Rossby and Kelvin waves is essential to the development of a realistic Undercurrent. Because the system supports the existence of these waves, low-order vertical modes can very nearly adjust to Sverdrup balance (defined below), which in a bounded ocean and for winds without curl is a state of rest. As a result, higher-order vertical modes are much more visible in the total solution. This property accounts for the surface trapping and narrow width scale of the equatorial currents. The high-order modes tend to be in Yoshida balance (defined below) and generate the characteristic meridional circulation pattern associated with equatorial Ekman pumping.