The study of self-similar solutions of multidimensional conservation laws leads to systems of equations that change type. Change of type occurs either across a transonic shock or at a sonic line. Often the sonic line appears as a free boundary in the formulation of the problem. Some recent numerical (and experimental) discoveries of a new kind of shock reflection ('Guderley Mach reflection') lead to interesting and still unresolved questions concerning the nature of the self-similar solutions in this generic case. In this talk, I will present some analysis of a simple model for this phenomenon, using the transonic small disturbance equation. The simplified problem seems amenable to analysis, but we are just beginning to make progress. This is a report on current joint work with Allen Tesdall and Kevin Payne.