A simple formula is derived for estimating the asymptotic solution for two‐dimensional diffraction when a cylindrical obstacle is illuminated by an anisotropic cylindrical wave. The result is applied to derive the asymptotic diffracted field by a half‐plane, by a wedge, or by a circular cylinder. The solution for a half‐plane has been derived by other methods and both results are found to agree. The numerical calculations of the asymptotic solutions for the diffracted field by a slit and by a circular cylinder are carried out for the case when the obstacle is illuminated by an isotropic cylindrical wave. These numerical results are then compared with the exact solutions. It is found that the asymptotic series solutions provide precise results for estimating a diffracted field.