Assuming linear wave theory for waves riding on a weak current of 0(&egr;) compared to the wave phase speed, and approximate dispersion relation is developed to 0(&egr;2) for arbitrary current U(z) in water of finite depth. The 0(&egr;2) approximation is shown to be a significant improvement over the 0(&egr;) result, in comparison with numerical and analytic results for the full problem. Various current profiles in the full range of water depths are considered. Comments on approximate action conservation and application to depth-averaged wave models are included. ¿ American Geophysical Union 1989