The spatial structure of beta-plane Rossby waves in a sinusoidal basic zonal flowU0cos(γ,y) is determined analytically in the (stable) asymptotic limit of weak shear,U0γ20/β≈1. The propagating neutral normal modes are found to take their greatest amplitude in the region of maximum westerly flow, while their most rapid phase variation is achieved in the region of maximum easterly flow. These results are shown to be consistent with what is obtained by ray-tracing methods in the limit of small meridional disturbance wavelength.