Stevenson's general theory of electromagnetic horns or flared wave guides, which is valid only for geometrical configurations whose cross sections are continuous along the direction of propagation together with their first and second derivatives, is generalized to include sudden jumps in the first derivatives of the cross sections. Thus the junction of wave guides with different flares and of arbitrary cross sections can be treated, provided that at the junction plane the cross sections on the two sides are identical. General expressions are derived for the reflection and transmission coefficients correct up to the second order in flare angle. Explicit results for a few configurations of practical interests are derived, including some second‐order ones. In cases where special theories exist, comparisons are made with favorable results. In addition, an experimental check is performed.