Eastward propagating wave centered about the equator with zonal but no meridional velocity. The zonal velocity has a Gaussian meridional structure centered about the equator with standard deviation equal to the equatorial Rossby deformation radius. In the shallow water approximation the waves are non-dispersive with frequency omega = +/- c k, in which k is the zonal wavenumber and the phase speed c= (gH)^(1/2) with g the acceleration due to gravity and H the mean fluid depth. The equatorial Rossby deformation radius in this case is R=(eta/c)^(1/2) in which eta is the meridional gradient of the Coriolis parameter at the equator.