Weak Diffracted Shocks Near Singular Rays

A weak shock incident upon an obstacle produces weak reflected and diffracted shocks.The resulting flow is analyzed near a point P where either the incident shock or a reflected shock meets a diffracted shock.The path of P is called a singular ray, and it is analogous to a ray on a shadow boundary.For self-similar problems, the flow near P is shown to be the solution of a nonlinear elliptic free boundary problem.This flow is regular, in contrast to the singular flow given by linear theory.An analogous problem is found for the interaction of a weak rarefaction wave with a weak shock.Both problems are also applicable to steady self-similar supersonic flows past bodies.Hunter's analysis (SIAM J. Appl.Math.48, (1988), pp.1-37.)implies that these problems are canonical, i.e. that they apply to general hyperbolic systems in any number of dimensions.Simplified forms of both problems are solved numerically.