We suggest a criterion to examine the range of validity of the ray method, based on the theory of asymptotic series. This criterion examines the ratio of the second term of the ray series over the first one. For that purpose, we develop an algorithm for the computation of the second term of the ray series, together with the resolution of a system of differential equations for auxilliary functions, the determination of initial data near the source, and boundary conditions on interfaces. We apply this criterion to two-dimensional gradient velocity models without an interface and with interfaces. Serious limitations for the use of ray methods arise from our results in at least three situations: (1) when the distribution of energy along the initial wave front is not uniform; (2) when the incident angle is relatively large (near grazing incidence); and (3) after a large number of reflections in a waveguide, with increasing distance from the source. ¿ American Geophysical Union 1996